Reprogrammable metamaterial robot with embodied versatile computation and mechanical intelligence

Domain: biology
Published: 2026-04-03
Editorial outcome: major_revision

Abstract
Due to the flexible and robust features, recent advances emerging from computing intelligence in mechanical systems have led to many innovative techniques from signal processing to robotics. However, conventional designs of mechanical computing are typically manifested as deterministic functions with single-task layouts, which show limitations in versatility. This work proposes a versatile mechanical computing architecture, which can integrate both analog and logic operations into a unified robotic system. Inspired by Fourier optics, elastic wave metamaterials are introduced to spatial analog computing with robot crawling, enabling flexible spatiotemporal modulations and self-regulating locomotion. Using serial/parallel strategies, the metamaterial allows nested and cascaded designs from combinational analogs to the extended binary logics. In particular, by encoding analog signals as mechanicCal qubit-like states, the system can realize higher dimensional logic mappings through reprogrammable matrix operations. Beyond single-task layouts, nonlinear harmonics are also introduced to realize the parallel computing of multiple tasks or their transformation to specific tasks. The versatile strategies are shown to support the parallelism, integration and transformation from analogs to logics, which wish to offer an effective approach for incorporating mechanical intelligence in robotics.

Introduction
Metamaterials are known as new kinds of composite materials and structures, which are designed artificially with unusual physical and mechanical properties. Since the emergence of band gap theory, elastic wave metamaterials have received extensive attention due to their special wave manipulation abilities [1–4]. The design inspirations of elastic wave metamaterials are diverse and range from phononic crystals, cellular materials to biomimetic structures, etc. Such versatility provides elastic wave metamaterials with rich mechanical properties, e.g., acoustic cloaking, negative refraction and topological insulation properties [5–8]. Recently, wave-based mechanical computing devices have attracted increasing attention due to their special signal processing and storage capabilities [9–13]. Mechanical computing can operate without electrons and its ability to emulate digital calculation has attracted growing interests in intelligent material and structure fields [14, 15]. In such framework, computations are directly performed by position, deformation or vibration signals with high interference resistance and environment adaptability [16, 17]. In particular, the inherent reprogrammable and embodied features provide possibilities for integrating computing autonomy and mechanical intelligence into smart robotic materials [18, 19].

Recent studies in mechanical/optical computing have led to many novel computational metamaterials from logic, analog to qubit computing [20–35]. As the basic units of logic operations, logic gates have been realized through multiple mechanical implementations, such as buckled beams, origami and waveguides [20–22]. Investigations about mechanical logics have developed many effective strategies. For instance, structural properties of metamaterials such as chirality and multi-stability have been employed to achieve logic encoding, signal processing and cascaded computing designs [23–25]. However, traditional mechanical logic processors often struggle with accuracy or efficiency when processing large-scale dynamic tasks. In contrast, wave-based analog computing [26–30] enables continuous signal processing in both time and space domains, which shows advantages for complex dynamic computing. Since the proposals of Green’s function method [26], spatial analog computing have achieved from convolution and calculus operations to equation solver and quantum search functions [27–30]. Compared to digital logics, the special amplitude and phase modulations offer spatial analog computing more flexible information storage and processing capabilities. In particular, the similarities between mechanical and optical waves in signal superposition and interference also provide platforms for the mechanical analogue of qubit-based optical operations [31–35].

In recent years, increasing attention has been paid to the versatile designs of computational systems, which highlights their abilities to process various tasks in a unified framework. Conventional designs of mechanical computing are mainly manifested as deterministic functions with single-task configurations, which show limitation in versatility. In order to address these challenges, increasing efforts have focused on the parallel strategy that can perform multitasking computing [36–38]. Relevant attempts range from spatial analogs to digital logics, such as reprogrammable sequential memory [36], cascaded logics in multi-stable beams [37], as well as parallel analogs with nonlinear harmonics [38]. Although many studies about versatile computing have been reported, the effective method that can integral analog, logic and qubit-like matrix computing into a unified framework is still sporadic. On the other hand, mechanical metamaterials have also inspired many interesting robotic designs from worm-like crawler, origami robot to metamaterial swimmer [39–42]. Previous studies about mechanical computing are mainly focused on the static and table-top demonstrations with prospects in smart and wearable devices. The inspiration of integrating mechanical intelligence and computational autonomy into robotic systems also deserves further explorations.

In this work, a versatile mechanical computing architecture is proposed, which can integrate mechanical intelligence into robot designs with both analog and logic functions. Inspired by Fourier optics, elastic wave metamaterials are introduced to spatial analog computing with robot crawling, which enables flexible spatiotemporal modulations and self-regulating locomotion (e.g., calculus, convolution and solving equations). Using the spatial serial/parallel strategies, the metamaterial robot allows the combinations of several analog functions from nested and cascaded designs (e.g., addition/subtraction and multiplication) to their extended binary digital logics (e.g., AND, OR and XOR gates). In particular, nonlinear harmonics are introduced to realize parallel computing of multiple tasks or their transformation to specific tasks. On the other hand, due to the similarity between mechanical and optical waves, the mechanical analog computing can be also applied to emulate the qubit matrix operation achieved in optical systems. By encoding analog signal pairs as combinations of mechanical qubit-like states |0〉 and |1〉, multiple linear and invertible matrix transformations can be achieved. The associated computing results can perform the logical mappings such as Hadamard, Pauli X and CNOT gates. Furthermore, analytical derivations and experiments are performed to verify the theory with reprogrammable versatile performances.

Results
Configuration and design principle. The metamaterial robot in Fig. 1(a) employs a periodic network of hexagonal blocks with two translational components u and u in the 2D plane. Each hexagonal block is composed of a tunable mass m(x, y) and a supporting base to behave as the robot foot. Adjacent moving blocks are connected by two curved metal sheets, which can be considered as leaf springs with shear (k), linear (k) and nonlinear (k ) tension stiffness. In addition, a linear vibration motor array is installed at the front edge of the robot to provide the spatial profile f(y) of incident waves. To enable the robot crawling, a sawtooth design of robot feet is introduced at the bottom of each moving block to generate asymmetric friction R± along the crawling direction (see Supplementary Figure S1 for details).

Inspired by previous studies about mechanical/optical computing [26–30], this work introduces the framework of elastic wave metamaterials to regulate the robot crawling with spatiotemporal modulations. As shown in Fig. 1(b), the metamaterial robot is divided into 3 sub-arrays with 2 Fourier transform arrays (ℱ , ℱ ) and 1 transfer function layer H. Based on Fourier optics, the spatial output g(y) associated with the incident profile f(y) can be determined by the integral relation g(y) = ℱ{H(k )ℱ[f(y)]} with k being the spatial frequency. Since the spatial frequency k is in the same domain of y, the transfer function with target operators can be expressed as H(k ) = H(y), in which y replaces the role of k . During the calculations, the metamaterial robot can be driven to move or steer in response to the spatial output g(y). In order to minimize the interference from incident signals, the robot feet near the input layers are designed with symmetric friction but the others remain asymmetric. Then, the metamaterial robot in Fig. 1(b) can switch different locomotion modes according to the output field distribution from g to g regions.

Based on the architecture above, Figs. 1(d)–1(f) provide multiple mechanical computing strategies from reprogrammable analog computing to their extended logic and matrix operations. Firstly, with the ℱ -H-ℱ layout in Fig. 1(b), the metamaterial can integrate the robot crawling with multiple computing functions by proper designs of transfer layers H, e.g., integral, differential, translation operations and their related motion modes in Fig. 1(d). Secondly, in addition to single-task computation, Fig. 1(e) introduces nonlinear sub- ω and super- ω harmonics into mechanical computing and combines them with harmonic band gaps. By tuning incident signals ω within different harmonic band gaps, the metamaterial allows parallel computing of multiple tasks or single task with specific frequency. Thirdly, the metamaterial robot can be also extended to digital logics by labeling the presence and absence of analog signals as binary ‘1’ and ‘0’, e.g., AND, OR and XOR operations in Fig. 1(f). In particular, by encoding the analog signal pairs {f , f }T and {g , g }T as the combination of qubit-like states |0〉 = {1, 0}T and |1〉 = {0, 1}T, the metamaterial can be designed to perform mechanical matrix operations. The associated results enable logical mappings such as Hadamard, Pauli X and CNOT gates [43, 44]. At last, using the spatial serial and parallel strategies, Fig. 1(g) further extends the metamaterial of several transfer functions H or inputs f = (f , f ) to their combinational multiplication H = H H …H and addition/subtraction operations g = f ± f .

Among these functions, the proposed versatile platform allows parallel computing in frequency (Case II), bitwise (Case III) and spatial (Case IV) domains with multitasking capabilities. The following discussion is focused on the computational functions achieved by the metamaterial robot. Details about the robot design and computing principle are demonstrated in Movies S1 and S2.

Reprogrammable analog computing. As discussed in Fig. 1(b), the metamaterial robot consists of 3 sub- arrays in series with the universal ℱ -H-ℱ layout. With elastic wave metamaterials, the Fourier ℱ and transfer function H layers are realized by 2 mechanical graded-index (GRIN) lenses and a reprogrammable spatial filter [26, 28]. Based on Fourier optics, the GRIN lens can be designed as an arc mass distribution along the y-axis in place of conventional refractive designs as [26]

m ( y ) m 0 = 1 − ( L F  F )  1 − ( 1 − y 2 L 2F ) 1 /2  , (1)

where Δ denotes the lens thickness and L refers to the focal length; m represents the mass at the central plane of the lens. As wave propagates through the GRIN lens ℱ , the incident profile f can be transformed as its spatial Fourier form F and exhibits as the mechanical focusing function. After ℱ focusing, the wave field F modulated by the transfer function layer H can be expressed as G = HF. Analogous to electromagnetic modulation [26, 28], when we consider a mechanical signal with wave length λ , the transfer function H can be expressed as function of mass density as

H ( y ) e x p  i k ( y )  e x p 2 i H m ( y ) ( m 0 0 )   =  =    , (2a)

m H ( y ) m 0 0 l n H ( y ) ( 2 i H )   =  , (2b)

where k(y) is the wave number; k = 2π/λ correspond to k at y = 0; Δ denotes the characteristic length. It is worth noting that the mass relation in Eq. (2) behaves as a complex variable m (y) = m (y)+im (y), which cannot be found in a nature material. In order to eliminate imaginary mass, we consider a general vibration equation (m +im )d2u/dt2+c du/dt+k u = 0 which contains real stiffness k , damping c and complex mass m +im . Assuming that the imaginary mass m can be characterized by the damping term c , the equation can degenerate as a free vibration system m a d2u/dt2+k a u = 0 with harmonic solution u = Aeiω a t. Then, substitution of the solution into the complex equation yields ω = (k /m )1/2 and the equivalent transfer function as

H ( y ) e x p  2 i H m a ( y ) i c a ( y ) m a ( y ) / k a ( 0 m 0 )    =   +  , (3)

where the imaginary mass m = c /ω exhibits as function of damping coefficient c . As a result, the transfer function H can be determined by both real mass m and damping coefficient c .

With the material relation above, Fig. 2 illustrates several mechanical realizations of spatial analog computing. As shown in Figs. 2(a) and 2(b), the metamaterial robot is designed with single and double GRIN lenses to perform the mechanical focusing and reflection functions. As incident waves f propagate, the output profiles g at once and twice the focal length L exhibit as their Fourier transform and mirror images, which agree with the relation ℱ[f(y)] = F(y) and ℱ{ℱ[f(y)]} = f(–y), respectively. In Figs. 2(c)–2(f), multiple computational functions are implemented by the ℱ -H-ℱ layout, including differential, integral, convolution and calculus equation solver. For instance, the transfer function associated with the first order differential and integral are derived as H(y) = iy and 1/iy, which can be achieved by the mass relation m (y) = ± m λ /(i2πΔ )ln(iy). Since the mass density is positive in a nature material, the mass relation needs to be further derived as m (y) = m λ /(i2πΔ )[i(π/2+2nπ)±lny] (n  ) using the Euler’s formulas. Similarly, the integral-differential equations (IDEs) in Fig. 2(f) with H(y) = (–LCy2+iRCy+1)/(iCy) can be solved by the mass variation m (y) = m λ /(i2πΔ )lnH(y).

Cooperative and transformable computing. During the derivations above, we have discussed the principle of mechanical computing and its application in wave modulation and analog operation. The reprogrammable functions from convolution and calculus to their combinations with robot crawling suggest the potential in complex working conditions. However, due to the single-task layout, the robot structure is required to be one-for-one matched with the computational function, which shows limitations in multitasking. In order to improve the computing performance, the following work introduces nonlinear sub- ω and super- ω harmonics into the robot to achieve the cooperation and transformation of multiple computing modes.

As shown in Figs. 1(e) and 3, the principle for implementing mechanical computing with nonlinear harmonic waves is illustrated. Based on the versatile ℱ -H-ℱ layout in Fig. 1(b), the previous GRIN lens ℱ is replaced by a Fresnel lens ℱ * and a periodic array of filtering layers (PLs) in Figs. 3(a) and 3(b). The Fresnel lens ℱ * employs a piecewise mass distribution as m*(y) = m (y)–n*T * (n* ∈ ℤ), which satisfies the constraint 0 < m*(y) < m and period T * = m λ /Δ . Compared to conventional GRINs, the Fresnel lens ℱ * can be designed thinner in Fig. 3(b) to achieve the same focusing function of ℱ in Fig. 3(c). After the Fresnel lens, the periodic filtering layers PLs are introduced to regulate the nonlinear harmonic components by their corresponding band gaps. The filtering layers PLs employ a periodic network of hexagons and leaf springs, which can exhibit multiple dispersion relation according to the nonlinear harmonic order, e.g. the 3rd super- and sub- harmonics in Fig. 3(e) (see Supplementary Note S6 for detailed derivations).

Based on the architecture above, Figs. 3(f)–3(h) provide a discrete metamaterial model with the layout of ℱ -H-ℱ *-PLs, which features both nonlinearity and dispersion relation. Firstly, the incident wave ω can propagate through the GRIN lens, transfer function layer and Fresnel lens in turn and then produces a series of harmonic components rω and ω /r (r = 1, 2, …). According to previous derivations in Eq. (2), the mass relation m H and transfer function H 0 of fundamental wave ω 0 satisfy H 0 = exp[2πiΔ H m H /(m 0 λ 0 )] with λ 0 being the wave length. For the same mass relation m H , the super- and sub- harmonics with wave lengths λ /r and rλ can generate different outputs with the following transfer functions:

H super ( y ) e x p 2 i H m H ( y ) r ( m 0 0 )   =    , (4a)

H sub ( y ) e x p 2 i H m H ( y ) ( r m 0 0 )   =    . (4b)

For instance, when we design the fundamental wave ω to perform n-th order differential operation g (y) = dnf (y)/dyn in Fig. 3(i), the harmonic outputs corresponding to rω and ω /r can be determined by the Fourier relation g(y) = ℱ{H(y)ℱ [f (y)]} as

g super ( y )  d n r – n g 0 ( y ) d y n r – n , (5a)

g sub (y)g 0 (y)dydy. (5b)

It is worth noting that Eq. (5) provides a general relation among the fundamental ω , super- rω and sub- ω /r harmonics with both differential and integral functions. This relation suggests that the mechanical system can support the cooperative propagation of multiple nonlinear harmonics and allow them to compute independently. Therefore, combining the nonlinear harmonics with the fundamental one and considering the phase relation as φ , we can derive the total output field of the computational system as

g ( y , t ) C r1 g 0 ( y ) c o s ( 0 t r1 ) r 2 C r 2 d n r n g 0 ( y ) d y n r n c o s ( r 0 t r 2 )      + +  − − + r 3 C r 3 g 0 ( y ) d y (n r – n d ) y r c o s ( 0 t r r 3 )   +     + , (6)

where C are coefficients to be determined. Then, the output will be injected into the periodic filtering layers PLs to further regulate the harmonic components. By tuning the incident frequency ω within different harmonic pass and stop bands, the mixed harmonics can be directly output as multi-task results in Fig. 3(j) (Mode I) or separated as single-task results in Figs. 3(k)–3(m) (Mode II). For Mode I, both fundamental and harmonic waves locate at pass bands and each component has its own computational result. It indicates that the system allows the cooperative computing of different components, which corresponds to the parallel computing in frequency domain. For Mode II, the periodic layers PLs can filter the mixed harmonics into single-task computing with specific frequency. In particular, it can transform different harmonic inputs into the same frequency while maintaining their computational results, which is considered as the transformable computing. Such a property provides possibilities for the subsequent operations requiring identical frequency condition, e.g., spatial addition/subtraction operations. Limited by space, details about the parallel and cooperative computing in the frequency domain are presented in Supplementary Figure S4. The following discussion is focused the transformable computing achieved by the elastic wave metamaterial.

Based on the principle above, examples of derivative operations are provided in Fig. 4 to illustrate the transformable computing from fundamental to subharmonic waves. Following the ℱ -H-ℱ *-PLs layout, the filtering layers PLs in Fig. 4(a) employs a periodic triatomic configuration with mass ratios m :m :m = 1:3:1. Then, multiple dispersion relation including linear, nonlinear fundamental and subharmonic waves can be derived and their band structures are presented in Figs. 4(b)–4(d). Since fundamental and subharmonic band gaps belong to different frequency regions, the mixed signals can be filtered as specific harmonic component with target output. In order to verify such property, Figs. 4(e)–4(h) illustrate the transition from fundamental to subharmonic waves and the evolution from differential to its integral. Based on the eigenstate diagram in Fig. 4(e), 3 incident frequencies f = 70, 140 and 300 Hz are selected in Figs. 4(f)–4(h), which belong to the 1st pass, 1st stop and 2nd stop bands of fundamental waves, respectively. The transfer layer H is designed as the 2nd derivative g (y) = d2f (y)/dy2 for fundamental waves and can be theoretically transformed to g (y) = ∫…∫g (y)dy…dy (2r–2)/r for the r-th order subharmonics.

Different from conventional single-tasking devices with sole response or multi-tasking devices with mixed responses, our system can present multiple independent results according to the incident frequency: (i) f = 70Hz: we can see from Fig. 4(f) that the metamaterial with frequency 70Hz can perform the 2nd differential operation, in which the input s and output s keep almost the same frequency in Fig. 4(o). It means that both s and s exhibit as fundamental waves, which agree with the fundamental pass band in Fig. 4(d). (ii) f = 140Hz: The incident frequency locates at the stop band of fundamental waves but falls into pass bands of subharmonics. As shown in Fig. 4(g), the metamaterial with frequency 140Hz can perform the 1st differential operation g (y) = df (y)/dy, which exhibits as the integral of the 70Hz case, i.e., g (y) = ∫g (y)dy. Further observation from the spectra in Figs. 4(o) and 4(p) shows that the incident signal s is gradually transformed to its subharmonics and the fundamental wave can be eliminated by its corresponding band gap. In particular, the output frequency is about half the incident one, which can generate the 2nd subharmonics during the computation. (iii) f = 300Hz: The frequency is far away from fundamental pass bands and the metamaterial keeps almost zero output in Figs. 4(h). In addition to the theoretical results above, Figs. 4(i)–4(k) provide the experimental responses of unit cells in the input f(y) and output g(y) layers. The bar charts in Figs. 4(i)–4(k) represent the normalized amplitudes of their velocity responses, in which the red and black curves are fitting curves of input and output responses, respectively. These results are used to the support the transformable computing for different exciting frequencies, which show good agreements with the theoretical results in Figs. 4(f)–4(h).

