Feedback-driven recurrent quantum neural network universality
Authors: Lukas Gonon
Rodrigo Martínez-Peña
Juan-Pablo Ortega
Venue: ICLR 2026 Poster (2026)
Keywords: quantum machine learning
quantum neural networks
recurrent neural networks
expressivity
universal approximation
state-space systems
quantum reservoir computing
Primary area: learning theory
TL;DR: The paper proves quantitative universal approximation results for learning state-space systems using recurrent quantum neural networks.

Abstract: Quantum reservoir computing uses the dynamics of quantum systems to process temporal data, making it particularly well-suited for machine learning with noisy intermediate-scale quantum devices. Recent developments have introduced feedback-based quantum reservoir systems, which process temporal information with comparatively fewer components and enable real-time computation while preserving the input history. 
Motivated by their promising empirical performance, in this work, we study the approximation capabilities of feedback-based quantum reservoir computing. More specifically, we are concerned with recurrent quantum neural networks, which are quantum analogues of classical recurrent neural networks. Our results show that regular state-space systems can be approximated using quantum recurrent neural networks without the curse of dimensionality and with the number of qubits only growing logarithmically in the reciprocal of the prescribed approximation accuracy. 
Notably, our analysis demonstrates that quantum recurrent neural networks are universal with linear readouts, making them both powerful and experimentally accessible. These results pave the way for practical and theoretically grounded quantum reservoir computing with real-time processing capabilities.

Published as a conference paper at ICLR 2026
FEEDBACK-DRIVEN RECURRENT QUANTUM NEURAL NETWORK UNIVERSALITY
Lukas Gonon
Rodrigo Martínez-Peña
School Computer Science University of St. Gallen, Switzerland lukas. lukas.gonon gonon @ unniig ionis..co ch
Donostia International Physics Center San Sebastián, Spain rodri 1o.rigo.wo ma rt ne @d www.ddddddoooooooooooo org
Juan-Pablo Ortega
School of Physical and Mathematical Sciences Nanyang Technological University. Singapore Juan-Pablo Ortega@ntu. edu. sg
ABSTRACT
Quantum reservoir computing uses the dynamics of quantum systems to pro- cess temporal data, making it particularly well-suited for machine learning with noisy intermediate-scall quantum devices. Recent developments have introduced feedback-based quantum reservoir systems, which process temporal information with comparatively fewer components and enable real-time computation while preserving the input history. Motivated by their promising empirical performance, this work, we study the approximation capabilities of feedback -based quantum reservoir computing. More specifically, tum neural networks, which are quantum analogues of classical recurrent neural networks. Our results show that regular state-space systems can be approximated using quantum recurrent neural networks without the curse of dimensionality and with the number of qubits only growing logarithmically in the reciprocal of the prescribed approximation accuracy. Notably, our analysis demonstrates that quan- tum recurrent neural networks are universal with linear readouts, making them both powerful and experimentally accessible. These results pave the way for prac- tical and theoretically grounded quantum reservoir computing with real-time pro- cessing capabilities.
concerned with recurrent quan-
INTRODUCTION
Recent advances in quantum computing have led to rapid development of quantum machine learn- ing methods. These methods aim to exploit the potential computational speed-up and reduced com- plexity offered by quantum computing for machine learning purposes. For learning problems with temporal structure, quantum reservoir computing (QRC) has emerged as promising approach for exploiting noisy intermediate-scale quantum (NISQ) technologies. In contrast classical machine learning methods based on bits valued in {0. 1}. quantum bits (qubits) can be in a continuum of states. QRC aims to exploit this fundamental difference to build efficient machine learning methods for time series prediction and learning.
In this paper, we are concerned with recurrent quantum neural networks (RQNN), a particular type of quantum reservoir computing method. RQNNs are a quantum analogue to classical recurrent neural networks. RQNNs are built from quantum neural networks (QNNs), with weights and biases typically realized via quantum circuits. Thus, these networks can be evaluated directly quantum computers. Thereby, quantum machine learning aims to achieve a significant increase in neural network expressivity and computational speed-up in inference and training.
Also affiliated as Honorary Senior Lecturer with the Department of Mathematics, Imperial College, Lon- don, United Kingdom

---

Published as a conference paper at ICLR 2026
Motivated by their promising empirical performance, in this work, we study the approximation capabilities of feedback-buse based quantum reservoir computing methods and, specifically, RQNNS. In particular, our work provides precise bounds on the number of qubits and the size of the underlying quantum circuit that is required to guarantee a prescribed approximation accuracy. Our results show that QRNNS can approximate regular state-space systems using quantum circuit with qubit number only growing logarithmically in the reciprocal of the prescribed approximation accuracy and with error rates not suffering from the curse of dimensionality. Thereby, our results pave the way for theoretically grounded quantum reservoir computing with real-time processing capabilities.
1.1 RELATED LITERATURE
Quantum reservoir computing methods have been extensively studied for a variety of time-series prediction and learning tasks, employing different architecture types such as online protocols (Mu- jal et al. 2023 Franceschetto et al. 2024), mid-circuit measurements and reset operations (Hu et al.,
2024 Murauer al. (2025). feedback protocols (Kobayashi et al. 2024). QRC with quantum mem-
ristors (Spagnolo et al. 2022) and hybrid QRC techniques (Piefter et al. 2022 2023).
We provide
a detailed discussion of QRC methods in Appendix A
Despite these promising developments, key questions regarding universal approximation capabilitiess and expressivity of feedback-driven QRC methods have not been addressed in the literature. For classical neural networks, qualitative and quantitative universal approximation theorems have been extensively studied, with seminal works including, e.g. [Horıik (199)); Barron (993);: Yarotsky 2017). Universality results for the dynamic reservoir computing setting have been obtained in Grigoryeva & Ortegal [ Gonon & Ortegal 2020 2021; Gonon et al. 2023) for echo state networks, state-affine systems and linear systems with polynomial neural network readouts. For (feedforward) QNNs first qualitative results on universal approximation properties of QNNs have been proved only very recently [Pérez-Salinas et al. (2020) (Schuld et al., (2021). Subsequently, quantitative approximation eтror bounds for feedforward QNNs were proved in Gonon & Jacquier 2025 Yu et al. (2024); Attab & Yang (2024).
For RQNNs, no quantitative approximation error bounds have been previously available in the lit- erature. Moreover, previous universality results concerning QRC models have relied on the use of polynomial output layers (Chen & Nurdin 2019] Chen et al., 2020; Nokkala et al.) 2021 Sannia jtt al. 2024b a). which yield a polynomial algebra that can then be used with the Stone Weterstrass theorem to obtain universality statements. Nevertheless, most numerical and experimental imple- mentations of reservoir computers use linear output layers due to their simplicity and fast training.
1.2 CONTRIBUTIONS
For applications of QRC methods learning tasks with temporal dependence, a precise understand- ing of RQNN approximation capabilities is essential. In this paper, we derive approximation error bounds and prove universality statements for RQNN families with a linear output layer and in the context of the feedback protocol. Universality refers to the ability of these families to uniformly approximate arbitrarily well a large category of dynamic processes, so-called fading memory in- put/output systems. Thereby, contribute to a precise understanding of RQNN approximation capabilities in several aspects.
We provide RQNN approximation error bounds for regular state-space systems. Our first main result, Theorem 4.6, shows that RQNNs are able to approximate regular state-space systems without the curse of dimensionality, using quantum circuits with qubit number only growing logarithmically in the reciprocal of the prescribed approximation accuracy.
In our second main result, Theorem 4.8 we prove that RQNNs can uniformly approximate the arbitrary fading memory, causal, and time-invariant filters. In particular, RQNNs have approximation properties
competitive as those of popular reservoir computing/state- space system families like echo state networks, state-affine systems, or linear systems with polynomial/neural network readouts.
To prove these results, we first derive novel qualitative and quantitative approximation error results for using feedforward QNNs to approximate functions and their derivatives (see Proposition and Corollary 4.5).
2