Mechanical matrix operations. In addition to spatial analog computing, the elastic wave system can be also applied to perform matrix operations on encoded mechanical signals. In the following discussion, we employ the versatile computational framework to realize multiple matrix operations which are commonly used in information processing. Different from spatial analog computing with single input f, the matrix operations are achieved by several encoded mechanical signals {f , f , …, f }T with a N-dimensional transformation matrix M(N). For example, we can encode the two-channel analog inputs {f , f }T and outputs {g , g }T as vectors in a classical 2D state space. Then, the metamaterial can achieve the linear transformation {g , g }T = M(2){f , f }T by a 2D matrix M(2). Such transformation can be employed to realize multiple fundamental 2×2 operations such as Hadamard or Pauli-X type matrices in linear algebra. Accordingly, this framework can be also extended to higher-dimensional state spaces by increasing the number of wave channels, e.g., the 4×4 controlled-NOT (CNOT) matrix. It is worth mentioning that many high-dimensional linear and invertible transformations can be decomposed into sequences of 2×2 matrix operations [33]. As a result, a robust and reprogrammable M(2) transformation device is essential to achieve more complex and scalable higher- dimensional operations.

As shown in Fig. 5(a), the metamaterial robot employs a similar multilayer ℱ -H-ℱ layout to spatial analog computing. A mass barrier array m b >> m 0 is placed along the axial centerline y = Y/2 which divides the input f into a signal pair {f , f }T. For the harmonic signals with identical frequency, the signal pair {f , f }T defines a 2D complex state vector {A *exp(iφ ), A *exp(iφ )}T with A *2 + A *2 = 1. This vector is characterized by the normalized amplitudes and phases A *, AL*, φ and φ of {f , f }T. We consider the state vector as a linear combination {α, β}T = α{1, 0}T + β{0, 1}T, in which the basis states {1, 0}T and {0, 1}T are defined by single-mode excitations (A * = 1, A * = 0 T ) and (A * = 0, A * = 1). In order to better describe the two-channel signals, we adopt the symbolic notation from acoustic phi-bit computing [32–34], which labels classical basis states {1, 0}T and {0, 1}T as mechanical qubit-like states |0⟩ and |1⟩. Then, a general input state is written as {α, β}T = α|0⟩ + β|1⟩ with α = A *exp(iφ ) and β = A *exp(iφ ). It is worth mentioning that this notation is a purely mathematical convention without relevance to quantum computing. With the normalization A *2 + A *2 = 1, the mechanical state can be fully characterized by the amplitude ratio A /A and the relative phase Δφ = φ − φ , Then, it can be visualized on a Bloch sphere by the mapping of two polar coordinates θ = 2arccos(A *) and ϕ = Δφ. Such representation is a classical analogous mapping to provide a visualization without quantum implication.

In order to realize the M(2) transformation function, Figs. 5(b)–5(d) provide the evolution from spatial analog to mechanical matrix computing. The metamaterials in Figs. 5(b) and 5(c) are designed with mass barriers along the axial and vertical centerlines y = Y/2 and x = X/2, respectively. In Fig. 5(b), the input f is divided into a signal pair {f , f }T by the axial barrier y = Y/2 and the logic states can be swapped after two Fourier lenses. In Fig. 5(c), a tunable slit is designed at the center of the vertical barrier x = X/2. Such a design allows the conversion of incident waves into point wave sources and presents the diffraction effect. After the Fourier layer ℱ , the point source can be further transformed as a parallel waveform and equally divided into {g , g }T as the superposition state |+〉 = {√2/2, √2/2}T, i.e., α = β. Then, combining Figs. 5(b) and 5(c), the incident waves with either |0〉 = {1, 0}T or |1〉 = {0, 1}T input in Fig. 5(d) can be transformed into their superposition states |+〉 = {√2/2, √2/2}T.

According to the principle above, the following discussion illustrates several mechanical realizations from fundamental 2×2 and 4×4 matrix operations to their combinational logic mappings. As shown in Fig. 5(e), the Hadamard operation [44] corresponds to the following transformation matrix:

H= 1 2 1 1  1 −1 , (7)

which enables the transformation from the basis states |0〉 = {1, 0}T and |1〉 = {0, 1}T to their superposition states |+〉 = {√2/2, √2/2}T and |–〉 = {√2/2, –√2/2}T as

H 0 = 1 2 1 1  1 1 2 1 = + , H 1 = 1 2 1 1  0 1 2 1  = − , (8a, b)

H + = 1 2 1 1  1 2 1 = 0 , H − = 1 2 1 1  1 2 1  = 1 . (8c, d)

Based on the previous discussion, the evolution |0〉 → |+〉 can be achieved by the robot configuration in Fig. 5(d). In order to obtain the |1〉 → |–〉 output, the transfer layer H in Fig. 5(e) is designed as a double-slit layout, in which the lower slit is arrange by two mass arrays with π phase difference. As the input |1〉 = {0, 1}T is applied, the incident wave f can be focused into the lower slit as F and divided into two separated wave fronts F ± with opposite phases. Based on the Huygens-Fresnel principle, the separated wave fronts can be considered as two half-space wave sources that inject into the Fourier layer ℱ . Then, they can generate two opposite output states as |–〉 = {√2/2, –√2/2}T, i.e., |1〉 → |–〉. Similarly, for the |+〉 = {√2/2, √2/2}T input, the output corresponding to the upper half-space g can be enhanced by two in-phase wave fronts F + but the lower output g can be eliminated by two out-phase F –. Then, they can yield the |0〉 ={1, 0}T output, i.e., |+〉 → |0〉. Similar phenomena can be observed from the numerical results and robot attitudes during experiments in Figs. 5(f) and 5(g), respectively. For comparisons, Supplementary Figure S5 and Movies S5 further provide several mechanical realizations of 2×2 matrix operations with experimental supports, including the Pauli X, Pauli iY and Hadamard transformations.

In addition to double-channel layout above, Fig. 5(h) provides a combinational architecture of multiple channels to show the serial and parallel performance in spatial domain. The first and second levels are composed of 2 and 3 Hadamard gates with staggered connections between levels. After the inputs {1, 0, 0, 1}T at channels 2–5, the combinational gate can generate the {√2/2, –√2/2, 0, 1, √2/2, √2/2}T outputs from channels 1–6 with consistent performances between theories and calculations. On the other hand, the system can be also extended to the M(4) transformation by encoding information into two independent signal pairs, enabling more complex logic operations, e.g., the CNOT-like operations in Fig. 5(i). The CNOT gate [43, 44] enables two-qubit operations |b b〉 = |b 〉⊗|b〉 of control |b 〉 and target |b〉 ones, which satisfies CNOT|00〉 = |00〉, CNOT|01〉 = |01〉, CNOT|10〉 = |11〉 and CNOT|11〉 = |10〉. The symbol “⊗” represents the tensor product of |b 〉 and target |b〉, and the corresponding 4×4 matrix of CNOT-like operations is

CNOT =  1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0  , (9)

where the basis states of the four-channel system are defined as:

0 0 = 0  0 =  1 , 0 , 0 , 0  T , 0 1 = 0  1 =  0 , 1 , 0 , 0  T , (10a, b)

1 0 = 1  0 =  0 , 0 , 1 , 0  T , 11 = 1  1 =0, 0, 0, 1T . (10c, d)

In our proposed CNOT gate, there are 3 double-channel wave splitters comprising 6 input and output channels. As shown in Fig. 5(i), the double-channel splitter is designed by the superposition principle in Fig. 5(d), which can transform the incident signal s into certain splitting ratio s /s . As the slit width increases, the output ratio s /s can be approximately tuned from 1 to 0. In order to implement the CNOT function, we adopt the 1:2 splitter (s /s = 1/2) and consider the measurement of the operation under the ZZ-basis [44]. The ZZ-basis is defined as |0〉 = Channel 2 and |1〉 = Channel 3 for the control bit as well as |0〉 = (Channels 4+5)/√2 and |1〉 = (Channels 5–4)/√2 for the target one. In order to verify the computing performance, Fig. 5(i) illustrates two examples of four-channel |00〉 and |01〉 inputs and yields |00〉 and |01〉 outputs respectively, which show good agreements with the theory.

Robotic crawling with embodied versatile computations. Based on the previous discussion, a versatile framework of mechanical computing is developed with both analog and digital logic functions. Beyond conventional single or serial strategies, the combinational, cooperative and transformable features enable the parallelism of the proposed functions in spatial, frequency and bitwise domains. Following the concept of mechanical intelligence in Fig. 1(d), this work introduces computational functions into robot designs and combines them with robot crawling. During the calculation, the robot can be driven to move or guided to steer in response to the modulated wave propagation. The following work will be focused on the principle for implementing robot crawling regulated by the embodied computational functions.

As shown in Figs. 6(a) and 6(b), the harmonic input is split into an 8-bit signal sequence that governs 8 exciters and injected into the phase regulator and amplifier arrays. After wave modulations, the incident signals can be transformed as a spatial input sequence with desired wave profiles. In order to realize the robot crawling, a saw-tooth design of robot feet is applied near the output layer with asymmetric frictions along forward and backward directions. Then, the metamaterial robot can switch multiple driving modes according to output distributions, such as g straight crawling, g clockwise and g counterclockwise turning as well as their combinations in Fig. 6(c). Following the principle, Fig. 6(d) illustrates several experimental realizations of the crawling robot, in which the output wave profiles are mainly distributed in the g , g , (g , g ), and (g , g ) regions. For each experimental realization, when the output field becomes asymmetric, the robot can be driven to steer as time increases. As a result, the metamaterial robot with g , g , (g , g ) and (g , g ) output regions can present the locomotion behaviors of right turning, straight crawling, fast crawling and left veering in Fig. 6(d), respectively.

On the other hand, when we consider the robot as a computational system, the proposed functions can be abstracted as combinations of several embodied computing modules. As shown in Fig. 6(e), the operator ℱ{ℱ[g(y)]} = g(–y) after two Fourier transform arrays can be considered as a bit shifter that performs signal inversion. When encoded inputs are injected, the incident bit signals can be directly transformed as analog wave inputs, which exhibits as a digital-to-analog (D/A) converter. Then, after an embodied calculator, the metamaterial robot can be actuated to move by the cooperation of analog outputs and asymmetric frictions.

In order to verify the embodied function above, Fig. 6(f) provides a computational application of differential operation in target pursuit. Based on the relation between the target and real crawling directions, the input sequence can be encoded as a step function, in which signals at left/right sides of the target are labeled as ±1. Due to the differential operation, the output sequence can be modulated as a peak signal with mirror position to the target. The peak signal can continually follow the target position and drive the robot toward the target as negative feedback. In order to validate this principle, Figs. 6(g) and 6(h) provide the experiments of single- and multi-gait target pursuits. The input signals s can be changed by the target positions and thus generate a series of input sequences from 11000000 to 11111100. We can see from Fig. 6(g) that as the input sequence s shifts from 11000000 to 11111100, the metamaterial robot can be guided to steer from left to right by 00000110 to 01100000 outputs. Similarly, for an arbitrary path, the target pursuit can be achieved by the multi-gait locomotion with embodied differential operation.

In addition to the differential function above, Fig. 7 further extends the embodied computations to their combinational serial and parallel operations. Different from previous parallel strategies in frequency (Fig. 3) and bitwise (Fig. 5), Fig. 7 employs the parallel layout of several inputs f to realize addition and subtraction in spatial domain. On the other hand, with multiple transfer layers H in serial, the proposed functions can be also extended to their multiplication in spatial frequency domain. Limited by space, detailed derivations about spatial serial/parallel computing are provided in Supplementary Note S1. The following discussion is focused on the evolution from simple to combinational operations with robotic crawling.

As shown in Fig. 7(a), the robot employs an extended ℱ -H -ℱ layout of Fig. 6(c) with multilayer transfer functions H . Based on the serial/parallel strategies, the previous single-task layout is extended to several transfer functions and inputs in Figs. 7(b) and 7(c). According to the derivations in Supplementary Note S1, the serial connecting of several transfer functions H can lead to their multiplication operations in spatial frequency domain as H = H H … H . On the other hand, the parallel layout of two sub-inputs f = (f , f ) enables the addition/subtraction operation g = f ± f by a specific transfer function H(y) = exp(iyY/4) ± exp(–iyY/4). As examples, Figs. 7(d)–(r) provide several implementations of simple, serial and parallel operations. The left, middle and right columns correspond to the integral/differential, solving calculus equations and addition/subtraction operations, respectively.

In addition to theoretical simulations, Figs. 7(p)–(r) also provide the experimental responses of robot locomotion to verify the spatial serial and parallel computing. The upper and lower figures in Figs. 7(p)–(r) correspond to the integral and differential operations in Figs. 7(g) and 7(j), solving calculus equations in Figs. 7(h) and 7(k) as well as addition and subtraction operations in Figs. 7(i) and 7(l), respectively. The correspondences between locomotion states and computational results are provided in Figs. 7(m)–(o) and Fig. 6(c).

Firstly, we can observe from the first column of Fig. 7 that the differential and integral exhibit as inverse operations, in which the input and output profiles can be swapped in proportion. With a symmetric input profile, the robot can present asymmetric output pattern in Fig. 7(g) and left turning in Fig. 7(p). As transfer layers increase, the robot in the second column can be employed to solve calculus equations with both differential and integral terms. Based on Fourier transform properties, the transfer function in Fig. 7 can be derived as H = H H with H = –LCy2+iRCy+1 and H = 1/(iCy). Using the serial strategy, the system with C:L:R = 1:2:±3 can yield opposite profiles in Figs. 7(h) and 7(k) and enables left and right turning in Fig. 7(q). In addition to serial functions, the left column provides the parallel addition/subtraction operations of two sub-inputs (f , f ) as g = f ± f . We can see from Figs. 7(i) and 7(l) that the outputs corresponding to f + f and f – f can be enhanced and eliminated, resulting in the motion and station of the robot, respectively. Furthermore, the parallel operations can be also extended to binary logics that encode robot station and motion as binary ‘0’ and ‘1’, e.g., AND, OR and XOR gates. For instance, the XOR gate can be achieved by the subtraction operation g = f –f which offsets two logic ‘1’ inputs as ‘0’. With a series of simple logics in serial and parallel, more complex and cascaded designs can be achieved. Limited by space, details about the evolution from analog to logic operations are provided in Supplementary Figure S3 and Movie S4.

In addition to fundamental straight and turning behaviors, the metamaterial robot also enables the combinations of multiple moving gaits with more complex performances. As shown in Fig. 8(a), an arbitrary crawling path with several obstacles is applied to show the multi-gait locomotion from S0 (start point) to S1 (stop point). After 2 right-angle, 1 C-shape, 1 obtuse-angle and 1 acute-angle turning, the metamaterial robot can accurately arrive at target position. The crawling trajectory and gait parameters in Figs. 8(b) and 8(e) show that the robot behaves as the steady-state locomotion with robust performances. In addition, Figs. 8(c) and 8(d) further illustrate the single- and multi-gait locomotion. From curved and O-shape paths to acute-angle and obtuse-angle turning, the metamaterial robot can present diverse crawling to satisfy different requirements.

Furthermore, Figs. 8(g)–(j) provide several multifunctional experiments from sharp turning and load carrying to gap crossing and slope climbing. Figs. 8(g) is used to show the flexibility of robot crawling and Fig. 8(h) aims to illustrate its motion robustness and load carrying capability. Figs. 8(i) and 8(j) are presented to demonstrate the gap crossing and slope climbing abilities with complex environment adaptivity. As we can observe from the discussion above, the diverse locomotion modes and versatile computational functions suggest its potential applications in complex and multitasking working conditions.

Finally, it is worth mentioning that the proposed metamaterial robot has the ability to integrate both robot crawling and computational functions into a united framework. Such a robotic device can crawl while performing mathematical operations and its moving state can be represented by the computational results. Different from conventional crawling robots requiring additional control devices, the proposed embodied functions enable fully mechanical regulations of robot crawling in place of electrons. These properties would provide potential applications to some special conditions in which electronic devices are limited, e.g., high temperature and ionizing radiation. In addition, different from wheeled and legged systems, the wave-driven devices can switch locomotion modes by tuning the driving wave frequency and amplitude. Such properties which make the robot more flexible and adaptive to different working requirements. As illustrated in Fig. 8, such flexibility and adaptability allow the metamaterial robot to show multifunctional performances from sharp turning and load carrying to gap crossing and slope climbing. Furthermore, Movies S8 and S9 provide the experimental videos of multi-gait and multifunctional applications mentioned above. As supplements, the load carrying and gap crossing functions are further investigated in Movie S9 with the max load of 4.28 kg and gap width of 1/2 robot length.

Methods
Summary of Supplementary Information. Details about the designs of metamaterial robot and principles for implementing mechanical computing are provided in Supplementary Note 1. Evolutions from the spatial analog computing to logic operations are illustrated in Supplementary Note 2. Theoretical derivations of nonlinear harmonic band structures are performed in Supplementary Note 3. Investigations about nonlinear harmonic behaviors and band gap properties are presented in Supplementary Note 4. Analytical derivations of steady-state crawling velocities are given in Supplementary Note 5. Numerical calculations and experimental comparisons of the crawling velocity and multifunctional locomotion are discussed in Supplementary Note 6. Furthermore, structural and material parameters used during experiments and numerical calculations are listed in Table S1. Design principles, experimental performances and computational functions are discussed and presented in Movies S1–S9.

Data Availability
All data supporting the findings of this study are available within the Article and its Supplementary Information. Additional data generated in this study are available from the corresponding author on request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Mechanical computing, which leverages structural architectures to perform calculations and respond to mechanical stimuli, offers a promising pathway toward realizing embodied computational autonomy and mechanical intelligence in smart, adaptive, and robotic materials. In this work, the authors propose a mechanical metamaterials platform capable of integrating a broad range of computational functions—including analog operations such as differentiation and integration, digital logic operations such as AND, OR, and XOR, as well as qubit-like logics—into robotic locomotion. The study is technically sound and interesting, and I recommend it for publication in Nature Communications. The following comments should be considered for further improving the manuscript:\n(1) In this paper, the term “universal” seems to be intended to highlight that mechanical analog, logic, and quantum-inspired computing can be integrated within a unified framework. However, in computational theory, a universal computing machine usually refers to a system that can, in principle, simulate any real-world computer given unlimited time and memory. Since the proposed metamaterial does not possess information storage capabilities, its computational power may not fully align with this definition. Thus, describing its computational capabilities as “universal” is potentially misleading to readers.\n(2) Could the authors provide a more detailed explanation of the claimed parallel computing in the frequency domain? In Fig. 4(f) and 4(g), the input frequencies are 70 Hz and 140 Hz, respectively, yet the corresponding computational outputs are both observed to be 70 Hz (Fig. 4o). If two inputs at different frequencies (70 Hz and 140 Hz) are applied simultaneously, it seems that the outputs would overlap at 70 Hz, making it difficult to distinguish the contributions of different inputs. Does this suggest that the system cannot simultaneously perform calculations at both 70 Hz and 140 Hz? If so, how to understand the parallel computing capability in the frequency domain emphasized in the paper?\n(3) On page 11 (second paragraph), the authors state: “Based on the eigenstate diagram in Fig. 4(e), three incident frequencies with f_e = 70, 140 and 300 Hz are selected in Figs. 4(f)–4(h).” However, the third paragraph on the same page lists a different incident-frequency value, which appears to be a clerical inconsistency. Please verify the intended frequencies and correct the manuscript so that the values reported in the text, figure captions, and Fig. 4 itself are consistent.\n(4) I agree that this paper provides an interesting perspective on the modulation of acoustic waves by metamaterials within the framework of quantum computing. Nevertheless, a more rigorous discussion of acoustic quantum computing could further enrich the work. For example, Ref. 32 reports the realization of acoustic quantum analogue computing based on acoustic phi-bits, which were argued to satisfy DiVincenzo’s five criteria for the physical implementation of a quantum computer (Runge, K., Deymier, P. A., Hasan, M. A., Lata, T. D. & Levine, J. A. Acoustic metamaterials for realizing a scalable multiple phi-bit unitary transformation, AIP Advances 14, 025010 (2024)). It would be helpful if the authors could clarify the distinction between the acoustic phi-bits and the mechanical qubits presented here, and discuss to what extent the proposed mechanical qubits may be compatible with or fall short of DiVincenzo’s criteria.", "reviewer_2": "Comments for Author\n• What are the noteworthy results?\nA mixed signal architecture that integrates analog and logic operations, allowing for nested and hierarchical operations. The architecture is also shown to be able to switch between multiple different modes providing multi-task ability. The implementations is based on electric wave metamaterials. The extensive experimentation demonstrates the various computing methodologies and implements a simple crawler robot.\n• Will the work be of significance to the field and related fields? How does it compare to the established literature? If the work is not original, please provide relevant references.\nThe work is similar to a broad swathe of literature in mechanical computing, which has seen a notable amount of popularity over the past few years. The core claim of mixed signal and multi-task has previously been shown to some extent in, e.g., https://www.pnas.org/doi/10.1073/pnas.2407431121. There is also a string of uncited works covering e.g., ‘ghost circuits’, etc. There’s a lot of work on the embodied side e.g. https://arxiv.org/abs/2110.15245, especially Helmut Hauser’s work e.g. https://link.springer.com/article/10.1007/s00422-012-0471-0 . The work is therefore partially original. It would likely be of interest to specialists in the field. It is too dense and specialised to be understandable by many non-specialists.\nOne weakness with this work is that the robotic implementation is simple – this is fine as a proof of concept but really shows the inability to turn this technology into something useful. Despite claiming to present a universal architecture, it is unclear why this is an important problem and the justification for using their implementation is weak/underdeveloped.\n• Does the work support the conclusions and claims, or is additional evidence needed?\nThe work would benefit from a degree of ‘hand holding’ – e.g. for each experiment, what are you showing, why are you showing it, etc? The paper needs to be navigable.\n• Are there any flaws in the data analysis, interpretation and conclusions? Do these prohibit publication or require revision?\nResults seem sound.\n• Is the methodology sound? Does the work meet the expected standards in your field?\nThere are large numbers of equations without much guidance between them. The figures are average quality. In general this paper was very hard to read and understand. There is a lot of quantum theory interspersed into large walls of text. It is therefore a huge effort, requiring a background in quantum, to see if the methodology is sound, and if the work is reproducible.\n• Concerns\nUnclear how this is novel compared to some of the recent literature that appears to do the same thing.\nAlthough I’ve been told not to comment overly on the language, use of language in this manuscript is of a poor standard in the introduction, and can lead to confusion. I frequently found myself becoming lost, or having to re-read a passage multiple times to try and decode what the authors are trying to say. I highly recommend the authors use a professional editing service. Many of the sentences are not legal sentences, or incredibly unclear, e.g.\n1. “The flexible designs from bioinspired materials to molecular crystalline structures endow metamaterials with extraordinary mechanical properties, such as acoustic cloaking, negative refractive and topological insulators [5–8].”\n2. “In comparison, novel progress in metamaterials has developed wave-based analog computing [26–30], which enables continuous processing of complex information in spatial.”\nFig 1 the caption does not explain the figure, which is very dense and needs much clearer explanation.", "reviewer_3": "Referee report for\nNature Communications\nManuscript NCOMMS-25-641411\nReprogrammable metamaterial robot with embodied universal mechanical computations\nby W. Zhou and Y.-Z. Wang\nThe manuscript presents a metamaterial robotic system that has three modes for computations. The first mode is conformal change in the metamaterial structure, a second mode is in the vibrational spectrum supported in the robotic system. Finally, the system can use locomotion as a means to accomplish computation. The authors claim that they have demonstrated binary and quantum logic operation using the robotic system. Further, they claim that the system is capable of both serial and parallel operation. Finally, the authors predict that the system will perform robustly in harsh environments. The current manuscript has a number of drawbacks including the lack of quantitative measures for the fidelity of the computations. Also, the concept of quantum computing is not well represented by this system as it lacks both scalability and flexibility with respect to input states. The length of supplementary material, both written and video, is an impediment to the assessment of this submission. I recommend resubmission of this manuscript after major revision.\nThe concept of using multiple aspects of the metamaterial robotic system for computational means is of interest and would be advantageous for exploration in harsh environments. It is, however, unclear how the authors would propose to accomplish inputs from the environment to adapt an instruction set for flexible operation. Further, attention must be paid to the accuracy of the computed result, the authors fail to report any measure of the uncertainty related to any of the computations performed. It is clear from the figures, e.g., Figure 2, that the accuracy of the computation is limited by the nature of the robotic system. A measure of uncertainty is required for each of the computations performed.\nWhile acoustic systems analogous to quantum computers have been reported in the literature and cited by the authors, some requirements for quantum computation are not met by the current system. In particular, following Divincenzo’s criteria, the system fails to be scalable. Recall that quantum advantage is usually considered to require 50 qubits. This robotic system does not scale to such a large number, or at least such scaling is not clear to this reviewer. Secondly, for general quantum computing the initial state must be able to accommodate a wide range of states available on the Bloch sphere. Again, it is not clear that the current system will support such diversity in specifying the initial state.\nWith regard to the supplementary material, it does not appear to this reviewer that the video material adds to the content of the manuscript. While the written supplementary material does support the manuscript, its length could easily be reduced without losing the vital supporting information the manuscript requires.\nI strongly recommend that the authors include quantitative measure of the uncertainty related to each of the computations performed in the manuscript. Further, references to quantum computation should be removed from the manuscript. Operations that have performed on so-called ‘few qubit states’ can be reported as matrix manipulations rather than quantum logic operations.\nI, therefore, recommend that the article be resubmitted after these major revisions.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Responses to Reviewers’ Comments\nJournal: Nature Communications\nManuscript number: NCOMMS-25-62680-T\nArticle title: Reprogrammable metamaterial robot with embodied versatile computation and\nmechanical intelligence\nPoint-by-point responses to Reviewer 1’s comments\nReviewer 1\nGeneral comment: Mechanical computing, which leverages structural architectures to perform\ncalculations and respond to mechanical stimuli, offers a promising pathway toward realizing\nembodied computational autonomy and mechanical intelligence in smart, adaptive, and robotic\nmaterials. In this work, the authors propose a mechanical metamaterials platform capable of\nintegrating a broad range of computational functions—including analog operations such as\ndifferentiation and integration, digital logic operations such as AND, OR, and XOR, as well as\nqubit-like logics—into robotic locomotion. The study is technically sound and interesting, and I\nrecommend it for publication in Nature Communications. The following comments should be\nconsidered for further improving the manuscript:\nResponses:\nThe authors greatly appreciate Reviewer 1 for your positive comments and high recognition of our\nwork. We wish that our work could provide some effective strategies for the applications of elastic\nwave metamaterials in mechanical computing and robotics in the future!\nWe also sincerely appreciate Reviewer 1 for your careful reading and helpful suggestions on our\nwork. We have carefully checked our manuscript and substantially improved the expression in the\nrevised version. Our responses to your valuable comments and suggestions are listed below with\npoint-by-point responses. We wish that these revised statements and improvements could better\nconvey the main features of our present work.\nComment 1: In this paper, the term “universal” seems to be intended to highlight that mechanical\nanalog, logic, and quantum-inspired computing can be integrated within a unified framework.\nHowever, in computational theory, a universal computing machine usually refers to a system that\ncan, in principle, simulate any real-world computer given unlimited time and memory. Since the\nproposed metamaterial does not possess information storage capabilities, its computational power\nmay not fully align with this definition. Thus, describing its computational capabilities as\n“universal” is potentially misleading to readers.\nResponses:\nThe authors greatly appreciate your careful reading and insightful comment about the use of\n“universal”. We quite agree with Reviewer 1 about that in computational theory, the “universal\ncomputing” is usually related to the Turing-completeness and ability to simulate any real-world\ncomputer in principle, which typically requires additional information storage and retrieval\nfunctions.\nOur main attention is focused on the ability to perform multifunctional and versatile computing,\nrather than implying its universality and completeness. Such a robotic system aims to integrate\nmultiple different computational functions within a unified framework, from mechanical analog\ncomputing to binary and quantum-inspired logic computing. It can crawl while performing\nmechanical computing and its locomotion states can be fully represented by computational results.\nWe quite agree with your valuable comments and sincerely appreciate the opportunity to further\nimprove this statement.\nRevisions:\nIn order to present more precise description, we use a more proper word “versatile” to replace the\nterm “universal” and revised the manuscript throughout. We wish that the statement “versatile\nmechanical computing” could better express our main idea of integrating multiple computational\nmodes with robotic locomotion. The title, abstract and relevant sections have been modified\naccordingly and please refer to blue sentences in the revised manuscript. We wish that the revised\nstatements could address most of your concerns.\nComment 2: Could the authors provide a more detailed explanation of the claimed parallel\ncomputing in the frequency domain? In Fig. 4(f) and 4(g), the input frequencies are 70 Hz and 140\nHz, respectively, yet the corresponding computational outputs are both observed to be 70 Hz (Fig.\n4o). If two inputs at different frequencies (70 Hz and 140 Hz) are applied simultaneously, it\nseems that the outputs would overlap at 70 Hz, making it difficult to distinguish the contributions of\ndifferent inputs. Does this suggest that the system cannot simultaneously perform calculations at\nboth 70 Hz and 140 Hz? If so, how to understand the parallel computing capability in the\nfrequency domain emphasized in the paper?\nResponses:\nThank you very much for your valuable comments on this work with your careful reading of our\nmanuscript. We apologize for not clearly conveying the principle of parallel computing in the\nfrequency domain. We have improved the statements and further explained the relation between\nparallel/cooperative computing and transformable computing in the following parts:\nAs mentioned in Fig. 3, the authors have proposed two kinds of computational modes with the\nℱ -H-ℱ *-PLs layout, including the cooperative computing in Fig. 3(j) as well as transformable\n1 2\ncomputing in Figs. 3(k)–3(m).\n(1) Cooperative computing\nFirstly, when both fundamental and harmonic waves locate at pass bands, the robot allows\npropagations of different harmonic components within the same structure, which corresponds\nto the “cooperative computing” or “parallel computing”. For this computational mode, all\nharmonic components are independent and their computational results behave as functions of\ntheir harmonic orders r. As a result, the total output fields g(y, t) can be expressed as the\nsuperposition of different harmonic results as illustrated in Eq. (6):\n3\ng ( y , t ) C\nr1\ng\n0\n( y ) c o s (\n0\nt\nr1\n)\nr 2\nC\nr 2\nd n r n g\n0\n( y ) d y n r n c o s ( r\n0\nt\nr 2\n)      + +  − − +\n+ C  g (y)dydycos(t r+ ) . (R.1)\nr3 r3 0  0 r3\n(nr–n) r\nWe can observe from Eq. (R.1) (i.e., Eq. (6)) that in the frequency domain, each harmonic\ncomponent has its own computational result and all components operate at the same time. It\nindicates that for this computational mode, the system allows parallel computing of different\nharmonic components, which corresponds to the “parallel computing in the frequency domain”\nin the previous version.\n(2) Transformable computing\nDue to the band gap property of periodic layers PLs, the mechanical metamaterial allows the\nfiltering of harmonic components and can transform them into different frequency regions. By\ntuning incident frequencies within different harmonic stop and pass bands, the mixed waves\ncan be directly output as multi-task results in Eq. (R.1) or separated as single-task results in\nFigs. 3(k)–3(m). The latter is considered as the “transformable computing”.\nFor example, as your valuable comments on Fig. 4, The two inputs with frequencies f = 70\ne1\nHz and f = 140 Hz are injected into the metamaterial but their corresponding outputs are\ne2\nboth 70 Hz. It is because that elastic waves with f = 140Hz locate at the stop band of\ne2\nfundamental waves but fall into the pass band of subharmonic waves. Due to the nonlinearity,\nthe inputs with f = 140 Hz can yield subharmonic components with frequency f = 70 Hz.\ne2 sub\nThen, the fundamental signal with f = 140 Hz can be eliminated by periodic layers PLs but\ne2\nthe subharmonic one with f = 70 Hz remains.\nsub\nIt is worth noting that there are some advantages for the “transformable computing”. Different\nfrom conventional single-tasking devices with sole response or multi-tasking devices with\nmixed responses, our system can present multiple independent results according to the\nincident frequency. On the other hand, it can transform different harmonic inputs into the\nsame frequency while maintaining their computational results. This function provides\npossibilities of subsequent operations requiring identical frequency condition, e.g., the spatial\naddition/subtraction operations in Fig. 7. In this case, two input signals f and f with different\n1 2\nfrequencies (e.g., 70 and 140Hz) can yields the results g = d2f /dy2 ± df /dy (70Hz), which is\n1 1\nnot possible for the frequency-domain parallel computing. As a result, Fig. 4 is used to\nillustrates the “transformable computing” instead of presenting the “parallel computing in the\nfrequency domain”.\nWe quite agree with your comments for not clearly explaining the “transformable computing” and\nfrequency-domain “parallel computing”. In the revised manuscript, we have added additional\nfigure and discussion to further illustrate the frequency-domain “parallel computing” function.\nRevisions:\nIn the previous version, we verified the “transformable computing” in Fig. 4 but discussion about\n“parallel computing” in frequency domain remains several sentences. In this revision, we add\nfigure and discussion to further illustrate the parallel and cooperative computing functions\nachieved in the frequency domain. As discussed above, the “cooperative computing” requires that\nall harmonic components locate at their corresponding pass bands. As a result, we consider the\nℱ -H-ℱ layout without periodic layers PLs. This layout can be seen as a degeneracy of the\n1 2\nℱ -H-ℱ *-PLs layout and is equivalent to the ℱ -H-ℱ *-PLs layout without stop bands. The\n1 2 1 2\n4\ntransfer layer H is designed as the 2nd derivative g (y) = d2f(y)/dy2 for 70 Hz waves and can be\n0\ntheoretically transformed to g(y) = ∫…∫g (y)dy…dy (2r–2)/r for the r-th harmonics.