---

Published as a conference paper at ICLR 2026
In comparison to Gonon & Jacquier (2025), our RQNNs introduce memory through a feedback loop. Mathematically analysing our RQNNs architecture hence requires a novel, intricate analysis of QNN approximations of functions jointly with their derivatives. Moreover, approximation analy- sis in the temporal domain is inherently much more challenging due to the feedback loop. Proving Theorems 4.6 and 4.8 thus requires new techniques specifically tailored deal with this situation (see Appendix C Most previous literature on RC and QRC universality (Grigoryeva & Ortega
2018a b; Gonon & Ortega 2020 2021; Chen & Nurdin 2019 Chen et al., 2020; Nokkala et al. 2021 Sannia et al. 20024b a) implicitly assumes the search for an optimal model within a class in which all parameters are estimated. Also our results are formulated for variational quantum cir- cuits for which all parameters are trainable. Nevertheless, the obtained results and developed proof techniques also promise to be useful for QRC systems in which certain parameters in the recurrent layer are randomly generated. Our RQNN architecture builds on and extends the feedforward QNN architecture introduced in Gonon& & Jacquier (2025), which also admits results for the randomized setting. Hence, combining the techniques developed here with these randomized architectures may provide fruitful for studying randomization in the dynamic quantum reservoir computing setting. Moreover, the obtained approximation error bounds may serve crucial ingredient for bounding the overall generalization error of QRC methods, by combining our results with suitable risk bounds as obtained in other contexts in Gonon et al. (2020) Chmielewski et al. (2025).
1.3 OUTLINE
The paper is structured as follows. Section 2 introduces background on filters, functionals, fading- memory and echo state properties. Section 3 describes the RONN model, a recurrent QNN with state feedback. Section 4.1 derives QNN approximation error bounds for functions and their first derivatives. We then use these results (see Proposition 4.4 and Corollary (.5) to study the properties of the RQNN state maps for approximating more general state equations. These results are then used in Section [4. to prove the universal uniform approximation properties of the filters associated with RQNN systems. More specifically, in Theorem 4.6 we provide filter approximation bounds that show that RQNNs can uniformly approximate the filters induced by any contracting Barron- type state-space system. Finally, Theorem 4.8 of Section 4.3 extends this universality property to the much larger category of arbitrary fading memory, causal, and time-invariant filters. The paper concludes with Section where the main contributions and outlook of the paper are summarized.
2 BACKGROUND ON FILTERS AND FUNCTIONALS
We start by introducing the input-ouiput maps to be learnt in the dynamic setting. In a static context, input-output maps are given by functions of the form f Rdd - I For learning with temporal dependence, the relevant input-output maps are filters and functionals defined sequences.
Specifically, let (R23)² denote the set of infinite real sequences of the form & (..., 2- 20, Z1, 2 2; R. Z; z-=)z- is the subspace consisting of left infinite sequences: (R²)²-= = {z = (..., 2-2, 2-1, _o) 2 E RR"", i E -_ Analogously, (D,)² and (D,))= stand for infinite and semi-infinite sequences, with elements in the subset D C R". Let D,, R and
BN C CRN. We refer to the maps of the type U (D,,)² - (Bx) )² as filters and to those like H: : =D,,)² - BN (or H (D,)²- + Bx) as functionals. A filter U (D,)² - (Bx)² 2 is called causal when for any two elements z, w E (Da)² that satisfy that Z, = w. for any T S L, for E Z be the time delay operator defined by T, (z); := 21-7. The filter U is called time-invariant when it commutes with the time delay operator, that is, Tr U U Tr. for any T E Z, with the two operators T, defined in the appropriate sequence spaces. Finally, there is a bijection between causal inne-invariant filters and functionals on (D,)² 2- and we can use them interchangeably (Grigoryeva & Ortega 2018b).
a given E Z. we have that Uz)): = U(w),. Let T, (D,,)² - +D
A specific class of filters is given by state-space systems (such as recurrent neural networks) deter- mined by two maps, namely the recurrent layer or the state map F; R x R" - RN n, N E N.
and readout or observation map h: : R - Rmm m E N, given by
T F(x yt h(x)):
where /. E Z,, z, denotes the input, x, N is the state vector, and y: Rm is the output vector.
3

---

Published as a conference paper at ICLR 2026
Consider now subsets BN C RN and Dn R" and a recurrent layer defined them, that is, F: :N x D,, - By and h : BN - R".. Denote by Dm := h(Bx) C R" The recurrent system F is said to have the echo state property with respect to inputs in (D,)² when for any E (D,.) there exists unique element x: E (Bx)² that satisfies the first equation in ( for each 1 E Z. When the echo state property holds, a unique filter U (D,)² _ (Bx)² can be associated to the recurrent system determined by F, namely, (=), := 2, E Bx. for all E Z. We will denote by U (D,)² - (Dm) )a the corresponding filter determined by the entire recurrent system, that is. (=), - h (x) :- E Dm for all / E Z. The filters U and Uh," are causal and lime-invariant by construction. The echo state property is much related with the so-called fading memory property defined as the continuity of U/2" with respect to weighted norms in its domain and codomain (Boyd& & Chua 1985) or the product topologies when Dn and Dm are compact (Griooryeva & (
It can be shown that when Dm is compact, the echo state property implies the fading memory property (Manjunath] 2020; Ortega & Rossmannek| (025b)); see Ortega & Rossmannek (2025c) for a comprehensive account of the dynamical implications of the fading memory property as well as [Ortega & Rossmannek (2025a) for a stochastic version.
RECURRENT QUANTUM NEURAL NETWORK ARCHITECTURE
Before going into details about the considered RQNN architecture, let us first explain the basic working principle of feedforward QNNs built in quantum circuits. A QNN is built by transforming
quantum bits (gubits) in a parametric quantum circuit. Each qubit is in state (v)) = (0) + 3 [1) for some a E C, 3 E C with |a|2 + |3]2 = and with elementary quantum bit states (0) and [1). For circuit with n qubits,
any given point in the circuit, the circuit state can thus be identified with vector in C for ny = 2n. The quantum state (&) can be transformed by applying a quantum gate, that is, unitary matrix U X A QNN now applies quantum gates U(x 0) that depend on the initial data and neural network parameters and transforms the circuit accordingly. The QNN output obtained by measuring the final quantum state after applying the circuit quantum gates.
Next, we introduce in detail the employed RQNN architecture. Our recurrent quantum circuit is constructed based on two parametric quantum gates and V, which now introduce. The con- struction extends the feedforward QNN architecture introduced in Gonon & Jacquier (2025) to a recurrent setting by feeding back the network's 's state.
Construction of U. For ó, ? E [0, 2T]] and a e R, denote by (5))), By (1), and Ra(a)) the rotations around the X-, Y-and the Z-axis, corresponding to angles 8, y and a, respectively, and obtained as the exponentials of the Pauli matrices:
cos Rx(() = (iisin isn
- -s (1) cos (9) (5)) (y()) = cos sin (7) sin
cos (
Ra(a) = (00
c e
0 el9
For a given accuracy parameter E N. consider weights = (a...... , a") E (Red+N)n b (61 (b'....,.6)) 6a)) E I8" and y = (7),..... (??......?"" 7 E [0, 2m)". For i = I,..., n, we define parametric gate maps uf"): R.V x Rd - C2x2 that map a current system state and a current observation to rotation gate. Gate map depends on parameters a' bf and is defined by
(z. 2) : H (-b+) R2 (-aiy4a=a) ...R, (-a'x+111) Ru (-alyax) ...RR, (-afz1) H for any (.1,...,, xN) N and = (z1,..., Rd, with the Hadamard gate. We may rewrite
(x. 2) (x) Rx (8'), 8 -b' a a d'+112 a af ai.
Moreover, we also define the gates (1) Ry (7*) and denote the circuit parameters by 0 (a', bb, ? E
(IR d
00, 2T))
With these notations, we are now ready to define the key element of our parametric quantum circuit, the gate U Up Ua( 2). U is defined as block matrix built from the gates ((() (1x, z) (x, 2) e
4