\nr 0\nSingle-frequency excitation Dual-frequency excitations\nf e = 70 Hz f e = 140 Hz f e = 70 & 140 Hz Full components\n1.2\ns\nin\nxam s H\nv 0\nInput sin Input sin Input sin v/ s\nout\n–1.2 df/dy + d2f/dy2\n–Y/2 0 Y/2\n(g) y\nsH sH sH\n70Hz component\n1.2\ns\n1.2 1.2 1.2 xam\nin, 1\n(\nv v / –\na\nxam 1\n) .\n–\n0 2\nY\nO\n/2\nu tpu\ny\n0\nt sou\nY\nt\n/2\nv v /\n(b\n– xam 1\n) .\n–\n0 2\nY\nO\n/2\nu tpu\ny\n0\nt sou\nY\nt\n/2\n(\nv v /\nc\n– xam\n) 1 .\n–\n0 2\nY\nO\n/2\nu tpu\ny\n0\nt sou\nY\nt\n/2 f\ng\nl\nn\na\ni\nn\nr e tli v v /\n(\n–\nh\n1\n) .2 0\n– Y\nd\n/2\n2f /dy2\n0\ny\ns ou\nY\nt,\n/\n1\n2\ng\niS\nFrequency responses of computational layers\n140Hz component\n1.2\n70Hz 140Hz 70Hz 140Hz s\nin, 2\nv v / xam 0\n1\n.\n. 0 6\n2\ns in s H s out v v / xam 0\n1\n.\n. 0 6\n2\ns in s H s out v v / xam 0\n1\n.\n. 0 6\n2\ns out s H s in v v / –\nxam\n1.2 0 df/dy s out, 2\n0 100 200 300 0 100 200 300 0 100 200 300 –Y/2 0 Y/2\n(d) Frequency (Hz) (e) Frequency (Hz) (f) Frequency (Hz) (i) y\nFig. R1. Parallel and cooperative computing in the frequency domain. (a) and (b) Velocity responses of unit\ncells under single-frequency excitation with (a) fe = 70 Hz and (b) fe = 140Hz. (c) Velocity responses of unit cells\nunder dual-frequency excitation with fe1 = 70 Hz and fe2 = 140Hz. (d)–(f) frequency responses of unit cells\ncorresponding to (a)–(c). The red sin, black sH and blue sout curves represent signals generated by input, transfer\nfunction and output layers. (g) Full-frequency input/output responses of (c) under dual-frequency excitation. (h)\nand (i) Filtered input/output responses of (c) corresponding to (h) 70 Hz and (i) 140 Hz components.\nAs shown in Fig. R1, the elastic wave metamaterials with single- and dual- frequency excitations\nare considered to illustrate the parallel and cooperative computing. Firstly, the single-frequency\nexcitation is performed in Figs. R1(a) and R1(b) with different frequencies 70 Hz and 140 Hz. It\ncan be observed from Figs. R1(a) and R1(b) that with the same input profile s and transfer layer,\nin\nthe outputs s for 70 Hz and 140 Hz behave as odd and even functions, respectively. This\nout\nphenomenon agrees with theoretical results of 2nd and 1st derivatives. In addition, the frequency\nresponses in Figs. R1(d) and R1(e) show that for both 70 Hz and 140 Hz excitations, the input s\nin\nand output s almost keep their original frequencies and do not change during wave propagation.\nout\ndifferent from the transformable computing in Figs. 4(f) and 4(g).\nIn order to support the function of simultaneously processing different tasks, Figs. R1(c) and R1(f)\npresents the spatial and frequency responses of the metamaterial under dual-frequency excitation.\nWe can see from Fig. R1(c) that the output profile s behaves as the superposition of two\nout\nharmonic outputs and becomes asymmetric to the midpoint y = 0. The frequency responses in Fig.\nR1(f) show that both 70Hz and 140Hz components exist in the output signals, which agree with\nthe input ones. In order to further verify the parallel property, the combined input s and output\nin\ns profiles in Fig. R1(c) are filtered into two harmonic components s and s in Figs.\nout in,1–2 out,1–2\nR1(g)–R1(i). It is obvious from Figs. R1(h) and R1(i) that the 70 Hz and 140 Hz components can\npresent almost the same computing results as those with single-frequency excitations. These\nphenomena indicate that the system allow the independent calculation of different frequency\ncomponents, which show the ability of parallel and cooperative computing in the frequency\ndomain.\nWe have added the figure and explanations above to Supplementary Information and modified\nstatements about the parallel computing in the revised Manuscript. Please refer to the blue\nsentences we modified in the second paragraph on page 5, the second paragraph on page 10 in the\nrevised Manuscript. And please also refer to Fig. S4 and the blue sentences we added on pages\n10 and 11 in revised Supplementary Information. We wish that these revised statements and\nimprovements could address most of your concerns.\nComment 3: On page 11 (second paragraph), the authors state: “Based on the eigenstate diagram\nin Fig. 4(e), three incident frequencies with f_e = 70, 140 and 300 Hz are selected in Figs. 4(f)–\n4(h).” However, the third paragraph on the same page lists a different incident-frequency value,\nwhich appears to be a clerical inconsistency. Please verify the intended frequencies and correct the\nmanuscript so that the values reported in the text, figure captions, and Fig. 4 itself are consistent.\nResponses:\nThe authors quite agree with Referee 1 and greatly appreciate your careful reading. We apologize\nfor the incorrect labeling of the incident frequencies, which is indeed a clerical inconsistency. We\nhave revised the frequencies in the discussion about Fig. 4 and checked other sentences carefully\nin the whole work. The modifications are provided as follows.\nRevisions:\nThe incident frequencies in the previous version are revised as 70, 140 and 300 Hz, respectively.\nPlease refer to the blue sentences in the first paragraph on page 12 in the revised manuscript.\nComment 4: I agree that this paper provides an interesting perspective on the modulation of\nacoustic waves by metamaterials within the framework of quantum computing. Nevertheless, a\nmore rigorous discussion of acoustic quantum computing could further enrich the work. For\nexample, Ref. 32 reports the realization of acoustic quantum analogue computing based on\nacoustic phi-bits, which were argued to satisfy DiVincenzo’s five criteria for the physical\nimplementation of a quantum computer (Runge, K., Deymier, P.A., Hasan, M.A., Lata, T.D. &\nLevine, J. Acoustic metamaterials for realizing a scalable multiple phi-bit unitary transformation,\nAIP Advances 14, 025010 (2024)). It would be helpful if the authors could clarify the\ndistinction between the acoustic phi-bits and the mechanical qubits presented here, and discuss to what extent\nthe proposed mechanical qubits may be compatible with or fall short of DiVincenzo’s criteria.\nResponses:\nThank you very much for your positive comments and constructive suggestions on our present\nwork. We quite agree with Reviewer 1 and perform more comprehensive discussion about the\nmechanical qubit-like computing.\n1. Comparisons between acoustic phi-bits and mechanical qubits\nFirstly, we sincerely appreciate Reviewer 1 for highlighting the pioneering contributions of Runge\net al. [R1] and their group’s related investigations on acoustic phi-bit computing [R2–R5]. These\nstudies have extended the applications of acoustic/elastic wave metamaterials to mechanical\ncomputing. The authors have followed your kind suggestions and summarized some similarities\nand differences between the acoustic phi-bit computing and mechanical qubit-like computing\npresented here.\n(1) Physical forms\nAccording to the studies of Runge, Deymier and Hasan et al. [R1–R5], their group members\nhave developed many effective strategies which can perform acoustic phi-bit unitary\ntransformations analogous to the qubit computing. As Runge et al. previously discussed [R1],\nthe logical phi-bit is “a two-state DOF in the spectral domain of nonlinear acoustic modes”. It\nis capable of operating in a coherent superposition state with complex amplitude coefficients.\nBased on the spectral responses in Refs. [R1, R2], the logical phi-bits are individual frequency\npoints in the spectral domain and their logic states are fully characterized by their relative\nphase relation φ and φ . However, in our present work, the mechanical qubit behaves as a\n21 31\npair of wave signals in the spatial domain and their logic states are represented by both wave\namplitudes {A , A }T and phase differences Δφ = φ – φ . Adjacent mechanical qubits are\n1 2 1 2\ndivided through spatially discrete arrays rather than spectral frequency intervals employed in\nthe phi-bit system.\n8\n(2) Excitations and measurements\nAs reported by Runge et al. [R1], the acoustic phi-bits which are represented by individual\nharmonic signals in the spectral domain are generated by a couple of excitation frequencies f\n1\nand f in different nonlinear waveguides. Then, the detected outputs are the Fourier sum of\n2\nacoustic modes with f and f and their harmonic components. Their logic states are measured\n1 2\nby detecting the phase differences of two nonlinear waveguides relative to the fundamental\none. Different from phi-bits, our mechanical qubit originates from a couple of parallel\ntraveling waves with identical excitation frequency. By measuring the amplitudes and relative\nphases of two traveling waves, the logic states can be determined.\n(3) Computational principles\nAs mentioned in Ref. [R1], the acoustic phi-bits are correlated via the nonlinear elasticity of\ntheir physical system. Accordingly, the coherent superposition of logical phi-bit states can be\nrealized by modulating physical parameters such as the frequency, relative phase or intensity\nof driving forces. For instance, in Ref. [R1], phase exchange and the associated several\nunitary transformations can be achieved by changing the driving parameters e.g., frequency\nand phase relation. However, in our present work, the computations are achieved by the wave\ninteraction of two incident signals within the architected structure, such as superposition,\ninterference, diffraction, as well as wave focusing, reflection and division. More complex\noperations can be achieved by the serial and parallel designs in the spatial domain.\nIt is worth mentioning that both computational methods have their own advantages. The acoustic\nphi-bits are distributed in several small intervals in the spectral domain, which have special\nadvantages of processing large information and multitasking. For multi-bit computations, it does\nnot require decomposition into smaller-scale bit gates. Our present designs allow versatile serial\nand parallel modes in the spatial domain and are compatible with previously mentioned analog\nand digital logic operations. In particular, the computational results can be directedly output as\nwaveform signals or robot locomotion states without additional Fourier transformations.\n2. DiVincenzo’s five criteria\nThe authors are also grateful for your kind suggestions of discussing what extent the proposed\nmechanical qubits may be compatible with the DiVincenzo’s criteria. It would help readers to\nhave a more thorough understanding of the mechanical qubit-like computing presented here. We\nhave followed your valuable suggestions and summarized as follows:\n(1) A scalable physical system with well-characterized qubit\nAs discussed above in the “Physical forms” section on page 8, our proposed mechanical\nqubits are characterized as a pair of traveling wave signals in the spatial domain and their\nlogic states can be represented by both the wave amplitudes {A , A }T and phase differences\n1 2\n9\nΔφ = φ – φ . As the exciting waveforms and phases change, the system can present different\n1 2\nqubit-like states as the superposition of two basis states |0〉 = {1, 0}T and |1〉 = {0, 1}T , i.e.,\n|ψ〉 = α|0〉+β|1〉 with |α|2+|β|2 = 1. The logic ‘0’ and ‘1’ states in |0〉 and |1〉 correspond to the\nexistence and nonexistence of traveling signals. And α and β are complex parameters related\nto {A , A }T and φ – φ . Since the metamaterial can be designed with multiple waveguides for\n1 2 0 1\ntraveling wave propagating, the physical system can be also scaled by the serial and parallel\nstrategies of fundamental qubit-like gates in the spatial domain.\n(2) The ability to initialize the state of the qubits to a simple fiducial state\nThe ability to initialize the state of the qubits to a simple fiducial state requires all qubits to be\na known, pure initial state before computations. As discussed above, the fiducial states |0〉 =\n{1, 0}T and |1〉 = {0, 1}T in our system are characterized by two single-mode excitations, i.e.,\n(A = 1, A = 0) and (A = 0, A = 1), respectively. For example, for single-qubit gates like\n1 2 1 2\nHadamard and Pauli X gates, the fiducial state |0〉 can be initialized by setting the wave\namplitude {A , A }T as {1, 0}T, which corresponds to wave propagations only along the upper\n1 2\nhalf-space.\n(3) Long relevant decoherence times\nSimilar to the acoustic phi-bits discussed in Rf. [R1], the coherent superpositions of phi-bit\nstates are acoustic complex amplitudes. These states are not subjected to decoherence, which\nare different from the probability amplitude of quantum systems composed of qubits. In our\npresent work, the physical forms for achieving mechanical qubit-like computing are elastic\nwaves with deterministic amplitude and phase information. Before computation completed,\nthe coherent superpositions of mechanical qubits can be sustained during wave propagation,\nwhich exhibits long relevant decoherence times.\n(4) A ‘universal’ set of quantum gates\nQubit-like gates: Our proposed qubit-like computing is achieved by the interactions of\ntraveling wave pairs, such as interference, diffraction, superposition, as well as wave focusing,\nreflection and division. As previously verified in Figs. 5 and S5, multiple fundamental and\ncombinational qubit-like operations are achieved, including the single-qubit Pauli X, Pauli iY\nand Hadamard gates, two-qubit CNOT gate, as well as more complex serial and parallel\noperations. Among these designs, the system behaves as a mechanical bit processor that can\nperform different qubit-like unitary operations.\nUniversal requirements: On the other hand, the ‘universal’ conditions require that the system\ncan theoretically perform arbitrary unitary operations. As reported by Runge et al. [R1],\nlargescale multiple-qubit matrix operations can be achieved by their decompositions of\nmultiple single- and two- qubit ones. It requires that our system accommodate a wide range of\nstates available on the Bloch sphere.\n11\nc o s ( / 2 ) 0 e i s i n ( / 2 ) 1     = + , (R.2)\nwhere θ and ϕ are Bloch sphere coordinates covering the entire sphere. As mentioned above,\nour presented qubit-like states can be characterized by the normalized wave amplitudes {A *,\n2 1 2\nA *exp(iφ ) and A *exp(iφ ) of the wave pair, we can obtain the equivalent qubit-like state as\n1 1 2 2\nA *1 e i 1 0 A\n2\n* e i 2 1    = + , (R.3)\nwhere A * and A * satisfy A *2 + A *2 = 1. Setting φ as the fiducial phase, the phase relation is\n1 2 1 2 1\nactually determined by the relative phase Δφ= φ – φ . We neglect the overall phase factor\n2 1\nexp(iφ ) and the qubit-like state can be written as |ψ⟩ = A *|0⟩ + A *exp(iΔφ)|1⟩. We assume\n1 1 2\nthat α = A * and β = A *exp(iΔφ), which results in the qubit-like state as\n1 2\n0 1    = + , (R.4)\nwhere α and β satisfy |α|2 + |β|2 = 1 and are equivalent to the Bloch sphere representation as α\n= cos(θ/2) and β = exp(iϕ)sin(θ/2). By tuning the wave amplitudes to satisfy A *2 + A *2 = 1\n1 2\nand adjusting relative phase Δφ= φ – φ , the system can accommodate a wide range of states\n2 1\non the Bloch sphere.\nDuring the computations, the wave splitter in Fig. 5(i) can adjust the wave amplitude ratio\nA /A , which corresponds to α and β varying in the real number range. The relative phase Δφ=\n1 2\nφ – φ can be adjusting by tuning the mass distribution m of transfer function layers to yield\n2 1 H\nphase differences, e.g., π for Hadamard gate in Fig. 5(e), which corresponds to β varying in\nthe imaginary range. Their combinations as well as spatial serial/parallel strategies provide\npossibilities of more complex unitary operations.\n(5) A qubit-specific measurement capability\nAs mentioned in Ref. [R1], measurements in conventional quantum computing approaches\nentail the collapse of the wavefunction. In contrast, measurements in our present mechanical\nqubit-like systems do not alter the wave function, which persists as long as the drivers are\napplied. In particular, the computational results can be directedly output as waveform signals\nand robot motion states without additional Fourier transformations and filtering devices.\nRevisions:\nFollowing the Reviewer 1’s valuable suggestions, we have added discussion about the mechanical\nqubit-like computing and compared with the well-established quantum computing and acoustic\nphi-bit computing. The response letter also discusses what extent this approach can satisfy the\nDiVincenzo’s five criteria for quantum computing. Limited by writing space, explicit discussion is\nlisted in the Supplementary Information. Please refer to the blue sentences we added on pages 13\nand 14 in Supplementary Information and Refs. [32–34] on page 26 in the revised Manuscript. We\nwish that these revised statements and improvements could address most of your concerns.\nOnce again, we sincerely appreciate your thorough review and valuable feedback that helps us to\ncomprehensively improve our manuscript!\nReferences:\n[R1] Runge, K., Deymier, P.A., Hasan, M.A., Lata, T.D., Levine, J.A., 2024. Acoustic\nmetamaterials for realizing a scalable multiple phi-bit unitary transformation. AIP Advances 14,\n025010.\n[R2] Hasan, M.A., Runge, K., Deymier, P.A., 2021. Experimental classical entanglement in a 16\nacoustic qubit-analogue. Scientific Reports 11, 24248.\n[R3] Deymier, P.A., Runge, K., Cutillas, P., Hasan, M.A., Lata, T.D., Levine, J.A., 2023. Scalable\nexponentially complex representations of logical phi-bit states and experimental demonstration of\nan operable three phi-bit gate using an acoustic metastructure. Applied Physics Letters 122,\n141701.\n[R4] Runge, K., Hasan, M.A., Levine, J.A., Deymier, P.A., 2022. Demonstration of a two-bit\ncontrolled-NOT quantum-like gate using classical acoustic qubit-analogues. Scientific Reports 12,\n14066.\n[R5] Cavalluzzi, D., Ige, A.S., Runge, K., Deymier, P.A., 2025. Realizing permutation gates with\nphi-bits: Acoustic quantum analogue computing. Journal of Applied Physics 137, 104901.\nReviewer 2\nComment 1: What are the noteworthy results?\nA mixed signal architecture that integrates analog and logic operations, allowing for nested and\nhierarchical operations. The implementation is based on electric wave metamaterials. The extensive experimentation demonstrates the various computing methodologies\nand implements a simple crawler robot.\nResponses:\nThank you very much for your thorough reading and comprehensive summary of our present work.\nWe are sincerely grateful for your recognition of the potential significance of this study.\nAs noted in your comments, the main idea of our present work is to develop a versatile mechanical\narchitecture of metamaterial robots which can integrate multiple analog and logic operations with\nrobot motion. This hybrid approach enables the nested and hierarchical operations within a single\nrobotic platform, allowing compatible and highly flexible designs. Moreover, the ability to\nreconfigure the system to different computational modes endows the metamaterial robot with\nparallel and multitasking functions, which is particularly promising for adaptive robot applications.\nIn addition, a proof-of-concept robot is designed to further verifies the feasibility of applying such\ncomputational designs to embodied robotic systems.\nWe wish that these clarifications could better convey the main idea of our work and align with\nyour understanding. Thank you again for your valuable feedback!\nComment 2: Will the work be of significance to the field and related fields? How does it compare\nto the established literature? If the work is not original, please provide relevant references.\n2.1. The work is similar to a broad swathe of literature in mechanical computing, which has seen\na notable amount of popularity over the past few years. The core claim of mixed signal and\nmulti-task has previously been shown to some extent in, e.g., https://www.pnas.org/doi/10.1073/\npnas.2407431121. There is also a string of uncited works covering e.g., ‘ghost circuits’, etc.\nThere’s a lot of work on the embodied side e.g. https://arxiv.org/abs/2110.15245, especially\nHelmut Hauser’s work e.g. https://link.springer.com/article/10.1007/s00422-012-0471-0. The\nwork is therefore partially original. It would likely be of interest to specialists in the field. It is too\ndense and specialized to be understandable by many non-specialists.\nResponses:\nThank you very much for your valuable comments on the mechanical computing and robotic\nimplementation presented in our work. We appreciate your careful reading and would like to\naddress your concerns in detail.\n2.1. Relationship to recent advances in mechanical computing\nFirstly, we sincerely appreciate Reviewer 2 for highlighting the pioneering contributions of Mousa\net al. (PNAS 2024) [R1], Nanayakkara et al. (Springer Nature 2024) [R2], Roy et al. (arXiv 2021)\n[R3] and Hauser et al. (Biological Cybernetics 2021) [R4], in multi-tasking processing, ghost\ncircuits, as well as embodied computing fields, respectively. These papers have significantly\nextended the applications of mechanical computing in embodied intelligence and robotics. The\nauthors have summarized some similarities and differences between their and our present works.\nA. Parallel and multi-tasking processing (Mousa et al., PNAS 2024)\nAs reported by Mousa et al. [R1], they have developed a time-modulated acoustic metasurface\nthat allows parallel and multitasking computing in the spectral domain. Due to the nonlinear\ncharacteristics, the incident signals generate several individual harmonic components that\ncalculate independently. Then, the received outputs are expressed by the Fourier sum of\nacoustic modes and their computational results are measured by filtering specific output\nfrequencies. In their work, the relation between computational functions and frequencies are\nestablished by designing the transmission responses of acoustic Helmholtz cavity at specific\nfrequencies.\nWe agree with Review 2’s valuable comments about that the work mentioned in PNAS is an\ninteresting study on mixed and multi-tasking computing. However, there are some obvious\ndifferences between our and their works, which include the working principles, functions\n14\nand architectures, application scenarios, as well as our core focus.\n(1) Differences between PNAS parallel computing and our transformable computing\nThe main focus of Mousa et al. [R1] is to validate the capability of acoustic metasurfaces to\nperform tasks parallelly in the frequency domain. However, in Figs. 3(k)–3(m) and 4, our\nsystem introduces the concept of “transformable computing” instead of previously mentioned\n“parallel computing”. With the nonlinear band gaps of periodic layers PLs, the system can\npresent filtering, tuning and transformable functions of different harmonic signals. Such an\narchitecture not only enables “parallel computing” (pass band properties) in Fig. S4, but also\nallows wave filtering and transformation of different harmonic inputs into a same frequency\n(stop band properties) in Fig. 4. This function allows algebraic superposition of different\ncomputational results.\nFor the conventional “parallel computing”, computational results are encoded across distinct\nharmonic components. Since these harmonics are physically decoupled, the signals cannot be\ndirectly output through elementary arithmetic operations. Conventional parallel strategies in\nfrequency domain usually require additional Fourier transforms and digital filtering devices.\nHowever, the “transformable computing” can integrate the multi-frequency information\ninto a single-frequency mode while maintaining computational results. Such a property\nprovides platforms for subsequent operations requiring identical frequency condition,\ne.g., spatial addition/subtraction operations.\nIn addition, as mentioned in Comment 2 of Reviewer 1, conventional “parallel computing”\ncan be considered as a special case of “transformable computing” in our system. Such a\ncomputing mode can be directly realized through the ℱ -H-ℱ layout without periodic layers\n1 2\nPLs, which is a degeneracy of the ℱ -H-ℱ *-PLs layout for “transformable computing”. As a\n1 2\nresult, this work extends Mousa et al.’s work [R1] from frequency-domain parallel computing\nto a hybrid cooperative and transformable computing mode.\n(2) Differences in core focus and system architecture\nThe paper of Mousa et al. [R1] and our work are not only different in functionality but also in\nsystem architecture. The acoustic metasurface of Mousa et al. exhibit as a “specialized analog\nprocessor”, which only focuses on the parallel function in the frequency domain. In contrast,\nour system aims to develop a “versatile computing architecture” that can integrate multiple\ncomputational modes within a single platform, including the following key aspects:\n(a) Versatility in computing modes. Our work can support spatial analog operations such as\nintegration and differentiation, binary logic operations including AND, OR and XOR gates, as\nwell as qubit-like operations like Hadamard, Pauli-X and CNOT gates. (b) Cross-domain\ncompatibility. The proposed platform allows serial, parallel and nested operations in different\ncomputational domains such as addition, subtraction and multiplication operations in spatial\n15\ndomain, parallel and transformable computing in frequency domain, as well as multilevel\nand combinational logics in bitwise domain. (c) Transformability. As discussed above, the\n\"transformable computing” serves as a bridge between spatial and frequency domains, which\ncan transform multi-frequency information into a single frequency. Such function provides\nplatforms for follow-up spatial serial/parallel operations.\nAs a result, the compatibility and flexibility suggest that our framework provides a hybrid and\nversatile computational platform instead a task-specific tool.\n(3) Differences in mechanism and engineering application\nThe physical mechanisms are different: The acoustic metasurface of Mousa et al. [R1] is\nrealized by acoustic waves in fluid media and its computational function is achieved by\ntransmission responses of acoustic Helmholtz cavity. Our present work is based on the\npropagation of elastic waves in solid media with periodic spring-mass structures, in which\nsignals are manipulated by tunning the mass, stiffness and damping coefficients. Different\nfrom acoustic systems, elastic wave metamaterials are more robust to large deformation and\nallow large-amplitude vibrations, which align with the working conditions of robotic systems.\nAs a result, this work shows the advantages of elastic wave metamaterials in robot designs\nand introduces the mechanical computing to robot crawling.\n(4) Realizations of robot crawling and embodied computing in our work\nA key attempt of our work is to achieve the metamaterial computing from static and\ntable-top demonstrations to dynamic crawling robots. Previous studies about metamaterial\nmathematical computing have reported many theoretical approaches from analog to logic\noperations (e.g., Science 343, 160–163, 2014 [R5]; Science 363, 1333–1338, 2019 [R6]).\nHowever, most of them are mainly based on fixed and stationary platforms which process\nsignals without interactions of physical motion. The integration of robotic locomotion with\nmathematical computing and especially with mechanical metamaterials, provides a new\nstrategy for embedding mechanic intelligence into robotics.\nDifferent from Mousa et al.’s paper [R1], this work not only focuses on the computing\nfunctions but also consider how to integrate them at the system level: (a) Firstly, this work\nestablishes a closer relation between computation and locomotion. Mousa et al.’s work [R1] is\nmore focused on isolated analog computing without actuation or behavioral output. However,\nour robot can crawl while performing mechanical computing and its locomotion states can be\nfully represented by the computational results. (b) Secondly, our approach is supported by\nanalytical derivations, numerical simulations, as well as experiments, which provides both\ntheoretical and engineering feasibilities. However, Mousa et al.’s [R1] paper is mainly\nverified by numerical simulations. As a result, we have further extended the theoretical\nwork to experimental validation and practical application.\nThe discussion above has illustrated the differences between our and previous works about\nparallel and multi-tasking processing. Next, we will focus on the following two concepts:\n“ghost circuits” and “embodied computing” [R2–R4].\nAs reported by Nanayakkara et al. [R2], the “ghost circuit” describes a form of distributed\ndynamic network which originates from dynamic coupling between a physical body and its\nenvironment. Its function is neither pre-programmed nor governed by a central controller. It is\ngenerated from the real-time interactions among the physical body, control system and\nneighboring environment. As a result, the “ghost circuit” is a system-environment interactive\nsystem in specific conditions, and it will emerge and dissipate when the interaction changes.\nAs mentioned by Nanayakkara et al. [R2], its function is analogous to a dead fish that can\nsway with water vortex, which forms an interactive system. When the vortex disappears,\nthe fish will no longer move [R2].\nTable R1. Comparisons with ghost circuits\nFeature Ghost Circuits [R2] Our Metamaterial Robot\nSystem Systematic and distributed Structural and integrated\narchitecture dynamic network physical body\nCoupled with specific environmental Active pre-design with programmability\nFormation\nconditions and multifunctionality\nCore idea System-environment coupling Computational-structural integration\nSystem Temporariness, low predictability, Sustainability, high predictability, high\ncharacteristics high environmental perception computational compatibility\nComputational Single mode: Analog operation in Hybrid mode: Analog, digital logic and\nmode coupled physical fields mechanical matrix operations\nRelationship with Dependent on specific environments Self-contained and adaptive to different\nenvironments (e.g., turbulence, spider webs) environments\nA dead fish swaying with flows A programmable computer (processing\nAnalogy function\n(leveraged by water vortex) tasks by structured hardware)\nInteractive systems: Integrated system:\nSystem level Focusing on how subsystems couple Integrating multiple subsystems into a\nand interact with each other unified and cooperative system\nIn our work, the robot is designed as a computational-structural integrated system that can\nintegrate multiple subfunctions into a unified physical body. Such a system employs a\nreprogrammable architecture, in which different sub cells cooperate to generate deterministic\nmathematical results. Before computations are completed, these results can be sustained and\n17\ndo not change by environments, exhibiting long relevant decoherence times. It does not mean\nthat our system is fully isolated, and we can also find the interaction between the robot and\nenvironment. One example is the ability of using differential operations to achieve target\ntracking in Fig. 6. The “ghost circuits” focuses on the dynamical analogue of coupled physical\nfields. But our system aims to provide a hybrid computational mode of analog, digital logic\nand matrix operations, as well as their transformations. Different from the environmental\ninteraction of “ghost circuits”, the concept of “embodied computing” [R3] in this work\nfocuses on the system itself serving as a signal processor to regulate its own locomotion.\nIn order to better compare the “ghost circuits” and our metamaterial robot, we have listed their\nmain features in Table R1 to further show the differences.\nC. Embodied morphological computing (arXiv, Biological Cybernetics 2011)\nSimilar to the “ghost circuits” mentioned above, the “morphological computing” [R4], as a\nkind of “embodied computing” [R3], has also served as a novel strategy for integrating\nembodied intelligence into mechanical systems. In Ref. [R4], Hauser et al. developed two\nmechanical filters which use “morphological computing” to simulate arbitrary nonlinear and\ntime-invariant dynamic systems. These systems are composed of two parts: (a) Physical body:\nA mechanical component includes mass-spring array/network, which is used to provide time\nintegration and memory, as well as nonlinear mapping functions. (b) Digital algorithm: A\nsubsequent linear regressor or digital neural network is used to process and iterate outputs,\nwhich requires additional digital training algorithm.\nWe quite agree with Reviewer 2 that the “morphological computing” is an interesting work\nwhich combines the mechanical system with digital computing. However, after carefully\nreading of Hauser et al.’s studies, we can find many differences between our work and their\npaper from operating mechanism, computational mode and analogy function to their\nengineering applications.\n(1) Differences in operating mechanisms\nIn Hauser et al.’s work, computations are achieved through the cooperation of mechanical\nmass-spring network and digital regressor/neural network. The mechanical network act as a\ndynamic feature extractor/filter, and the digital regressor processes and outputs signals after\ntraining. The function of the mechanical mass-spring network is similar to the convolutional\nlayers of neural networks which can extract information from complex inputs. In contrast, our\nrobot is a computational-structural integrated system, in which computations can be\nperformed by the robot itself without additional training and iteration. Its function is similar to\na movable mathematical calculator which can process tasks by the inner structured hardware.\n(2) Differences in computational modes\nIn Hauser et al.’s work, the inputs to the mechanical mass-spring network are time-domain\n18\nvibration signals. The mechanical mass-spring network functions as a time-domain analog\noperator, in which dynamic responses are described by the ordinary differential equations\n(i.e. vibration equations). Then, signals after the mass-spring network behave as implicit\nresults which require additional training and iteration to generate target results. In our work,\nthe inputs to the metamaterial robot are spatial-domain waveform signals f(y), which is\nobviously different from the time-domain vibration signals in vibration system. When\nwaves propagate through different processing components in our work, dynamic behaviors\ncan be described by the partial differential equations (i.e. wave equations). Our system can\nsupport three computational modes, including the spatial-domain analog computing, binary\nlogic operations, as well as qubit-like logic operations. After calculations, it can generate\nexplicit waveform output that directly drive and regulate the robot motion.\n(3) Differences in engineering applications\nAs described by Hauser et al., they have demonstrated how mechanical mass-spring systems\ncan be employed to emulate arbitrary, time-invariant and nonlinear filters with fading memory.\nSuch system has theoretically verified the ability of morphological computing to simplify\ncalculating tasks. Their design is more suitable to applications requiring a fixed-parameter\nprocessor for dynamic control, in which morphological computing can effectively reduce the\ncomputational complexity. As a result, their paper provides an effective method to design\nspecialized and dynamic feature extractors.\nTable R2. Comparisons with morphological computing\nFeature Morphological Computing [R4] Our Metamaterial Robot\nSystem\nDynamic feature extractor/filter Computational-structural integrated robot\narchitecture\nComputational Hybrid mode: Spatial-domain analog,\nTime-domain analog operations\nmode binary logic and qubit-like operations\nComputational Implicit results that require training Explicit results that can directly drive and\nresult and iterations modulate locomotion\nSolving ordinary differential Solving partial differential equations\nMechanical model\nequations (vibration equation, ODE) (wave equation, PDE)\nSystem Random and unchangeable: Deterministic and reprogrammable:\nparameters mass/stiffness is unchangeable mass/stiffness varies as function changes\nSystem Partially complete: Requiring Complete: the computation and robotic\ncompleteness subsequent digital regressors locomotion are integrated\nNeural network (feature extracting, Mathematical calculator (processing\nAnalogy function\ntemporal relevance and nonlinearity) tasks by structured hardware)\nmathematical functions: including spatial analog operations (differential/integral), binary\nlogics (AND, OR, XOR), as well as mechanical qubit-like operations (Hadamard, Pauli X,\nCNOT gates). Such a system can support hybrid computational modes with reprogrammable\nfunctions, which is more suitable to create a versatile and multitasking robotic device.\nWe have listed their main features in Table R2 to further show these differences. We wish that\nthese statements and comparisons could address most of your concerns.\n2.2. One weakness with this work is that the robotic implementation is simple – this is fine as a\nproof of concept but really shows the inability to turn this technology into something useful.\nDespite claiming to present a universal architecture, it is unclear why this is an important problem\nand the justification for using their implementation is weak/underdeveloped.\n2.2 Robotic implementation and experimental verification\nThe authors also appreciate Reviewer 2 for you concerns about the validity and justification\nfor the robotic implementation. Following your valuable comments, we have improved the\nstatements about robotic implementation and added more explanations about the applications\nof our computational architecture. We wish that these statements and improvements could\nbetter clarify the significance of our work.\n(1) The “simple” proof of robotic implementation\nThe authors acknowledge that our demonstrated robotic model is simple in experimental\nverification. However, as explained in Comment 1, the main idea of this work is to design a\nsimple and effective framework that can perform diverse computing functions with robotic\nlocomotion. The prototype is used to verify whether the elastic wave metamaterials can be\neffectively integrated with robotic motion, instead of testing its functional complexity. During\nthe experiments, such a simple framework can show its advantage in visualizing outputs.\nIt can present a clear and explicit relation between the computational result and robotic\nlocomotion. As a result, the simple and direct experimental demonstration is a more robust\nmethod to verify the basic computing principles.\nIn addition, it is worth mentioning that our robot is a highly scalable platform that is not\nlimited to our present function. (a) For example, our robot can be applied to emulate the\nparallel computing which is achieved by acoustic Helmholtz cavity (Mousa et al. PNAS 2024,\n[R1]). Because their function is only verified by simulations without experiment, our robot\ncan extend their work to engineering applications with robotic crawling. (b) Similarly, both\nthe “ghost circuits” [R2] and morphological computing [R4] have verified that such\nsimple mass-spring frameworks can be used for complex computational behaviors.\n20\n(2) Significance and justification of universal architecture\nRecently, a lot of attention has been focused on the multifunctional computing which can\nintegrate multiple computational modes into a single platform [R7–R10]. These designs allow\nhardware to dynamically adapts the target function, which changes from specialized to general\ncomputing. Similar conclusion is highlighted by Mousa et al. [R1] as “while there is added\ncomplexity to enable the multiplexing, the extra cost, space and error rate of multiple\nsingle-task components has exponentially exceeded the complexity brought upon by\nmultitasking”.