---

Published as a conference paper at ICLR 2026
follows:
(x,,)) x)
O 01x4
01x1
O1x1
O1xno
04x4
(ax)
04x4
04x4
:
(a( ==
04x4
04x4
(a.
04x4 ((n) (x,
04x4
04x4
04x
0nox4
Ongx4
In X
Here, no is chosen as the smallest natural number such that the matrix dimension Ry = 4n + no is power of 2, that is, ny 2". It can be casily shown that no = 4x with E N. since 4n + no and 2n + n must be even for n 2 2. Then, U E Ch a unitary quantum gate operating on (og)(ng) = 2 log2(n x) (logg(2n) qubits with a diagonal-bloct structure:
n-1
n+k-1
(a 2)
Σ
(t) (il & T 2x)
Σ
(t) (i) 14x4-
These unitary operators with a block structure are known as uniformly controlled quantum gates. They are present in many quantum algorithms and are used to decompose general unitary gates and locally prepare arbitrary quantum states (Möttönen al., 2004, Mottonen et al. 2004] [Bergholm et al. 2005; AArrazola et al., 2022) Park et al.j 2019). They are defined as multi--onttollle unitaries where each unitary block targets a set of qubits, two qubits in this case, while the other loga (n + k) qubits act as control qubits. Multi-controlled unitaries are applied depending on the state of the con- trol qubits, which are unchanged, and only modify the target qubits. These operations generalize the CNOT gate for two qubits, in that can now have several control and target qubits. Notice that the block structure of the unitary Un arises from indexing the targets as the lowest-order bits. Recently, efficient decompositions of multi-controlled unitaries have been proposed in terms of the number of single-qubit and two-qubit gates (Zindorf& & Bose 2024) (2025). ) as well as for approximations of the multi-controlled gate (Silva et al. 2024). Code implementations of these quantum gates can be found in the Qclib library (Araujo et al. 2023).. Finally, the identity blocks 14x4 do not introduce additional gates into the quantum circuit, so the effective circuit can be reduced to the application of the ((() gates. However, the number of control qubits is fixed by logg(n + x)) and we need all of them to compute the output probabilities, as we will see below.
Construction of V. Next, let V E C be any unitary matrix mapping (0] Sn to the state (w) =
1 VR. EE= |4i) which, for 2 2, is also explicitly given as (&b) = v Er= (i) & (00). Note that
different choices of V are possible and the only required property is v 10) &n Iv). We refer to Appendix for an example.
Mcasuring circuit outputs. We can now measure the state of the n-qubit system after applying the gates and U. The possible states that we could measure are given by O..... ny - 1 (in binary). obtain (up well-controlled Monte Carlo error, Appendix E) the probabilities IPn, that the measured state is in {m, 4 + m, 4(n - 1) + m}. for m {0, 3,, where m is the binary state of the last two qubits (the target qubits).
By running the circuit repeatedly,
More formally, consider the unitary gate map C(x, z) = Cn. 2) := Uø( z)V acting on n = 2 log2 (n + k)) qubits. This circuit acts on the initial state 10) @n via the quantum gates V and U as
(z,0(x,)) x) 10) 00
1)
(+ (2
(z, x) 10) & of 10).
Then, measure
p pm.0 (2, (e,211 2) := (" ez, x) 10) on E {m, 4 m. 4(n 4(n-(nn)))))) 1) + m)")
(2)
This is the sum of the probabilities of being in the states (i) @ |m), where i = 0...., 1. That is,
p=.0 m (xx 2) =
(ml (u (z. x) 10) @ u.! (0)
5

---

Published as a conference paper at ICLR 2026
Parallel circuits. With (or equivalently n) fixed, the quantum circuit introduced above is uniquely defined by the choice of circuit parameters 0 E O We will now run N such circuits in parallel, each representing component of the state map F in IT. Each circuit is described by its parameters 0¹ @.j e (I....,N} NJ. The circuit outputs then induce maps pm,0' IRN x xR [0. 1]
by the circuit output probabilities 2
with parameters for the j-th circuit given by 0 = 01.
State and inpur vectors
Parameters
...... Rt) (3) .......... 2c)
0 (a b, of Ien....
(0)
Control qubiis
(0]
(0]
pel fl (8
(0]
aad 1
(1)
Targer qubits
(0)
U (.)
( (8,
100
Feedback: & F..M (8)-112))
Figure 1: Schematic representation of the j-th circuit given by parameters Oi O.. [I....,NN:
Recurrent quantum neural networks (RQNN). With these ingredients, we can now define the RQNN that we will consider. Given the gate map Cn and 0, we define Fr.0 RR R. + R. by its component maps R (FR. R........ FR,)). For = 1, I,...,N, the j-th component map
F.n. RN Rd + R is defined by
p (a, 2),, F a z) R 2R(PT. (z, 2) + P
(ar, 2) E TR. x TR..
with 0 (0) ........e NN) N Our recurrent quantum neural network (RQNN) is then defined by the state-space system associated to the state map Fn.0
# Fi (i:-1, t ZE
(4)
Figure [ provides schematic representation of how the RQNN acts each time step for the j- th circuit: at any time I, the system is initialized, the gates V and Ugs (#.- z1) are applied, and
the system is measured. This process is repeated to estimate the probabilities PT p (#t-1. z.) and
Fr En.00 (#,-1; 2:), which are aggregated into the network output Fn. (1,-1, 2.) according to (3) Once this is done for all j E [I....,NN] NY. the network state & stored to be used as feedback for the next time step f. + 1.
In the next paragraphs, we aim to address the following questions:
Can we choose the parameters 0 in such a way that the system determined by satisfies the echo state property?
Can the family of systems determined by equations of the type 4 approximate general state-space systems arbitrarily well? More specifically, given arbitrary state-space map x, = F(x 22)) with F: N x I - IR as general as possible, can be approximated by equations of the type
RECURRENT QUANTUM NEURAL NETWORK UNIVERSALITY
This section contains approximation guarantees and universality results for the recurrent quantum neural network (RQNN) family. To achieve this. in Section 4.1 we first prove refined approxi- mation error bounds (that generalize those in Gonon & Jacquier (2025)) for feedforward quantum