\nFor instance, a recent work by Zhao et al. (Nature 640, 931–940) [R7] employs modular\nchiral metamaterials to design multiple deformable devices with multimodal, multi-stable and\nreconfigurable functions. In their work, some simple prototypes of metamaterials are used to\ndemonstrate a variety of complex robotic and computing functions. Their applications range\nfrom robotic morphogens, thermal regulation, mechanical memory to information encryption\nand energy absorption. Similar strategies have also been reported by El Helou et al. [R9]\n(Nature 608, 699–703), Mei et al. [R8] (Nature Communications 12, 7234) and Hu et al. [R10]\n(Nature Communications 14, 6709), etc. In their papers, the universal architectures of\ncomputing metamaterials are introduced, which are extended to various programmable\nmechanical logics and verified by simple prototypes.\nIn our work, the mechanical spatial analog, binary logic and qubit-like operations are not\nisolated functions. They can cooperate and transform through serial, parallel, nested, as well\nas transformable computing. Such advantages have not been reported by conventional parallel\ncomputing and references about analog computing [R1, R5, R6]. In particular, the capability\nto integrate a wide range of computing functions into a single robotic system is also a key\nindicator for assessing its versatility and intelligence level. For example, based on the\nparallel addition/subtraction operations, the universal architecture can emulate digital logic\nsignals to achieve the binary logic AND, OR and XOR operations. Furthermore, our system\ncan also emulate the acoustic “parallel computing” (Mousa et al. PNAS 2024, [R1]) by the\nuniversal architecture. Based on the serial modular design, the “parallel computing” can be\nfurther extended to “transformable computing”, which is also a justification of the universal\narchitecture.\nFinally, we would like to further convey our main ideas. This work aims to provide a versatile\ncomputation framework which not limited to our present designs. Our robot implementation is\nused to help readers to better understand the integrated multiple functions embedded in a\nsingle system. In our future work, we wish to design more intelligent robotic devices with\nintegrated, adaptive and multitasking capabilities.\n[R1] Mousa, M., Nouh, M., 2024. Parallel mechanical computing: metamaterials that can\nmultitask. Proceedings of the National Academy of Sciences of the United States of America\n2024, e2407431121.\n[R2] Nanayakkara, T., Laschi, C., Mazzolai, B., Godage, I., Walker, I., Srivastava, M., Hirai,\nS., Duriez, C., Rojas, N., Sung, C., Paik, J., 2024. Handbook on soft robotics. Springer\nNature, Switzerland.\n[R3] Roy, N., Posner, I., Barfoot, T., Beaudoin, P., Bengio, Y., Bohg, J., Brock, O., Depatie,\nI., Fox, D., Koditschek, D., Lozano-Perez, T., Mansinghka, V., Pal, C., Richards, B., Sadigh,\nD., Schaal, S., Sukhatme, G., Therien, D., Toussaint, M., Van de Panne, M, 2021. From\nmachine learning to robotics: challenges and opportunities for embodied intelligence. ArXiv,\n2110.15245.\n[R4] Hauser, H., Ijspeert, A.J., Füchslin, R.M., Pfeifer, R., Maass, W., 2021. Towards a\ntheoretical foundation for morphological computation with compliant bodies. Biological\nCybernetics 105, 355–370.\n[R5] Silva, A., Monticone, F., Castaldi, G., Galdi, V., Alù, A., Engheta, N., 2014. Performing\nmathematical operations with metamaterials. Science 343, 160–163.\n[R6] Estakhri, N.M., Edwards, B., Engheta, N., 2019. Inverse-designed metastructures that solve\nequations. Science 363, 1333–1339.\n[R7] Zhao, T., Dang, X.X, Manos, K., Zang, S.X., Mandal, J., Chen, M.J., Paulino, G.H., 2025.\nModular chiral origami metamaterials. Nature 640, 931–940.\n[R8] Mei, T., Meng, Z.Q., Zhao, K.J., Chen, C.Q., 2021. A mechanical metamaterial with\nreprogrammable logical functions. Nature Communications 12, 7234.\n[R9] El Helou, C., Grossmann, B., Tabor, C.E., Buskohl, P.R., Harne, R.L., 2022. Mechanical\nintegrated circuit materials. Nature 608, 699–703.\n[R10] Hu, X.Y., Tan, T., Wang, B.L., Yan, Z.M., 2023. A reprogrammable mechanical\nmetamaterial with origami functional-group transformation and ring reconfiguration. Nature\nCommunications 14, 6709.\nComment 3: Does the work support the conclusions and claims, or is additional evidence\nneeded?\nThe work would benefit from a degree of ‘hand holding’ – e.g. for each experiment, what are you\nshowing, why are you showing it, etc.? The paper needs to be navigable.\nResponses:\nThe authors greatly appreciate your careful reading and valuable comments, which help us to\nimprove our work. We quite agree with Reviewer 2 that more explicit guidance of experiments\nwould help readers to better understand this work. Following your suggestion, we have added\nmore introduction and explanation to each experiment and carefully revised the statements in the\nwhole manuscript. We wish that these revised statements and improvements could better address\nmost of your concerns.\nRevisions:\nThe authors have improved descriptions about all experiments in our work, which includes the\nspatial analog computing in Fig. 2, transformable computing in Fig. 4, robotic crawling with\nembodied computing in Fig. 6, spatial serial and parallel operations in Fig. 7, multifunctional\nperformance during locomotion in Fig. 8, combinational analog computing in Fig. S2, binary\nlogic operations in Fig. S3, mechanical matrix (qubit-like) computing in Fig. S5, robotic crawling\nwith band gap properties in Fig. S6, crawling velocity for different exciting frequencies in Fig. S8,\nmulti-gait crawling and turning behaviors in Figs. S9, as well as effects of wave amplitudes and\nmass ratios on the robot crawling velocity in Fig. S10.\nPlease refers to the captions and sentences we marked in the Manuscript: the caption of Fig. 2 on\npage 7, the blue sentences for Fig. 4 in the first paragraph on page 12, the blue sentences for Fig. 6\nin the last paragraph on page 16, the blue sentences for Fig. 6 in the second paragraph on page 18,\nthe blue sentences for Fig. 7 in the first paragraph on page 20, the blue sentences for Fig. 8 in the\nlast paragraph on page 20.\nPlease also refers to the captions and sentences we modified in the Supplementary Information:\nthe caption of Fig. S2 on page 7, the caption of Fig. S3 on page 9, the caption of Fig. S5 on page\n12, the blue sentences for Fig. S8 in the last paragraph on page 19, as well as the blue sentences\nfor Figs. S9 and S10 on page 26.\nComment 4: Are there any flaws in the data analysis, interpretation and conclusions? Do these\nprohibit publication or require revision?\nResults seem sound.\nResponses:\nThe authors greatly appreciate Reviewer 2 for your thorough reading and positive feedback on our\npresent work.\nFollowing the revisions and improvements to previous comments, we have further clarified the\ncorrelations between the theoretical results and experimental validations in the revised manuscript.\nThe data analysis, interpretation and conclusions are rechecked to ensure that they are consistent\nand well supported by the evidence. We wish that the revised manuscript could better convey the\nmain features of our work!\nComment 5: Is the methodology sound? Does the work meet the expected standards in your field?\nThere are large numbers of equations without much guidance between them. The figures are\naverage quality. In general this paper was very hard to read and understand. There is a lot of\nquantum theory interspersed into large walls of text. It is therefore a huge effort, requiring a\nbackground in quantum, to see if the methodology is sound, and if the work is reproducible.\nResponses:\nWe sincerely appreciate Reviewer 2 for your insightful comments about the formula derivations\nand concept introductions. We quite agree with your valuable suggestions and carefully revise\neach of these points. Your feedback has helped us substantially improve our work and further\nclarify the logical structure of the manuscript. We wish that the revised manuscript could better\nconvey the main features of our work.\nRevisions:\nThe following illustrates the key changes we have modified in response to your feedback:\n(1) Firstly, we have deleted a large number of repeated and complex derivations which may\nmake readers confused. Some key steps and explanatory notes are retained.\n(a) Some derivations about spatial analog computing in previous Supplementary Note 2 have\nbeen deleted. Then, only computational principles of fundamental analog operations, serial\nand parallel operations are retained and integrated into the revised Supplementary Note 1.\n(b) The nonlinear dynamic equations and their related vectors and matrices in previous\nSupplementary Note 4 have been simplified. The remaining equations have been integrated\ninto Supplementary Notes 3 and 5.\n(c) Derivations of nonlinear harmonic band structures in Supplementary Note 3 and robot\ncrawling velocity in Supplementary Note 5 have been simplified and integrated with the\nnonlinear dynamic equations mentioned above.\nAfter the simplification, the equation number has been reduced from 55 to 30 and the chapter\nnumber of Supplementary Information has been reduced from 8 to 6. The numbers of figures\nand pages have been reduced from 12 to 10 and 34 to 29, respectively. The revised equations\nare integrated into the following three parts: (i) computational principles, (ii) harmonic\nband structures and (iii) robot crawling velocity. These equations have been collected into\nSupplementary Notes 1, 3 and 5 with their corresponding theoretical and experimental results\nin Supplementary Notes 2, 4 and 6. The revised equation number of each part is about 10\nwithin 3 pages to improve readability.\n25\n(2) We have also improved the expressions of theoretical derivations and discussions. More\nexplanations are introduced to adjacent equations and sections, as well as equations are\nprovided with clearer guidance and supplementary references.\n(a) Manuscript: Please refer to Eqs. (1) and (2) with the corresponding blue sentences on\npages 5 and 6; Eqs. (4) and (5) with the corresponding blue sentences on page 8; the blue\nsentences to explain Eq. (6) on page 10; as well as Eqs. (7)–(10) we added to explain the\nHadamard and CNOT operations on pages 14–16 in the revised Manuscript.\n(b) Supplementary Information: Please also refer to the blue sentences before Eqs. (S1) and\n(S3) on page 4; the blue sentences before Eq. (S5) on page 5 and before Eq. (S8) on page 6;\nthe blue sentences before Eqs. (S11), (S17) and (S21) on pages 15–17; the blue sentences\nbefore Eqs. (S23), (S24), (S26), (S27) and (S29) on pages 22 and 23 in the revised\nSupplementary Information.\n(3) Furthermore, we have also thoroughly modified the discussion about the mechanical\nqubit-like computing and added more explanations about the principles for achieving\nmulti-bit matrix operations.\n(a) We have removed the abstract concepts related to quantum theory and introduced our ideas\nfrom the classical matrix operations. The principle for encoding spatial analog signals into the\nmechanical multi-bit states has been further discussed in the revised manuscript.\n(b) In order to avoid unfamiliar concepts of qubit-like states |0⟩ and |1⟩, we have provided\ntheir states {α, β} = α|0⟩ + β|1⟩ with classical wave representations α = A exp(iφ ) and β =\n1 1\nA exp(iφ ). The proposed qubit-like matrix operations are also added with equations to\n2 2\nexplain how the logical mappings perform.\n(c) Furthermore, we have also presented the comparisons between the mechanical qubit-like\nmatrix operations and acoustic phi-bit computing in Supplementary Note 2 - Part E in the\nrevised Supplementary Information.\nPlease refer to the blue sentences about mechanical matrix operations from pages 12 to 16 and\nEqs. (7)–(10) in the revised Manuscript. And please also refer to the blue sentences from\npages 13 and 14 in the Supplementary Information to see the comparisons between our\npresent work and conventional acoustic computing.\nComment 6: Concerns\n1. Unclear how this is novel compared to some of the recent literature that appears to do the same\nthing.\n2. Although I’ve been told not to comment overly on the language, use of language in this\nmanuscript is of a poor standard in the introduction, and can lead to confusion. I frequently found\nmyself becoming lost, or having to re-read a passage multiple times to try and decode what the\nauthors are trying to say. I highly recommend the authors use a professional editing service. Many\nof the sentences are not legal sentences, or incredibly unclear, e.g.\n(1) “The flexible designs from bioinspired materials to molecular crystalline structures endow\nmetamaterials with extraordinary mechanical properties, such as acoustic cloaking, negative\nrefractive and topological insulators [5–8].”\n(2) “In comparison, novel progress in metamaterials has developed wave-based analog computing\n[26–30], which enables continuous processing of complex information in spatial.”\n(3) Fig 1 the caption does not explain the figure, which is very dense and needs much clearer\nexplanation.\nResponses:\nThe authors great appreciate Reviewer 2 for your comprehensive reading and valuable comments\nabout the novelty and English expression. We quite agree with you about that the language used in\nthe introduction is proper. We have carefully checked our manuscript and substantially improved\nthe English expression in the revised version. Furthermore, we have added comparisons between\nour work and recent studies about mechanical computing. The caption of Fig. 1 is modified with\nclear explanation. We wish the revised manuscript could better convey the main features of our\nwork.\n1. Comparisons between recent literatures\nIn the previous responses to Comment 2, we have illustrated detailed comparisons between\nrecent literatures about mechanical computing and our present work. The similarities and\ndifferences between our and their works are listed with point-to-point responses.\nIn response to Mousa et al.’s work (PNAS 2024) [R1] about parallel computing, we have\ncompared our differences in working principles, engineering applications and core computing\nfunctions. The parallel computing function reported by Mousa et al. can be considered as a\nspecial case of our proposed “transformable computing”. The ability to integrate waves of\ndifferent frequencies into a single frequency and maintain their computational results has not\nbeen reported in their work. Different from conventional single-tasking devices with sole\n27\nresponse or multi-tasking devices with mixed responses, our system can present multiple\nindependent results according to the incident frequency without additional filtering devices.\nOn the other hand, this work not only focuses on the basic analog functions, but also considers\nbinary logic and matrix operations with robotic motion. Mousa et al.’s paper is more focused\non the isolated analog computing without actuation and behavioral output. However, our robot\ncan crawl while performing mechanical computing, and its locomotion states can be\nrepresented by the computational results. In addition, our experiments extend their purely\ntheoretical verification of acoustic metasurfaces to the practical applications in elastic wave\nrobotics.\nFurthermore, the authors also compare our work with studies of Nanayakkara et al. [R2], Roy\net al. [R3] and Hauser et al. [R4] about ghost circuits, embodied and morphing computing.\nWe have explained our differences in engineering applications, computational functions and\nthe core principles. The “ghost circuits” emphasize on the interaction between the individual\nand the environment as a whole system. However, our work mainly focuses on the individual\nitself which can present different computational functions with integrated subsystems. As for\nthe “morphing computing”, our core working principles are different. Their work is similar to\na mechanical neural network that requires additional digital filter to iterate outputs. In\nparticular, the inputs to “morphing computing” are time-domain analog signals. However,\nour work is similar to a movable mathematical calculator that can directly generate waveform\nsignal and control robotic locomotion. Our system can process the spatial-domain analog\nsignals as well as the encoded binary and qubit-like signals.\nDetails about the comparisons and related tables are illustrated in the previous responses to\nComment 2. We wish that these statements and explanations could address most of your\nconcerns.\n2. Grammar revision\nThe authors also appreciate your valuable comments about the English expression. We are\npleased to have the opportunity to comprehensively improve our work. Following your\nsuggestions, we have carefully checked our grammar in the introduction and improved the\nEnglish expressions. The previous statements about mechanical metamaterials and spatial\nanalog computing have been replaced by more precise descriptions. Furthermore, we have\nalso checked the whole manuscripts and revised the discussion. The caption of Fig. 1 is added\nwith more explanations.\nPlease refer to the introduction with the corresponding blue sentences on pages 1–4 in the\nrevised manuscript. Please also refer to the caption of Fig. 1 on pages 3 and 4 in the revised\nmanuscript for further illustrations.\nhelped us to comprehensively improve our manuscript!\nReferences:\n[R1] Mousa, M., Nouh, M., 2024. Parallel mechanical computing: metamaterials that can\nmultitask. Proceedings of the National Academy of Sciences of the United States of America\n2024, e2407431121.\n[R2] Nanayakkara, T., Laschi, C., Mazzolai, B., Godage, I., Walker, I., Srivastava, M., Hirai,\nS., Duriez, C., Rojas, N., Sung, C., Paik, J., 2024. Handbook on soft robotics. Springer\nNature, Switzerland.\n[R3] Roy, N., Posner, I., Barfoot, T., Beaudoin, P., Bengio, Y., Bohg, J., Brock, O., Depatie,\nI., Fox, D., Koditschek, D., Lozano-Perez, T., Mansinghka, V., Pal, C., Richards, B., Sadigh,\nD., Schaal, S., Sukhatme, G., Therien, D., Toussaint, M., Van de Panne, M, 2021. From\nmachine learning to robotics: challenges and opportunities for embodied intelligence. ArXiv,\n2110.15245.\n[R4] Hauser, H., Ijspeert, A.J., Füchslin, R.M., Pfeifer, R., Maass, W., 2021. Towards a\ntheoretical foundation for morphological computation with compliant bodies. Biological\nCybernetics 105, 355–370.\nReviewer 3\nGeneral comment: The manuscript presents a metamaterial robotic system that has three modes\nfor computations. The first mode is conformal change in the metamaterial structure, a second\nmode is in the vibrational spectrum supported in the robotic system. Finally, the system can use\nlocomotion as a means to accomplish computation. The authors claim that they have demonstrated\nbinary and quantum logic operation using the robotic system. Further, they claim that the system\nis capable of both serial and parallel operation. Finally, the authors predict that the system will\nperform robustly in harsh environments. The current manuscript has a number of drawbacks\nincluding the lack of quantitative measures for the fidelity of the computations. Also, the concept\nof quantum computing is not well represented by this system as it lacks both scalability and\nflexibility with respect to input states. The length of supplementary material, both written and\nvideo, is an impediment to the assessment of this submission. I recommend resubmission of this\nmanuscript after major revision.\nResponses:\nWe sincerely appreciate Reviewer 3 for your thorough review and thoughtful comments on our\npresent work. We have carefully considered your comments and suggestions and subsequently\nrevised our manuscript. For instance:\n1. Quantitative measures: we have improved the quantitative indicators for both theoretical\ncalculations and experimental measurements. Each of the computational results is marked\nwith computational uncertainty which is represented by the Pearson correlation coefficient r.\nDetails are provided in the following responses to Comments 1 and 4. Please refer to Figs.\n2, 4, 5 and 7 as well as Figs. S2, S3 and S5 in the revised version.\n2. Mechanical matrix operations: we have removed the concepts related to quantum theory\nand introduced our ideas from the classical matrix operations. The principle for encoding\nspatial analog signals into mechanical multi-bit states has been further discussed in the\nrevised manuscript. Details are provided in the following responses to Comments 2 and 4.\nPlease refer to the “mechanical matrix operations” section modified in the revised\nmanuscript.\n3. Simplified contents: we have reduced complex derivations and movies which would make\nreaders confused. More explanations are introduced to adjacent sections and equations with\nclear guidance and references. Details are provided in the following responses to Comment\n3. Please refer to Movies 1–9 and Supplementary Notes 1, 3 and 5 modified in the revised\nSupplementary Information.\nadditional changes are presented in the revised Manuscript and Supplementary Information\nand please refer to the blue sentences. We wish that these revised statements and improvements\ncould address most of your concerns. Once again, we truly appreciate your valuable feedback and\nthe time you have dedicated to our work.\nComment 1: The concept of using multiple aspects of the metamaterial robotic system for\ncomputational means is of interest and would be advantageous for exploration in harsh\nenvironments. It is, however, unclear how the authors would propose to accomplish inputs from\nthe environment to adapt an instruction set for flexible operation. Further, attention must be paid\nto the accuracy of the computed result, the authors fail to report any measure of the uncertainty\nrelated to any of the computations performed. It is clear from the figures, e.g., Figure 2, that the\naccuracy of the computation is limited by the nature of the robotic system. A measure of\nuncertainty is required for each of the computations performed.\nResponses:\nThank you very much for your positive feedback and recognition on our metamaterial robot.\nWe sincerely appreciate Reviewer 3 for your valuable comments on the environmental input and\ncomputational accuracy.\nSignal generator Experiments & Electrical modules\nExperiment setup\nCameras\nPower Oscilloscope\nRobot\nPC e d To robot\nu\nRobotic module\ntitta\nMonitor device\nRegulator & Amplifier arrays to\nb\no\n0 0 1 1 0 0 0 0 R\n0 0 0 0 0 1 1 0\n0 0 0 1 1 0 0 0 a\n1 1 0 0 0 0 0 0 re\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n0\n1\n1\n0\n1\nm\na C\n(a) Bit encoding Calculating Crawling (b) Signal generator Regulator & Amplifier arrays\nFig. R2. (a) Experimental setups and working principle for computational crawling and (b) electrical modules\nand monitor devices during experiments.\nFirstly, we would like to explain the principle for flexible operations according to the environment.\nAs shown in Fig. R2, our robotic system is composed of the following four parts: (a) a computer\nwith signal generator to produce harmonic inputs; (b) the regulator and amplifier arrays to adjust\nand amplify signals; (c) the robot with an excitation array on its head; (d) the monitor device to\nrecord the robot attitudes and transmit them to the computer. These devices form a closed-loop\nsystem, in which the environment inputs are represented by the recorded robot attitudes. The\nmonitor device in our present work is a set of cameras on the top and side of the experiment\nplatform, which is used to capture both the real-time position and crawling direction. It is worth\nmentioning that in addition to our present design, the environmental inputs can be detected by\nmore integrated strategies.\nits own weight. A micro camera or sonar device can be installed on the robot, which can provide\nposition information with less interference. The micro camera and sonar devices are both well-\nestablished systems which can be applied to robot vision and path planning fields. Using\nintegrated and onboard monitor devices, the robot can generate instructions itself for subsequent\ncomputing based on environments. Since our attention is mainly focused on the theoretical design\nand robot validation, more integrated designs for monitor systems are not discussed in detail. We\nwill focus on more integrated and adaptive robotic devices in our future work.\nRevisions:\nSecondly, following your scientific suggestion, we have added quantitative indicators for both\ntheoretical calculations and experimental measurements to show their computational accuracy.\nEach result is associated with computational uncertainty which is represented by Pearson\ncorrelation coefficient r. We know that Pearson correlation coefficient r is a statistical measure\nused to quantify the correlation between two data sets {x , x , …, x } and {y , y , …, y } as\n1 2 n 1 2 n\n33\nr =\nn\nn\nj=\n1\nx 2j\nn\n−\nn\nj=\n1\nn\n (\nj= \nx\n1\nj\nx\ny\nj\nj\n)\n−\n2 \nn\nj=\n1\nn\nx\nj\nn\nj=\nn\nj=\n1\ny\n1\ny\n2j −\nj\n(\nn\nj=\n1\ny\nj\n) 2\n, (R.5)\nwhere r ∈ [–1, 1]; n = 2, 3, … is the sample size.\nIn this work, the correlation coefficient r is employed to indicate the degree of linear agreement,\nwhich serves as a representation for the calculating uncertainty. An r closer to 1 denotes a stronger\nagreement. The computational accuracy with r ∈ [0.8, 0.9] indicates that two curves exhibit\nsimilar trends and those with r > 0.9 exhibit almost the same trend. The following discussion\nillustrates the specific data in terms of the computational accuracy:\n1. Manuscript:\n(1) In Fig. 2, the correlation coefficients corresponding to the theoretical simulations of focusing,\nreflection, integral, differential, convolution and solving equation are calculated as r =\n0.8182, 0.9154, 0.9445, 0.9566, 0.9614 and 0.9665, respectively. The correlation coefficients\nfor experimental measures are derived as r = 0.8045, 0.9605, 0.9601, 0.9259, 0.9083 and\n0.9365, respectively. Most of the theoretical and experimental results agree well with the\ntheory.\n(2) In Fig. 4, the correlation coefficients corresponding to the 2nd differential of fundamental\nwaves are calculated as r = 0.9045 for simulations and r = 0.8957 for experiments. The\ncorrelation coefficients corresponding to the 1st differential of subharmonic waves are\nderived as r = 0.9656 for simulations and r = 0.9028 for experiments. Both simulations and\nexperiments show reasonable accuracy.\n(3) In Fig. 5, the correlation coefficients corresponding to Hadamard operation are derived as r\n= 0.9187, 0.9717 and 0.9281 for |0〉, |+〉 and |1〉 inputs, respectively. The correlation\ncoefficient corresponding to the arbitrary operation in Fig. 5(h) is calculated as r = 0.9607.\nThe correlation coefficients corresponding to CNOT operations in Fig. 5(i) are derived as r =\n0.9409 and 0.9192 for |00〉 and |01〉 inputs, respectively. These results show good agreements\nwith theoretical ones.\n(4) In Fig. 7, the correlation coefficients for the integral, differential, solving equation I,\nsolving equation II, addition and subtraction operations are calculated as r = 0.9880,\n0.9363, 0.9297, 0.9714, 0.9028 and 0.9452, respectively. Furthermore, the robot crawling\nbehaviors during experiments also agree with the theoretical results.\n2. Supplementary Information\n(5) In Fig. S2, the correlation coefficients corresponding to theoretical simulations of addition I,\naddition II, subtraction I, subtraction II, serial I and serial II operations are calculated as r\n= 0.9486, 0.9044, 0.9429, 0.9250, 0.9293 and 0.8996, respectively. In addition, the correlation\ncoefficients for experiment measurements are calculated as r = 0.9497, 0.8809, 0.8921,\n0.9033, 0.9127 and 0.8869, respectively. Both simulations and experiments show reasonable\naccuracy.\n(6) In Fig. S3, the correlation coefficients corresponding to binary AND operations are\ncalculated as r = 0.8575, 0.8712 and 0.9940 for (1, 0), (0, 1) and (1, 1) inputs. The correlation\ncoefficients corresponding to binary OR operations are calculated as r = 0.9437, 0.9362 and\n0.8974 for (1, 0), (0, 1) and (1, 1) inputs. The correlation coefficients corresponding to binary\nXOR operations are calculated as r = 0.9102, 0.9348 and 0.9063 for (1, 0), (0, 1) and (1, 1)\ninputs, respectively.\n(7) In Fig. S5, the correlation coefficients corresponding to Pauli X operations are calculated as\nr = 0.9663, 0.9698 and 0.9393 for |0〉, |1〉 and |+〉 inputs. The correlation coefficients\ncorresponding to Pauli iY operations are calculated as r = 0.9296, 0.9258 and 0.9305 for (1,\n0), (0, 1) and (1, 1) inputs. The correlation coefficients corresponding to Hadamard\noperations are calculated as r = 0.9242, 0.9761 and 0.9765 for (1, 0), (0, 1) and (1, 1) inputs,\nrespectively.\nPlease refer to Figs. 2, 4, 5 and 7 in the revised Manuscript, as well as Figs. S2, S3 and S5 in\nrevised Supplementary Information for further illustrations.\nComment 2: While acoustic systems analogous to quantum computers have been reported in the\nliterature and cited by the authors, some requirements for quantum computation are not met by the\ncurrent system. In particular, following DiVincenzo’s criteria, the system fails to be scalable.\nRecall that quantum advantage is usually considered to require 50 qubits. This robotic system does\nnot scale to such a large number, or at least such scaling is not clear to this reviewer. Secondly, for\ngeneral quantum computing the initial state must be able to accommodate a wide range of states\navailable on the Bloch sphere. Again, it is not clear that the current system will support such\ndiversity in specifying the initial state.\nResponses:\nThank you very much for your careful reading and insightful comments about the mechanical\nqubit-like computing. The following responses aim to discuss the scalability of our work and the\nability to cover the Bloch sphere.\n1. Scalability in mechanical computing\nIn our previous version, we have reported the single-qubit operations and their combinations to\nmore complex multi-qubit logics. For example, based on the serial and parallel strategies,\nthe\nmulti-qubit operations are illustrated in Figs. R3(d) and R3(e) to verify its theoretical scalability.\nHowever, during the experiments, our metamaterial robot is designed with 9 and 8 unit cells along\nthe x and y directions, respectively. Such a system is mainly used to illustrate the feasibility of\nqubit-like computing instead of showing its scalability. Although it can be scaled with the spatial\nserial and parallel strategies, the large size for a 50-qubit system can lead to giant spatial\noccupation and energy consumption.\nFollowing your valuable suggestion, we have removed the concepts related to quantum computing\nand introduced our ideas from classical matrix operations. According to DiVincenzo’s criteria,\nquantum computing systems require at least 50 qubits to achieve the computing advantage.\nThe scalable dimensions for classical matrix operations are relatively flexible without\nrigorous requirements. Previous single-qubit operations, such as Hadamard, Pauli X and iY\ngates, can be considered as 2×2 mechanical matrix operations. The two-qubit CNOT gate\nachieved by several 1:2 wave splitters in Fig. R3(e) can be considered as 4×4 matrix operations\nwith 4 computational channels. In addition, the combinational serial/parallel operations of several\nHadamard gates in Fig. R3(d) can be considered as the 4×6 matrix operations. Furthermore, the\nprinciple of encoding spatial analog signals to mechanical matrix operations is further discussed in\nthe revised manuscript.\n35\n(a) Principle for Hadamard operation (b) Mechanical computational results (c) Robotic attitude\nd\nH(y) r a\nn o ita\nre\np o\nx irta\nm\ny tilib\na b o rP –\n2\n0 2\n| 1 〉\n|0〉\n|0〉\n|0〉\n|1〉 F\nF\nL\n1\n2\nF\nF\n2\n1\n+\n–\nn o ita\nre\np o\nd ra\nm\n|\n|\n+\n0\n〉 |\n|\n0\n+\n〉\n〉\n|0〉 |+〉\ntfe\nL\nw r o F\ne lp m iS e re h p s h c o ȁ0〉 ξ + 2 |1〉 ȁ0 ξ 〉– 2 |1〉 F 2 – a d a H |1〉 |–〉 | +〉 |0〉 d r a w\nlB Barrier r o\n|1〉 |1〉 |–〉 F\n(d) Arbitrary combinational matrix operation (e) Combinational CNOT matrix operation\nn o ita\nre\nSer\nH\nia\na\nl\nd\na\na\nn\nm\nd\na r\np\nd\na rallel operations\n2\nv v (\n)\n/\nxam 1\n0 1 2 C\nT\nh\nh\n3 a\ne\nn\no\nn 4\nr\ne\ny\nl 5 6\nr e ttilp\ns e v a W\ns i1\ns\no2\n≈\ns\ns s\no\no\no\n1\n2\ns\no2\n> s\no1\nL 2\ns\no2\n>> s\no1\nL\n3\nla\np o x irta\nm\n2\nv v (\n)\n/\nxam 1\nCh\nZ\na\n-\nn\nb\nn\na\ne\ns\nl\ni s\n1\n: C\nS\n0\np\n=\nl i\nC\ntte\nh\nr\n. 2\nIn\nC\np\n1\nu\n=\nt\nCh. 3 T0 = Chs. 4+5 T1 = Chs. 5–4\nn\no\nita Level 1 Level 2\n1 2\nC h\n3\na n n\n4\ne l\n5 6\nChannel 2 s\n|\n|\n0\n0\n0\n1\n〉\n〉\nn\nib\nm Channel 1 Channel 3\no Max\nC Channel 2\n|\nI\n1\n〉\nn\np\n,\nu\nt\ns Ch a nnel 3 Channel 4 y tic\nCh annel 4 Channel 5 Output o le\nCh annel 5 Outputs Channel 6 |00〉 V\nChannel 6 |–〉, |1〉, |+〉 s |01〉 Min\nFig. R3. Mechanical realizations of reprogrammable matrix operations. (a)–(c) Principle for mechanical\nHadamard operation and the computational results of |0〉, |+〉 and |1〉 inputs. (d) Combinational matrix operation\nwith several Hadamard operators and (e) the combinational four-channel CNOT operation. The correlation\ncoefficients are calculated as r = 0.9187, 0.9717 and 0.9281 for |0〉, |+〉 and |1〉 inputs in (b), r = 0.9607 in (d),\nas well as r = 0.9409 and 0.9192 for |00〉 and |01〉 inputs in (e).\nFor more complex matrix operations, we can employ a small-scale experimental model by\ndecreasing the unit size. During our experiments, the side length of hexagonal blocks a and the\ninitial length of springs b are 1.5 and 3.2 cm, which will be obviously large as the unit number\nincreases. As reported by Zhang et al. [R1] and He et al. [R2], they have designed some\nsuper-compact quantum logic gates in which the footprints can be minimized to 20~200 μm2.\nAlthough our purpose is not to design such a small-scale robot, we can apply similar ideas to\nminimize the robot size. In our future applications, the robot framework will be directly fabricated\nby small-scale 3D printing. It can reduce the unit size to a millimeter level and significantly\n36\nenhance the scalability of matrix operations. It is worth mentioning that this work is focused on a\ngeneral strategy from elastic wave metamaterials to crawling robots with embodied computing\nfunctions. Such strategy is not limited to our present design and can be extended to more versatile\nand integrated designs in future applications.\n[R1] Zhang, M., Feng, L.T., Li, M., Chen, Y., Zhang, L., He, D.Y., Guo, G.P., Guo, G.C., Ren,\nX.F., Dai, D.X., 2021. Supercompact photonic quantum logic gate on a silicon chip. Physical\nReview Letters 126, 130501.\n[R2] He, L., Liu, D.N., Gao, J.X., Zhang, W.X., Zhang, H.Z., Feng, X., Huang, Y.D., Cui, K.Y.,\nLiu, F., Zhang, W., Zhang, X.D., 2023. Super-compact universal quantum logic gates with\ninverse-designed elements. Science Advances 9, adg6685.\n2. Completeness of initial states\nThe authors are also very grateful for your kind suggestions to discuss whether the initial state\ncould accommodate a wide range of states on the Bloch sphere. It would help readers to have a\nmore thorough understanding of the matrix computing. We have followed your suggestion and\nsummarized as follows:\nMechanical matrix operator Evolution from analog to matrix computing\nle n t n o u a w p h n c - i f f 2 1 g g 2 1 le n n o a t t u u h o p c - (b) e g n a h c r e tn i e v a W o y f f 2 1 x ℱ 1 ℱ 2 Y (c) n o itc a r ffid e v a W o y f x ℱ 1 H(y ℱ ) 2 δ H (d)e c n e r e fr e e t l n b i tils- o y f f 2 1 x ℱ 1 H (y ℱ ) 2 g g 2 1\nT w u\ny (a) T o D\no x Lens+ Lens–\n|0〉 = {1, 0}T, |1〉 = {0, 1}T |1〉 → |0〉 |0〉 → |1〉 |+〉 → |+〉 –|+〉 → –|+〉 |1〉 → |+〉 |0〉 → |+〉\nBasis states\nFig. R4. Principle for achieving mechanical matrix operations. (a) Robot configuration for achieving\nmechanical matrix operations. (b)–(d) Evolution from analog to matrix computing: the matrix operation in\n(d) is achieved by the double-slit interference effect with combinations of (c) analog wave interchange and\n(d) wave diffraction.\n(1) Definition of mechanical bits\nAs shown in Fig. R4(a), the metamaterial employs a similar multilayer ℱ -H-ℱ layout to the\n1 2\nspatial analog computing. A mass barrier array m >>m is placed along the axial centerline y =\nb 0\nY/2 which divides the input f into a signal pair {f , f }T. For harmonic signals with an identical\n1 2\nfrequency, the signal pair {f , f }T defines a 2D complex state vector {A *exp(iφ ), A *exp(iφ )}T\n1 2 1 1 2 2\nwith A *2 + A *2 = 1. This vector is characterized by the normalized amplitudes and phases A *, A *,\n1 2 1 2\nφ and φ of {f , f }T. We consider the state vector as a linear combination {α, β}T = α{1, 0}T +\n1 2 1 2\n37\nβ{0, 1}T, in which the basis states {1, 0}T and {0, 1}T are defined by the single-mode excitations\n(A * = 1, A * = 0) and (A * = 0, A * = 1). In order to better describe the two-channel signals, we\n1 2 1 2\nadopt the symbolic notation from qubit computing, which labels the classical basis states {1, 0}T\nand {0, 1}T as the mechanical qubit-like states |0⟩ and |1⟩. Then, a general input state is written as\n{α, β}T = α|0⟩ + β|1⟩ with α = A *exp(iφ ) and β = A *exp(iφ ). It is worth mentioning that this\n1 1 2 2\nnotation is only a purely mathematical convention without quantum implication. Similar\ndiscussion is added in the manuscript and the responses to Comment 4 of Reviewer 1.\n(2) Ability to cover the Bloch sphere\nIn the quantum computing theory, an arbitrary qubit state can be expressed as\n38\nc o s ( / 2 ) 0 e i s i n ( / 2 ) 1     = + , (R.6)\nwhere θ and ϕ are Bloch sphere coordinates covering the entire sphere. As mentioned above,\nour mechanical states are characterized by the normalized wave amplitudes {A *, A *}T and phases {φ ,\n1 2 1\nφ }T of the traveling wave pair. Considering the complex amplitudes A *exp(iφ ) and A *exp(iφ )\n2 1 1 2 2\nof the wave pair, we can obtain the equivalent qubit-like state as\nA *1 e i 1 0 A\n2\n* e i 2 1    = + , (R.7)\nwhere A * and A * satisfy the relation A *2 + A *2 = 1. Setting φ as the fiducial phase, the phase\n1 2 1 2 1\nrelation is actually determined by the relative phase Δφ= φ – φ . We neglect the overall phase\n2 1\nfactor exp(iφ ) and the qubit-like state can be written as |ψ⟩ = A *|0⟩ + A *exp(iΔφ)|1⟩. Assume that\n1 1 2\nα = A * and β = A *exp(iΔφ), the qubit-like state can be expressed as\n1 2\n0 1    = + , (R.8)\nwhere α and β satisfy |α|2 + |β|2 = 1 and are equivalent to the Bloch sphere representation α =\ncos(θ/2) and β = exp(iϕ)sin(θ/2). With the normalization A *2 + A *2 = 1, the initial state can be\n1 2\nfully characterized by the amplitude ratio A */A * and relative phase Δφ = φ − φ . Then, it can be\n1 2 2 1\nvisualized on a Bloch sphere by the following mapping relation:\n=2arccos ( A*) or 2arcsin ( A*) , 0,  , (R.9)\n1 2\n= −, 0, 2) . (R.10)\n2 1\nDuring our experiments, both wave amplitudes and phases can be adjusted by the arbitrary\nwaveform generator. By tuning the wave amplitudes to satisfy A *2 + A *2 = 1 and relative phase\n1 2\nΔφ= φ – φ , the initial state can accommodate a wide range of states on the Bloch sphere.\n2 1\nReviewer 3\nComment 3: With regard to the supplementary material, it does not appear to this reviewer that\nthe video material adds to the content of the manuscript. While the written supplementary material\ndoes support the manuscript, its length could easily be reduced without losing the vital supporting\ninformation the manuscript requires.\nResponses:\nThe authors greatly appreciate Reviewer 3 for your careful reading and insightful comments on\nour work. We quite agree with your comments about that the length of supplementary materials\naffects the readability of our work. We have followed your suggestions and substantially revised\nour manuscript. The following discussion illustrates the detailed changes we modified in response\nto your suggestions. We wish that the revised version could better convey the main features of this\npresent work.\nRevisions:\n1. Firstly, we have removed several movies in which the contents are repeated in the text\nmaterials. The remaining movies illustrate the key functions of our metamaterial robot\nand its experimental performances.\n(a) The principle of robotic crawling in previous Movie S2 is deleted, which has been\nexplained in Fig. 1 and Supplementary Note 1, Part A.\n(b) The principle of spatial serial operations in previous Movie S4 is deleted, which has been\nexplained in Fig. 7 and Supplementary Note 1, Part C.\n(c) The principle of spatial parallel operations in previous Movie S6 is deleted, which has been\nexplained in Fig. 7 and Supplementary Note 1, Part D.