---

Published as a conference paper at ICLR 2026
neural networks (QNNs) that allow to control the enror committed when approximating a fune- tion and its derivatives simultaneously, a crucial ingredient for analysing the RQNN feedback loop. These error bounds show how recurrent QNNs can be used to approximate state-space maps F arbitrarily well as long as these are sufficiently smooth and satisfy Barron-type integrability condi- tions like, for example, JRMXR' (E) < oc, for i 1....,N N + d and j 1..... N. or
lg JR xx | (E) < 0o, for some 2 2 (see Proposition 4.2 and Corollary 4.3 below); RQNN state maps are hence universal in that category. These bounds are devised with respect to d and -tyt norms. As we shall prove, in the - case, the universality of the RQNN family still holds with respect to state maps that do not necessarily satisfy the Barron condition, even though in that framework we do not formulate approximation bounds. Finally, in the last two sections, we exploit all these results on the approximation of state maps to obtain universality statements and error bounds for the approximation using a modified recurrent QNN. In addition to the tools developed here, our proofs of these results rely on techniques from Gonon & Ortega (2020; 2021) and the overall strategy is reminiscent of the so-called internal approximation approach introduced in Grigoryeva & Ortegal (2018b Theorem 3.1 (iii) for echo state networks, which consists of obtaining approximation results for filters out of statements of that type for the state maps that generate them.
arbitrary causal, time-invariant, and fading memory filters
The approximation rate in all our results is free from the curse of dimensionality: the error decays as yn we increase n, with this rate of decay being independent of the input dimension d and the state space dimension N. Moreover, the required number of qubits n = [2n]] is only grow- ing logarithmically in the accuracy parameter n. Put differently, our circuit requires only O( 2) weights and O ([[logy(s qubits suffices to achieve approximation error & > 0 when approxi- mating functions with sufficiently integrable Fourier transforms.
4.1 RQNN APPROXIMATION OF STATE-SPACE MAPS AND THEIR DERIVATIVES
As a first step, we aim to establish RQNN approximation results for a function jointly with its derivatives. Denote by Fn the class of integrable functions f: RR. x Rd -> R with Fourier integral bounded above by constant R. > 0, that is,
F = { RRRN Rd + +R: E (RN x R nL' (RN x RT), lifl < oo).
(5)
Fn = (IEF, with 11 S R).
for R > 0.
(6)
Here, for a continuous and integrable function f: RN x Rd + R we denote its Fourier transform by f() == JRNXRE
fy) ye dyy dy2. with (EI, E RN x Rd.
Our first result derives a representation for the QRNN output.
Proposition 4.1. For any n N. j 1, 1,..., N, 0 (0) ..... ON) E C with 0i
(at), bi., 72) ) E E, the RQNN introduced in (3) can be represented as
Fn. (x, 2)
(I,z)
Reos (7²))) cos (b a'd (z, for all (a, 2) E Σ n2
x
Our next result provides an approximation error bound for the QRNN state map jointly with its derivatives. The proof is provided in Appendix B.2 Let μ be an arbitrary probability measure on (R& Rd BBB. x R..... Recall the notation
proof
If - oll = -V X 1f(x, 2) - g(x, x) u(dx, dz)
1/2
Proposition 4.2. Let R > O and suppose F = (FI....,FN RN x Rd + RN is continu- ously differentiable and satisfies F, FR and O,F, F and JaN EIFF, < co for i 1,,..., N + and j = 1,...,N. N. Then, for any n E N, there exists O E ON such that N+d
1 - P3 BB13 LP(2)) RJ
+
Σ
O,F, (2(1))
S
C
n
1=1
for any j E (1...., N), where C = ||
+ 4 21
112 See xxx
INNtd
<2 (E)
7

---

Published as a conference paper at ICLR 2026
Next, show that it is also possible to also obtain approximation results for QNNs with bounded network coefficients. The proof is provided in Appendix B.3
Corollary 4.3. In the setting of Proposition 4.2 assume, in addition, that fax x [E] 00 for some q 2 Then, for any N, there exists 0 O such that for any (I,...,
N+d
P F3 13122 12(1))
C. 0,F; (2( n
where Cj. = 3 Moreover, we can choose 0 = (0) ,,,,,, ON)
E
N
with AJ =
(a's, , in such way that for all i 1.....,.. n. = I..., N.
/ RN XB=
| || S 2π
3n F2
(E)
(8)
Next, we complement the LIR x IR (,) bound in Proposition 4.2 with uniform error bound on compact sets. For M and f, g C(RN x Rd) denote
Ilf - glloo, ==
sup (a.z)E(- MJ x M.
|f 2) x(x,2)].
Proposition 4.4. Let R. M > O and suppose F (Fi,..., FN) is continuously differentiable and satisfies F3 E FR and F,F, E F and Sgx xRd N, there exists
(E))(CE < oo for I....,N. Then, for any
O such that for any j E (1....,.) N).
N+d
C
F2
+
O F. RJj
0,F₁
S
(9)
10.M
Do,M
Vn'
where C3o 2(TT + 1)1l + (8T 4m (N + +d)
|
1/2
166
d)
/2
for 14.3 = -RN X 9 (F) co.
The proof can be found in Appendix B.4 Finally, we obtain a qualitative universal approximation result for QNNs jointly with their derivatives. The proof can be found in Appendix B.5
Corollary 4.5. Let F (Fi...., FN) be continuously differentiable. Then for any > and X C RN x Rd compact there exist n E N, R > O and 0 E O such that for any j E [l....,N N}
R..j satisfies
sup [2, x) (m,=)ex
Fn.0 FR. (x, x)1 + (x, VFR. (2, 2) SE.
(10)
4.2 RECURRENT QNN APPROXIMATION BOUNDS FOR STATE-SPACE FILTERS
The results in the previous section show that the family of RQNNS that were introduced in (3) capable of approximating arbitrarily well the very general class of continuously differentiable state- space maps with bounded Fourier transform, together with their derivatives. These approximations hold with respect to both the L22 norm (Proposition 4.2 and Corollary 4.3) and the - norm on compacta (Proposition 4.4 and Corollary 4.5). We will now use the uniform RQNN approximation results for the state maps conclude similar uniform approximation results for the corresponding filters under additional hypotheses that guarantee that those exist.
Consider state-space system
= F(F t Z_
(11)
with state process (mx) valued in N input process (zz)1Ez. valued in Ted and F: R x Rd R. We work under the assumption that F is contractive and satisfies Barron-type integrability (218)) Then, e.g., Proposition and Remark in Gonon et al. (2020) imply that, for any compact Da C Rdd the associated filter UF; (D,)² (Bx)²= induced by the restriction of F to By x Da is well-defined and continuous.
conditions (Barron 1992] 1993] Barron & Klusowski 2018).
8