\n(d) The principle of mechanical matrix operations in previous Movie S8 is deleted, which has been\nexplained in Fig. 4 and Supplementary Note 2, Part D.\n(e) The experiment of Pauli X operation in previous Movie S10 is deleted, which has been\ndemonstrated in Fig. S5 and Supplementary Note 2, Part D.\n(f) The experiments of amplitude dependency and parameter regulating effects in previous\nMovie S13 are deleted, which has been demonstrated in Fig. S9 and Supplementary Note 6.\nAfter the simplification, the movie number has been reduced from 15 to 9. The remaining\nmovies mainly illustrate the following contents, i.e., (i) Movie S1: fundamental principle of\nreprogrammable mechanical computing; (ii) Movies S2–S5: experiments of reprogrammable\nanalog computing, spatial serial computing, spatial parallel computing, as well as mechanical\nmatrix operations; (iii) Movies S6 and S7: band gap property and velocity response during\nrobot crawling; as well as (iv) Movies S8 and S9: multi-gait locomotion and multifunctional\napplication.\n2. We have also deleted several complex derivations and repeated figures that would make\nreaders confused. Some key steps and explanatory notes are retained in the revised\nSupplementary Information.\n(a) Some derivations of spatial analog computing in Supplementary Note 2 have been deleted.\nThen, only fundamental principle of reprogrammable analog computing, serial and parallel\noperations are retained and integrated into the revised Supplementary Note 1.\n(b) The nonlinear dynamic equations and their related complex vectors and matrices in the\nprevious Supplementary Note 4 have been simplified. The remaining equations have been\nintegrated into the revised Supplementary Notes 3 and 5.\n(c) Derivations of nonlinear harmonic band structures in Supplementary Note 3 and robot\ncrawling velocity in Supplementary Note 5 have been simplified and integrated with the\nnonlinear dynamic equations mentioned above.\n(d) Repeated figures and discussion about eigenstate diagram and robot multifunctional\nillustrations in previous Supplementary Figures 6 and 11 have been removed. An additional\nfigure has been added to show the parallel computing in the frequency domain in the revised\nSupplementary Figure 4.\nAfter the simplification, the equation number has been reduced from 55 to 30 and the chapter\nnumber of Supplementary Information has been reduced from 8 to 6. The numbers of figures\nand pages have been reduced from 12 to 10 and 34 to 29, respectively. The revised equations\nare integrated into three parts, i.e., computational principles, harmonic band structures\nand robot crawling velocity. These equations are collected into Supplementary Notes 1, 3\nand 5 with their corresponding discussion in Supplementary Notes 2, 4 and 6. The revised\nequation number of each part is about 10 within 3 pages to improve readability. Similar\nrevisions are added in the responses to Comment 5 of Reviewer 2.\n3. Furthermore, we have improved the expressions of theoretical derivations and their\ndiscussion. More explanations are introduced to adjacent sections and equations with\nclearer guidance and references.\n(a) Manuscript: Please refer to Eqs. (1) and (2) with the corresponding blue sentences on\npages 5 and 6; Eqs. (4) and (5) with the corresponding blue sentences on page 8; the blue\nsentences to explain Eq. (6) on page 10; as well as Eqs. (7)–(10) we added to explain the\nHadamard and CNOT operations on pages 14–16 in the revised Manuscript. Please also refer\nto other blue sentences we modified in the whole manuscript.\n40\n(b) Supplementary Information: Please also refer to the blue sentences before Eqs. (S1) and\n(S3) on page 4; the blue sentences after Eq. (S5) on page 5 and before Eq. (S8) on page 6; the\nblue sentences before Eqs. (S11), (S17) and (S21) on pages 14–16; the blue sentences before\nEqs. (S23), (S24), (S26), (S27) and (S29) on pages 22–23 in the revised Supplementary\nInformation.\nIn addition, in order to improve readability, Supplementary Figures S1–S9 and Table S1 are\nembedded in their corresponding text materials of Supplementary Notes S1–S6.\nIn summary, we have deleted a large number of complex derivations and figures which would\naffect the readability of this work. More explanations are introduced to adjacent sections and\nequations with clearer guidance and references. In addition, we have removed several movies in\nwhich the contents are repeated in the text materials. Please refer to the revised Supplementary\nFigures S1–S10, Notes S1–S6 and Movies S1–S9 for further illustrations. We wish that these\nimprovements and modifications could address most of your concerns.\nComment 4: I strongly recommend that the authors include quantitative measure of uncertainty\nrelated to each of the computations performed in the manuscript. Further, references to quantum\ncomputation should be removed from the manuscript. Operations that have performed on so-called\n‘few qubit states’ can be reported as matrix manipulations rather than quantum logic operations.\nI, therefore, recommend that the article be resubmitted after these major revisions.\nResponses:\n1. Quantitative measure of uncertainty\nThe authors sincerely appreciate your thorough review and constructive suggestions which help us\nbetter improve our present work. Following your valuable suggestions, we have carefully checked\nthe manuscript and added quantitative indicators for both theoretical and experimental results.\nEach computational result is marked with computational uncertainty, which is represented by the\nPearson correlation coefficient r. The Pearson correlation coefficient r is a statistical measure used\nto quantify the degree of linear correlation and the calculating uncertainty. Detailed revisions of\ncomputational accuracy have been illustrated in the responses to Comment 1. The relevant data\nfor each simulation (Sim.) and experiment (Exp.) results are relisted in the following tables:\nTable R3. Quantitative measure of uncertainty in Manuscript\nFigure Computing function Correlation coefficient r\n(i) focusing (i) 0.8182 (Sim.), 0.8045 (Exp.)\n(ii) reflection (ii) 0.9154 (Sim.), 0.9605 (Exp.)\nFig. 2 (iii) integral (iii) 0.9445 (Sim.), 0.9601 (Exp.)\n(Sim. & Exp.) (iv) differential (iv) 0.9566 (Sim.), 0.9259 (Exp.)\n(v) convolution (v) 0.9614 (Sim.), 0.9083 (Exp.)\n(vi) solving equation (vi) 0.9665 (Sim.), 0.9365 (Exp.)\nFig. 4 (i) 2nd differential of fundamentals (i) 0.9045 (Sim.), 0.8957 (Exp.)\n(Sim. & Exp.) (ii) 1st differential of subharmonics (ii) 0.9656 (Sim.), 0.9028 (Exp.)\n(i) Hadamard operations (i) 0.9187, 0.9717 and 0.9281 (Sim.)\nFig. 5 (Sim.) (ii) arbitrary operations (ii) 0.9607 (Sim.)\n(iii) CNOT operations (iii) 0.9409 and 0.9192 (Sim.)\n(i) integral, (ii) differential, (iii) solving (i) 0.9880 (Sim.), (ii) 0.9363 (Sim.),\nFig. 7 (Sim.) equation I, (iv) solving equation II, (v) (iii) 0.9297 (Sim.), (iv) 0.9714 (Sim.),\naddition and (vi) subtraction (v) 0.9028 (Sim.), (vi) 0.9452 (Sim.)\nFigure Computing function Correlation coefficient r\n(i) addition I (i) 0.9486 (Sim.), 0.9497 (Exp.)\n(ii) addition II (ii) 0.9044 (Sim.), 0.8809 (Exp.)\nFig. S2 (iii) subtraction I (iii) 0.9429 (Sim.), 0.8921 (Exp.)\n(Sim. & Exp.) (iv) subtraction II (iv) 0.9250 (Sim.), 0.9033 (Exp.)\n(v) serial I (v) 0.9293 (Sim.), 0.9127 (Exp.)\n(vi) serial II (vi) 0.8996 (Sim.), 0.8869 (Exp.)\n(i) binary AND operations (i) 0.8575, 0.8712 and 0.9940 (Sim.)\nFig. S3 (Sim.) (ii) binary OR operations (ii) 0.9437, 0.9362 and 0.8974 (Sim.)\n(iii) binary XOR operations (iii) 0.9102, 0.9348 and 0.9063 (Sim.)\n(i) Pauli X operations (i) 0.9663, 0.9698 and 0.9393 (Sim.)\nFig. S5 (Sim.) (ii) Pauli iY operations (ii) 0.9296, 0.9258 and 0.9305 (Sim.)\n(iii) Hadamard operations (iii) 0.9242, 0.9761 and 0.9765 (Sim.)\nPlease refer to Figs. 2, 4, 5 and 7 in the revised Manuscript, as well as Figs. S2, S3 and S5 in\nrevised Supplementary Information for further illustrations.\n2. Mechanical matrix operations\nFurthermore, following your constructive suggestions, we have removed the concept related to\nquantum computing and introduced our ideas from classical matrix operations. More explanations\nabout the principles of mechanical matrix operations have been added. Please refer to the modified\n“mechanical matrix operations” section in the revised manuscript.\n(1) The concept related to quantum computing has been removed and the previous qubit\ncomputing has been revised as classical matrix operation. The principle for encoding spatial\nanalog signals into mechanical multi-bit states has been further discussed in the revised\nmanuscript.\n(2) In order to avoid unfamiliar concepts of qubit-like states |0⟩ and |1⟩, we provide their states {α,\nβ}T = α|0⟩ + β|1⟩ with classical wave representations α = A exp(iφ ) and β = A exp(iφ ). The\n1 1 2 2\nmechanical matrix operations are added with equations to explain their logical mapping\nrelation.\n(3) Furthermore, we also present the comparisons between the mechanical matrix operations and\nconventional acoustic computing in Supplementary Note 2 in the revised Supplementary\nInformation.\nThe following discussion illustrates detailed modifications and improvements related to the\n“mechanical matrix operations” in the revised Manuscript:\nMechanical matrix operator Evolution from analog to matrix computing\nle n t n o u a w p h n c - i f f 2 1 g g 2 1 le n n o a t t u u h o p c - (b) e g n a h c r e tn i e v a W o y f f 2 1 x ℱ 1 ℱ 2 Y (c) n o itc a r ffid e v a W o y f x ℱ 1 H(y ℱ ) 2 δ H (d)e c n e r e fr e e t l n b i tils- o y f f 1 2 x ℱ 1 H (y ℱ ) 2 g g 2 1\nT w u\ny (a) T o D\no x Lens+ Lens–\n|0〉 = {1, 0}T, |1〉 = {0, 1}T |1〉 → |0〉 |0〉 → |1〉 |+〉 → |+〉 –|+〉 → –|+〉 |1〉 → |+〉 |0〉 → |+〉\nBasis states\nFig. R4. Principle for achieving mechanical matrix operations. (a) Robot configuration for achieving\nmechanical matrix operations. (b)–(d) Evolution from analog to matrix computing: the matrix operation in\n(d) is achieved by the double-slit interference effect with combinations of (c) analog wave interchange and\n(d) wave diffraction.\n(1) Paragraph 2 on Page 12\n“Mechanical matrix operations. In addition to spatial analog computing, the elastic wave\nmetamaterial can be also applied to matrix operations on encoded mechanical signals. In the\nfollowing discussion, we employ the versatile computational framework to realize multiple\nmatrix operations which are commonly used in information processing. Different from spatial\nanalog computing with single input f, the matrix operations are achieved by several encoded\nsignals {f , f , …, f }T with a N-dimensional transformation matrix M(N). For example, we can\n1 2 N\nencode the two-channel analog inputs {f , f }T and outputs {g , g }T as vectors in a classical\n1 2 1 2\n2D state space. Then, the metamaterial can achieve linear transformation {g , g }T = M(2){f ,\n1 2 1\nf }T by a 2D matrix M(2). Such transformation can realize fundamental 2×2 operations such as\nextended to higher-dimensional state spaces by increasing the number of wave channels, e.g.,\nthe 4×4 controlled-NOT (CNOT) matrix. It is worth noting that many high-dimensional linear\nand invertible transformations can be decomposed into sequences of 2×2 operations. As a\nresult, a robust and reprogrammable M(2) transformation device is essential to achieve more\ncomplex and scalable higher-dimensional operations.”\n(2) Paragraph 1 on Page 14\n“As shown in Fig. R4(a), the metamaterial robot employs a similar multilayer ℱ -H-ℱ layout\n1 2\nto spatial analog computing. A mass barrier array m >>m is placed along the axial centerline\nb 0\n44\ny = Y/2 which divides the input f into a signal pair {f , f }T. For the harmonic signals with\n1 2\nidentical frequency, the signal pair {f , f }T defines a 2D complex state vector {A *exp(iφ ),\n1 2 1 1\nA *exp(iφ )}T with A *2 + A *2 = 1. This vector is characterized by the normalized amplitudes\n2 2 1 2\nand phases A *, A *, φ and φ of {f , f }T. We consider the state vector as a linear combination\n1 2 1 2 1 2\n{α, β}T = α{1, 0}T + β{0, 1}T, in which the basis states {1, 0}T and {0, 1}T are defined by\nsingle-mode excitations (A * = 1, A * = 0) and (A * = 0, A * = 1). In order to better describe the\n1 2 1 2\ntwo-channel signals, we adopt the symbolic notation from acoustic phi-bit computing, which\nlabels classical basis states {1, 0}T and {0, 1}T as mechanical qubit-like states |0⟩ and |1⟩.\nThen, a general input state is written as {α, β}T = α|0⟩ + β|1⟩ with α = A *exp(iφ ) and β =\n1 1\nA *exp(iφ ). It is worth mentioning that this notation is a purely mathematical convention\n2 2\nwithout relevance to quantum computing. With the normalization A *2 + A *2 = 1,\nthe\n1 2\nmechanical state can be fully characterized by the amplitude ratio A /A and relative phase Δφ\n1 2\n= φ − φ , Then, it can be visualized on a Bloch sphere by the mapping of two polar\n2 1\ncoordinates θ = 2arccos(A *) and ϕ = Δφ. Such representation is a classical analogous\n1\nmapping to provide visualization without quantum implication.”\n(3) Paragraph 2 on Page 14\n“In order to realize the M(2) transformation function, Figs. R4(b)–R4(d) provide the evolution\nfrom spatial analog to mechanical matrix computing. The metamaterials in Figs. R4(b) and\nR4(c) are designed with mass barriers along the axial and vertical centerlines y = Y/2 and x =\nX/2, respectively. In Fig. R4(b), the input f is divided into a signal pair {f , f }T by the axial\n1 2\nbarrier y = Y/2 and the logic states can be swapped after two Fourier lenses. In Fig. R4(c), a\ntunable slit is designed at the center of the vertical barrier x = X/2. Such a design allows the\nconversion of incident waves into point wave sources and presents the diffraction effect. After\nthe Fourier layer ℱ , the point source can be further transformed as a parallel waveform and\n2\nequally divided into {g , g }T as the superposition state |+〉 = {ξ2/2, ξ2/2}T , i.e., α = β. Then,\n1 2\ncombining Figs. R4(b) and R4(c), incident waves with either |0〉 = {1, 0}T or |1〉 = {0, 1}T\ninput in Fig. R4(d) can be transformed into their superposition states |+〉 = {ξ2/2, ξ2/2}T.”\n(4) Paragraph 1 on Page 15\n“Based on the discussion above, the evolution |0〉 → |+〉 can be achieved by the robot\nconfiguration in Fig. R4(d). In order to obtain the |1〉 → |–〉 output, the transfer layer H in Fig.\nR5(a) is designed as a double-slit layout, in which the lower slit has two mass arrays with π\nphase difference. When the input |1〉 = {0, 1}T is applied, the input f can be focused into the\n2\nlower slit as F and divided into two separated wave fronts F ± with opposite phases. Based on\n2 2\nthe Huygens-Fresnel principle, the separated wave fronts can be considered as two half-space\nwave sources which input into the Fourier layer ℱ . Then, they can generate two opposite\n2\noutput states, i.e., |–〉 = {ξ2/2, –ξ2/2}T. Similarly, for the |+〉 = {ξ2/2, ξ2/2}T input, the\n45\noutput corresponding to the upper half-space g can be enhanced by two in-phase wave fronts\n1,2 2 1,2\nthe |0〉 ={1, 0}T output, i.e., |+〉 → |0〉. Similar phenomena can be observed from the\nnumerical results and robot attitudes during experiments in Figs. R5(f) and R5(g), respectively.\nFor comparisons, Supplementary Figure S5 and Movies S5 further provide several mechanical\nrealizations of 2×2 matrix operations with experimental supports, including Pauli X, Pauli iY\nand Hadamard transformations.”\n(a) Principle for Hadamard operation (b) Mechanical computational results (c) Robotic attitude\nd\nH(y) r a\ny tilib\na\nb o rP –\n2\n0\nF1\nF\n+\n2 + n o ita\n|0 |+〉\n|0〉 |+〉\nw r o\nF\n|1〉 |1〉 F1 re\n|0〉\n|0〉\n|0〉\nF2\nF1 –\np o\nd ra\nm\n|+〉 |0〉\ntfe\nL\ne\nre h p s h c o ȁ0〉 ξ + 2 |1〉\nȁ0\nξ\n〉–\n2\n|1〉\nF2 –\na d\na H |1〉 |–〉\n| +〉 |0〉\nd r a w\nlB Barrier r o\n|1〉 |1〉 |–〉 F\nFig. R5. Mechanical realizations of reprogrammable matrix operations. (a)–(c) Principle for mechanical\nHadamard operation and the computational results of |0〉, |+〉 and |1〉 inputs.\nPlease refer to the blue sentences about mechanical matrix operations from pages 12 to 16 and Eqs.\n(7)–(10) in the revised Manuscript. In addition to the modifications above, the relationship\nbetween the mechanical matrix operation and acoustic phi-bit computing are also discussed in\nSupplementary Note 2, Part E. Limited by space, please refer to the blue sentences on pages 13\nand 14 in the revised Supplementary Information. We wish that these modifications and\nimprovements could better convey the main features of our present work.\nOnce again, we sincerely appreciate your thorough review and valuable feedback that has helped\nus to comprehensively improve our manuscript!\nReviewers: 4 & 5\nGeneral comment: I co-reviewed this manuscript with one of the reviewers who provided the\nlisted reports. This is part of the Nature Communications initiative to facilitate training in peer\nreview and to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nResponses:\nThe authors greatly appreciate Reviewers 4 and 5 for your thorough review and constructive\nfeedback on our manuscript. We are sincerely grateful for the time and effort you have dedicated\nto the insightful comments, which have helped us significantly improve our work.\nFollowing your valuable suggestions, we have carefully checked our work and comprehensively\nimproved the theoretical framework, result presentation, as well as our English expression. A\npoint-by-point response has been provided to each of your comments and the places we modified\nhave been marked in the revised manuscript. Furthermore, the writing logic from spatial analog\ncomputing to matrix and logic operations has been revised with more explanations. We wish that\nthese improvements and revisions could align with your expectations and better convey the main\nfeatures of our work.\nFinally, we thank all the reviewers and editors for encouraging us to substantially improve our\nmanuscript with your valuable comments and kind suggestion!\nReviewer 1\nAll my concerns have been addressed in the revised paper. It is now recommended for\npublications in NC.\nThe authors sincerely appreciate Reviewer 1 for your positive feedback and valuable suggestions\nduring the review, which have enabled us to comprehensively improve our manuscript!\nReviewer 2\nI would like to thank the authors for their thoughtful response to the reviewers comments. From\nreading the revised manuscript, I can see that the authors have taken steps to make the article\neasier to follow, and to more acceptably place their work within the wider related literature.\nThe authors sincerely appreciate Reviewer 2 for your valuable suggestions and thoughtful\ncomments during the review. We wish that the revised manuscript could make it easier for readers\nto understand the main feature of this work!\nReviewer 3\nWe appreciate the revisions made by the authors and now find the manuscript appropriate for\npublication.\nThe authors sincerely appreciate Reviewer 3 for your thorough review and constructive comments\non our work, which have helped us to better combine the computational functions with our robot!\nReviewers: 4 & 5\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriate recognition for Early Career Researchers who co-review manuscripts.\nWe are sincerely grateful for the time and effort you have dedicated to the insightful comments,\nwhich have helped us significantly improve our work!\nFinally, we thank all the reviewers and editors for encouraging us to substantially improve our\nmanuscript with your valuable comments and kind suggestion!\n1"}, {"round": 2, "reviewer_comments": {"reviewer_1": "All my concerns have been addressed in the revised paper. It is now recommended for publications in NC.", "reviewer_2": "I would like to thank the authors for their thoughtful response to the reviewers comments. From reading the revised manuscript, I can see that the authors have taken steps to make the article easier to follow, and to more acceptably place their work within the wider related literature.\nI believe the revised manuscript to be suitable for publication.", "reviewer_3": "We appreciate the revisions made by the authors and now find the manuscript appropriate for publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

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