---

Published as a conference paper at ICLR 2026
Our next result shows that among the RQNNS that we discussed in Proposition 4.4 there exist sys- tems that have the echo state property and hence have a filter associated. More importantly, those filters can be used to uniformly approximate any of the filters corresponding to the general systems introduced above in (TT) as long as they satisfy a Barron-type integrability condition and are suffi- ciently contractive. The proof can be found in Appendix C.1 Here, || ||2 is the spectral norm. In particular, this result shows that the error rate is free from the curse of dimensionality: the error de- cays as J as we increase n, with this rate of decay being independent of the input dimension d and yn the state space dimension N. Thus, the RQNN requires only O weights and ((1log2(= qubits to achieve approximation error e > 0 for the considered state-space systems.
Theorem 4.6. Suppose F in (T1) is continuously differentiable with |V F(r, (),22)1l2 X for all x E R.,. E Da for some X E 1) and, moreover, satisfies Fj E Fr. O,Fj E F and
SaNxES [|E|) associated to (TI]. Then for any N with > no there exists I E 2 such that the system 4 has the echo state property and the associated filter U: (Da)² + (RN)² satisfies
oo for j .....,N. Denote by UF: (Da)² - +Bx)² the filter
1
VN max C3
sup sup | (z), -((=))) - zE(Da)² 1EZ_
1
(12)
1->
vn
N2 (max I..... C2
Here, ny may be chosen as no
(1-1)2
Notice that N represents the state space dimension of the target F, which is matched by the QRNN dimension to obtain the approximation error bound. Theorem 4.6 also proves advantage of QRNNS over classical RNNs. RNN approximation bounds for state-space systems driven by Barron- type functions were obtained in (Gonon al., 2023] Theorem 3). While the approximation rate in Theorem 4.6 is the same ( in both cases), the Fourier integrability condition required in the quan-
tum case strictly weaker. Specifically, the condition Jexxe [F) < oo implies that the smoothness assumption (Gonon et al., 2023 Definition 2) required for (Gonon al. 2023, Theo- rem 3) is satisfied. For example, consider a Sobolev function F E H (I x ) Then, the integra- bility condition for the QRNN approximation result is satisfied for any > N+d + 4 (by (Folland, 2020 Lemma 6.5) and its proof). In contrast, the integrability condition for the RNN approximation result in (Gonon et al., 2023 Theorem 3) would require the stronger condition s > N + d +
4.3
UNIVERSALITY
In the previous section, we proved error bounds for the approximation using recurrent QNNs of the filters induced by contractive state-space targets with Barron-type integrability conditions. These bounds show, in passing, the universality of the family of RQNN filters in that category. We now extend this universality statement (without formulating error bounds) to the much larger family of fading memory filters by introducing a modification in the RONN reservoir. We define Fm.0 RN x Rd - R.N by its component maps F =
FR..). For j = 1,...,, N, the j-th
component map F.n RN x Rd + R is defined by
(x. 2) = 2R(P (P,x, 2) + + F R.j
ph. (P,x,2)]
x R
(13)
with 0 = (0) 0N) E N and Pl...., PN E N linear preprocessing maps. Our modified RQNN is then defined by the state-space system associated to the state map E.
a. Fi ( t Z_.
(14)
The next lemma (with proof provided in Appendix C.2) shows that adding linear preprocessing maps to reservoir equations can lead to the echo state property without contraction assumptions.
Lemma 4.7. Let F = =Fi...., be reservoir map where each component F, .R.
x R,
jor = 1,..., N, is defined as
Fy(x,z) = g, g,
(15)
where Pi...., PN E N are linear preprocessing maps for any maps 9j FR R2d + R, j =
1,..., N. Define arbitrary partition of the state vector i; [a: (1)
...., 2 (K)
R x... RR.x

---

Published as a conference paper at ICLR 2026
such that Σ k==1 Is = N 0 and 1. 2 1 for all Z_. We define the index 12 Σ. -==1 I, for
K
+........ ......,kh 1,,, select Pj as the matrix with zero entries, except for (Pz)t,+4, = for I = 1,..., Eamk+ Lg and let P3 = 0 for j = -k-1+ I,..,,, Then, the map has the echo state property for any N E ENt.
k = 1..... K. For = 1, E (1.....4), and = 2...., -, E
j
Notice that Lemma 4.7 provides the echo state property by imposing a finite memory of K - time steps on the reservoir. Let Da C RR Bm R be compact. For readout W E RRxN denote
Yt W xt;
(16)
the output process associated to the recurrent QNNs
4 and 14
Our next result proves universality
of RQNNs. The proof is provided in Appendix C.3
Theorem 4.8. Let U: (Da)² + (Bm)² be a causal and time-invariant filter that satisfies the fading memory property (that is, it continuous with respect to the product topology). Then. for any
& > O there exist n, N E N, preprocessing matrices P1 Pi..... : PN E TRNNN readout W & RRxN, and circuit parameters A @ such that the RQNN (14) has the echo state property and the filter pa
Uw: (Da)a- + (Bm) )- associated to the output process (16) satisfies
sup sup [[U(2), Uw (=)e]l S. =E(D,)²-
(17)
5 CONCLUSIONS
Approximation bounds and universality properties are part of the theoretical cornerstone machine learning models. While some studies have addressed the question of universality for QRC models, the combination of the two had not previously been explored in the context of recurrent QNNs. In this paper, derived approximation bounds and universality statements for recurrent QNNs based on the circuit implementation presented in Gonon & Jacquier (2025), which is compatible with hardware deployment and whose implementation with Rydberg atoms has been already discussed in AAarwal et al.] (2024). This circuit uses a uniformly controlled quantum gate to apply multi- controlled rotations
of control and target qubits, and has been recently shown that can be efficiently implemented (Zindorf & Bose 2024 Silva et al., 2024 Zindorf & Bose 2025).
To prove our results, we first derived approximation bounds for the static version of the QNN and its derivatives. These results are used in Theorem 4.6 to provide filter approximation bounds that show that RQNNs are able to uniformly approximate the filters induced by any contracting Barron- type state-space system. Finally, Theorem (.8 extends this universality property to the much larger category of arbitrary fading memory, causal, and time-invariant filters. In this last result, neither Barron-type integrability nor contractivity conditions are needed for the target filter. While our results apply to variational systems in which all parameters are trainable, they pave the way for results on quantum reservoir systems in which some parameters in the recurrent layer are randomly generated and only the output layer weights are tuned. Which strategy is best in terms of speed and accuracy will depend the number of blocks of the circuit, the intrinsic noise of the hardware, and the target task. Future research will focus on implementing and comparing the variational and reservoir approaches.
This work paves the way for extending the theoretical analysis of QRC models beyond the state- affine system (SAS) paradigm (Martínez-Peña & Ortegal 2023). It is important to understand in which situations the feedback approach is preferable to other protocols. Questions such as the
exponential concentration of observables (Sannia al. 2025 Xiong al. 2025) and the suitability
of QRC models for learning quantum temporal tasks (Tran & Nakajima 2021 Nokkala 2023) are fundamental to discerning the conditions that render QRC models more useful than classical machine learning approaches.
While our paper obtains approximation bounds for Barron-type sate-space systems, an important direction of future research will consist in studying approximation error rates for systems with high degrees of roughness or non-contractive dynamics. Furthermore, our paper focuses on approxima- tion properties of RQNNs. Gradient- -baed training approaches for optimizing RQNN parameters have been proposed, e.g.. in Bausch (2020): Li et al. (2023); [Siemaszko et al. (2023). ) Quantum cir-
cuit training may face Barren plateaus McClean al. (2018)) 8); Larocca et al. (2025). , flat parameter optimization landscapes for large number of qubits. Developing efficient training algorithms and studying these effects in detail will be further important direction for future research.
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
The authors acknowledge partial financial support from the School of Physical and Mathematical Sciences of the Nanyang Technological University through the SPMS Collaborative Research Award 2023 entitled "Quantum Reservoir Systems for Machine Learning.. RMP acknowledges the QCDI project funded by the Spanish Government. JPO wishes to thank the hospitality of the Donostia International Physics Center and LG and RMP that of the Division of Mathematical Sciences of the Nanyang Technological University, during the academic visits in which some of this work was developed.
REFERENCES
Junaid Aftab and Haizhao Yang. Approximating korobov functions via quantum circuits. arXiv preprint arXiv:2404. 14570, 2024.
Ishita Agarwal, Taylor L Patti, Rodrigo Araiza Bravo, Susanne F Yelin, and Anima Anandkumar. Extending quantum perceptrons: Rydberg devices, multi-class classification, and error tolerance. arXiv preprint arXiv:241 1.09093, 2024.
Osama Ahmed, Felix Tennie, and Luca Magri. Optimal training of finitely sampled quantum reser- voir computers for forecasting of chaotic dynamics. Quantum Machine Intelligence, 7(1):1-16, 2025.
Israll F Araujo, Ismael C Araújo, Leon D da Silva, Carsten Blank, and Adenilton da Silva. Quan- tum computing library. 2023. httts://githuk com/ htt.cccomcmcco/e qclib/
Juan Miguel Arrazola, Olivia Di Matteo, Nicolás Quesada, Soran Jahangiri. Alain Delgado, and
Nathan Killoran. Universal quantum circuits for quantum chemistry. Quantum, 6:742, 2022.
Andrew R. Barron. Neural net approximation. In Yale Workshop on Adaptive and Learning Systems, volume 1, pp. -2, 1992.
Andrew R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory, 39(3):930 1993. ISSN 0018-9448. doi: 10.1 (0.1109/18.256000. URL htt.::::/doo..gg///00 111.1....1111...11 256500
Andrew R. Barron and Jason M. Klusowski. Approximation and estimation for high-dimensional deep learning networks. Preprint, arXiv 1809.03090, 2018.
Johannes Bausch. Recurrent quantum neural networks. Advances in neural information processing systems, 33:1368 13:1-68--2 1379, 2020.
Ville Bergholm, Juha Vartiainen, Mikko Möttönen, and Martti M Salomaa. Quantum circuits with uniformly controlled one-qubit gates. Physical Review A A-ttomic, Molecular, and Optical Physics, 71 (1550005 2005.
S. Boyd and L. Chua. Fading memory and the problem of approximating nonlinear operators with 1985.
Volterra series. IEEE Transactions on Circuits and Systems,
Rodrigo Araiza Bravo, Khadijeh Najafi, Xun Gao, and Susanne F Yelin. Quantum reservoir com- puting using arrays of rydberg atoms. PRX Quantum, 3(3):030325, 2022.
Jiayin Chen and Hendra I Nurdin. Learning nonlinear input- maps with dissipative quantum systems. Quantum Information Processing, 18:1-36, 2019.
Jiayin Chen, Hendra I Nurdin, and Naoki Yamamoto. Temporal information processing on noisy quantum computers. Physical Review Applied, 14(2):024065, 2020.
Naomi Mona Chmielewski, Nina Amini, and Joseph Mikael. Quantum reservoir computing and risk bounds. arXiv preprint arXiv:2501.0866 2025.
Saud Čindrak. Brecht Donvil, Kathy Lüdge, and Lina Jaurigue. Enhancing the performance of quantum reservoir computing and solving the comp-lxiyy problem by artificial memory re- striction. Physical Review Research, 60 6(1):01305) 2024.

---

Published as a conference paper at ICLR 2026
Samudra Dasgupta, Kathleen E Hamilton, and Amab Banerjee. Characterizing the memory capacity of transmon qubit reservoirs. 2022 IEEE International Conference on Quantum Computing and Engineering (QCE). pp. 162-166. IEEE, 2022.
Gerald B. Folland. Introduction to Partial Differential Equations: Second Edition. Prince- doi: (oi::100151339786991113333 URL https://doi. org/ 11w1151199777 1511/978066112133333
ton University Press, 2020. ISBN 9780691213033.
Giacomo Franceschetto, Marcin Plodzień, Maciej Lewenstein, Antonio Acfn, and Pere Mujal. Har- nessing quantum back-action for time-series processing. arXiv preprint arXiv:2411.03973 2024.
Jorge Garcfa-Beni, Gian Luca Giorgi, Miguel C Soriano, and Roberta Zambrini. Scalable photonic platform for real-time quantum reservoir computing. Physical Review Applied, 20(1):014051, 2023.
Lukas Gonon. Random feature neural networks learn black-scholes type pdes without curse of di- mensionality. J. Mach. Learn. Res., 24(189):1-51, 2023. URL URW http:httpt/mmlll wttpp/appssa.../////////
v24/21- 0987 html
Lukas Gonon Deep neural network expressivity for optimal stopping problems. Finance and Stochastics, 28:865-911 2024.
Lukas Gonon and Antoine Jacquier. Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs. Preprint, arXiv 2307. 12904; to appear IEEE TNNLS, 2025.
Lukas Gonon and Juan-Pablo Ortega. Reservoir computing universality with stochastic inputs. IEEE Trans. Neural Netw. Learn. Syst., 31(1):000-112, 2020. ISSN 2162-237X,,162-2388.
Lukas Gonon and Juan-Pablo Ortega. Fading memory echo state networks are universal. Neural Netw,, 138: 13-110, 2021.
Lukas Gonon, Lyudmila Grigoryeva, and Juan-Pablo Ortega. Risk bounds for reservoir computing. J. Mach. Learn. Res., 21:Paper No. 240, 61, 2020. ISSN 15333-493
Lukas Gonon, Lyudmila Grigoryeva, and Jubn-Pablo Ortega. Approximation bounds for ran- dom neural networks and reservoir systems. Ann. Appl. Probab., 33(1):28 69, 2023. ISSN 1050-5164,2168-8733
doi: 10.1214/22-ang aa888...
URL https:/ http::///co.orggg/// 1214/
22 22-aap1806,
Lyudmila Grigoryeva and Juan-Pablo Ortega. Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems. Journal of Machine Learning Research, 19(24):1-40, 2018a. URL /tt.ttpp//aciii. wwtp://..rgaab///////aaa
arxiv.
00754
Lyudmila Grigoryeva and Saan- Ortega. Echo state networks are universal. Neurul Networks, 108:495-508, 2018b.
Lars Hörmander. The analysis of linear partial differential operators I. Springer, second edition edition, 1990.
Kurt Hornik. Approximation capabilities of muitilayer feedforward networks. Neural Networks, (1989):251-257, 1991. doi: 10.101 1111166089936080090 19000--T..
Fangjun Hu, Saced Khan, Nicholas Bronn, Gerasimos Angelatos, Graham E Rowlands, Guil- hem Ribeill, and Hakan E Türeci. Overcoming the coherence time barrier in quantum machine learning on temporal data. Nature Communications, 15(():7491, 2024.
Kaito Kobayashi, Keisuke Fujii, and Naoki Yamamoto. Feedhack-driven quantum reservoir com- puting for time-series analysis. PRX Quantum, 5(4):040325, 2024.
Milan Korjjača, Hong- Ye Hu, Chen Zhao, Jonathan Wurtz, Phillip Weinberg, Majd Hamdan, Andrii Zhdanov, Sergio Cantu, Hengyun Zhou, Rodrigo Araiza Bravo, et al. Large-scale quantum reservoir learning with analog quantum computer. arXiv preprint 24520.2407.02553 2024.
12

---

Published as a conference paper at ICLR 2026
Tomoyuki Kubota, Yudai Suzuki, Shumpei Kobayashi, Quoe Hoan Tran, Naoki Yamamoto, and Kohei Nakajima. Temporal information processing induced by quantum noise. Physical Review Research, 5(2):023057, 2023.
Martin Larocca, Supanut Thanasilp. Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J Coles, Lukasz Cincio, Jarrod R McClean, Zoë Holmes, and Marco Cerezo. Barren plateaus in variational quantum computing. Nature Reviews Physics, pp. 1-16, 2025.
Michel Ledoux and Michel Talagrand. Probability in Banach Spaces. Springer Berlin Heidelberg, 2013.
Yanan Li, Zhimin Wang, Rongbing Han, Shangshang Shi, Jiaxin Li, Ruimin Shang, Haiyong Zheng, Guoqiang Zhong, and Yongjian Gu. Quantum recurrent neural networks for sequential learning. Neural Networks, (66:148-161) 2023.
Chen-Yu Liu, En-Jui Kuo, Chu-Hsuan Abraham Lin, Jason Gemsun Young, Yeong-Jar Chang, Min- Hsiu Hsieh, and Hsi-Sheng Goan. Quantum- train: Rethinking hybrid Antantum-classice. machine learning in the model compression perspective. Quantum Machine Intelligence. 7(2):80, 2025.
Manjunath. Stability and memory-loss go hand-in-hande three results in dynamics & compu- tation. Proceedings of the Royal Society London Ser. A Math. Phys. Eng. Sci., 476(2242):1-25, 2020. doi: 10. (099///pa.2000.00555 URL httt:htt///aaee org/ /tw//////////ooo//e/////e//eaae/ abs 2001 007 66
Rodrigo Martínez-Peña and Juan-Pablo Ortega. Quantum reservoir computing in finite dimensions. Physical Review E, 107(3):035306, 2023.
Jarrod McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Bar- plateaus in quantum neural network training landscapes. Nature communications, 9(1):4812, 2018.
Eduardo Miranda and Hari Shaji. A quantum reservoir computing approach to computer-aided music composition. Academia Quantum, 2(2), 2025.
Zoubeir Mlika, Soumaya Cherkaoui, Jean Frédéric Laprade, and Simon Corbeil-Letourreauu User trajectory prediction in mobile wireless networks using quantum reservoir computing. IET Quan- fum Communication, 4(3):125-1355 2023.
Riccardo Molteni, Claudio Destri, and Enrico Prati. Optimization of the memory reset rate of quantum echo-state network for time sequential tasks. Physics Letters A, pp. 128713, 2023,
Tomoya Monomi, Wataru Setoyama, and Yoshihiko Hasegawa. Feedback-enhanced quantum reser- voir computing with weak measurements. arXiv preprint arXiv:2503 2025.
Mikko Möttönen, Juha Vartiainen, Ville Bergholm, and Martti M Salomaa. Quantum circuits for general multiqubit gates. Physical review letters, 93(13): 2004.
Mikko Mottonen, Juha Vartiainen, Ville Bergholm, and Martti M Salomaa. Transformation of quantum states using uniformly controlled rotations. arXiv preprint quant-ph/0407010. 2004.
Pere Mujal, Rodrigo Martínez-Peña, Johannes Nokkala, Jorge Garcfa-Beni, Gian Luca Giorgi, Miguel C Soriano, and Roberta Zambrini. Opportunities in quantum reservoir computing and extreme learning machines. Advanced Quantum Technologies, 4(8):2100027. 2021.
Pere Mujal, Rodrigo Martínez-Peña. Gian Luca Giorgi, Miguel C Soriano, and Roberta Zambrini. Time-series quantum reservoir computing with weak and projective measurements. npj Quantum Information, 91)):1 2023.
Jakob Murauer, Rajiv Krishnakumar, Sabine Tomnow, and Michaela Geierhos. Feedback nections in quantum reservoir computing with mid-circuit measurements. arXiv preprint 2233: 2025.
Johannes Nokkala. Online quantum time series processing with random oscillator networks. Scien- fific Reports, 13(1):7694, 2023.
13

---

Published as a conference paper at ICLR 2026
Johannes Nokkala, Rodrigo Martinez-Peih Peña, Gian Luca Giorgi, Valentina Parigi, Miguel C Soriano, and Roberta Zambrini. Gaussian states of continuous- variiau quantum systems provide universal and versatile reservoir computing. Communications Physics, 4(1):53, 2021.
Juan-Pable Ortega and Florian Rossmannek. State-space systems dynamic generative models. Proceedings of the Royal Society A, 481(2300)))024000888 2025a. ISSN 1471-2946. doi: 10. 1098/spa.2024..3000 URL /ttpo/////////a.../o//////0////0
2024. 0308
Juan-Pablo Ortega and Florian Rossmannek. Stochastic dynamics learning with state-space systems.
Preprint, 2025b.
Luan-Pable Ortega and Florian Rossmannek. Echoes of the past: unified perspective on fading memory and echo states. Preprint, 2025c.
Iris Paparelle, Johan Henaff, Jorge Garcia-Beni, Emilie Gillet, Gian Luca Giorgi, Miguel C Soriano, Roberta Zambrini, and Valentina Parigi. Experimental memory control in continuous variable optical quantum reservoir computing. arXiv preprint arXiv:2506.07777 2025.
Daniel K Park, Francesco Petruccione, and June- Koo Kevin Rhee. Circuit-based quantum random access memory for classical data. Scientific reports, 9(1)3994,, 2019.
Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I. Latorre. Data re-uploading for a universal quantum classifier. Quantum, 4:226, February 2020. ISSN 2521-327X. doi: 10. 2233 1/q- 2020-02-00-0226 URLaatss:://aag wrg/o0gg//oooo 2232222333222222 2020 02-06-229
Philipp Pfeffer, Florian Heyder, and Jörg Schumacher. Hybrid classiiall reservoir comput- ing of thermal convection flow. Physical Review Research, 4(3):0331 162,00331,,,1000 2022.
Philipp Pfeffer, Florian Heyder, and Jörg Schumacher. Reduced-order modeling of two-dimensional turbulent rayleigh- bénard flow by hybrid quantum-classical reservoir computing. Physical review research, 544)0433 2023.
Jun Qi, Chao-Han Huck Yang, Pin- Yu Chen, and Min-Hsiu Hsich. Theoretical error performance analysis for variational quantum circuit based functional regression. npj Quantum Information, (1):4, 2023.
Antonio Sannia, Gian Luca Giorgi, Stefano Longhi, and Roberta Zambrini. Skin effect in quantum neural networks. arXiv preprini arXiv:2406. 141 12, 2024a.
Antonio Sannia, Rodrigo Martínez-Peña. Miguel C Soriano, Gian Luca Giorgi, and Roberta Zam- brini. Dissipation resource for quantum reservoir computing. Quantum, 8:1291, 2024b.
Antonio Sannia, Gian Luca Giorgi, and Roberta Zambrini. Exponential concentration and symme- tries in quantum reservoir computing. arXiv preprint arXiv:2505. 10062, 2025.
Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expres- sive power of variational quantum-manchin--llarnng models. Phys. Rev. A, 103:032430, Mar 2021. doi: 10. 20.100//hhyseea..10...22222 I1C3/PhysRevd. 103. 032430
URL https://ling: aps opo.o////ooo0/0/0000000/0//
Mirela Selimović, Iris Agresti, Michal Siemaszko, Joshua Morris, Borivoje Dakić, Riccardo Albiero, Andrea Crespi, Francesco Ceccarelli, Roberto Osellame, Magdalena Stobińska, et al. Experimen- tal neuromorphic computing based on quantum memristor. arXiv preprint arXiv:2504. 18694, 2025.
Michal Siemaszko, Adam Buraczewski, Bertrand Le Saux, and Magdalena Stobińska. Rapid train- ing of quantum recurrent neural networks. Quantum Machine Intelligence, 2023.
Jefferson DS Silva, Thiago Melo D Azevedo, Israel F Araujo, and Adenilton da Silva. Lin- ear decomposition of approximate multi-controlled single qubit gates. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2024.
Michele Spagnolo, Joshua Morris, Simone Piacentini, Michael Antesberger. Francesco Massa, An- drea Crespi, Francesco Ceccarelli, Roberto Osellame, and Philip Walther. Experimental photonic quantum memristor. Nature Photonics, 16(4):318-33 2022.
14

---

Published as a conference paper at ICLR 2026
Yudai Suzuki, Qi Gao, Ken C Pradel, Kenji Yasuoka, and Naoki Yamamoto. Natural quantum reservoir computing for temporal information processing. Scientific reports, 12(1):1-15, 2022.
Quoc Hoan Tran and Kohei Nakajima. Learning temporal quantum tomography. Physical review letters, 127(26):260401, 2021.
Hassler Whitney. Analytic extensions of differentiable functions defined in closed sets. Transactions of the American Mathematical Society, 36(1):63 1934.
Weijie Xiong, Zoë Holmes, Armando Angrisani, Yudai Suzuki, Thiparat Chotibut, and Supanut Thanasilp. Role of scrambling and noise in temporal information processing with quantum sys- tems. arXiv preprint arXiv:2505. 2025.
Dimitry Yarotsky. Error bounds for approximations with deep ReI.U networks. Neural Networks, 94:103-1 2017.
Toshiki Yasuda, Yudai Suzuki, Tomoyuki Kubota, Kohei Nakajima, Qi Gao. Wenlong Zhang, Satoshi Shimono, Hendra I Nurdin, and Naoki Yamamoto. Quantum reservoir computing with repeated measurements on superconducting devices. arXiv preprint arXiv:2310.06706, 2023.
Zhan Yu, Qiuhao Chen, Yuling Jiao, Yinan Li, Xiliang Lu, Xin Wang, and Jemry Zhijian Yang. Non-asymptotic approximation error bounds of parameterized quantum circuits. In The Thirty- eighth Annual Conference Neural Information Processing Systems, 2024. URL https://
Ben Zindorf and Sougato Bose. Efficient implementation of multi-controlled quantum gates. arXiv preprint arXiv:2404.0222 2024.
Ben Zindorf and Sougato Bose. Multi-controlled quantum gates in linear nearest neighbor. arXiv preprint 1xXiv:25506006995 2025.
APPENDIX
A QUANTUM RESERVOIR COMPUTING PROTOCOLS
For learning problems with temporal structure, quantum reservoir computing (QRC) has emerged as a promising approach for exploiting noisy intermediate-scnne quantum (NISQ) technologies. These include ion traps, nuclear magnetic resonance, cold atoms, photonic platforms, and superconducting qubits (Mujal et al. 2021 When implementing QRC models experimentally, it is necessary to consider the backaction and statistical effects introduced by quantum measurements. Backaction refers to the modification of quantum state after monitoring, also known as wavefunction collapse. Due to the probabilistic nature of quantum theory, measurements must be repeated compute the expected values of observables, which introduces a statistical component in all these methodologies. Most available experimental implementations rely on the quantum computer paradigm (Dasgupta
et al. 2022,
Mlika et al.) 2023, Suzuki et al., 2022 Yasuda et al. 2023 Chen et al. 2020; Kubota
al.
2023 Molteni et al.) 2023 Pfeffer et al.) 2022 Ahmed et al. 2025 Hu et al., 2024, Miranda
& Shaji 2025). However, there is an increasing interest in extending this technique new settings,
such as optical pulses (García-Bece et al. 2023 Paparelle et al. (025). Rydberg atoms (Bravo et al,
2022
Korjača et al. 2024). and quantum memristors (Spagnolo al. 2022 Selimović et al. 2025)
Early QRC model implementations relied on the simplest possible approach, namely, the restarting protocol (Dasgupta al. 2022 Suzuki et al. 2022 Kubota al. 2023 Chen et al.) 2020 Molteni
et al. 2023). In this approach, the expected values of observables are obtained by rerunning the algorithm from the first time step each subsequent time step. This avoids the backaction effect of quantum measurements. However, the complexity of this protocol scales quadratically with the length of the input sequence, making very time-consuming. A faster alternative the rewinding Čindrak protocol (Mujal al. 2021 2indaCindra et al. 2024), where the fading memory of the quantum reservoir
is exploited to restart the algorithm with a fixed window of past time steps. This reduces the com-
plexity of the algorithm to linear in terms of input length. Originally proposed in Chen et al. 2020
this protocol has thus far only been considered numerically (Mujal et al. 2023 Čindrak al. 2024
15

---

Published as a conference paper at ICLR 2026
Both the restarting and rewinding protocols repetition of previous time steps reproduce the dynamics of the theoretical model and avoid the disruptive effect of projective measurements used to extract output information. This comes the cost of halting the quantum dynamics at each time step and the need to buffer the input sequence. Consequently, these approaches lack one of the most important features of traditional reservoir computing, namely, the ability to process information in real time.
New protocols have been proposed circumvent this problem. The online protocol (Mujal al, 2023 Franceschetto et al. 2024) uses weak measurements to find a balance between erasing and
extracting information. Mid-cireuit measurements and reset operations (Hu al. 2024) can split the reservoir into two parts: memory and readout. The memory retains previous inputs, while mea- surements only affect the readout part. The feedback protocol (Kobayashi et 2024), which can be traced back to QRC with quantum memristors (Spagnolo et al., 2002) and hybrid QRC techniques (Pfeffer af., 2022 2023), reinjects the measured observables at each time step as parameters of an input quantum channel. This ensures that no backaction effects are present and that past input information is preserved. Note that in order to compute the observables in real time, these proto- cols all require several copies of the system to be in parallel. Furthermore, these protocols be combined with each other. For instance, the feedback protocol has been combined with both the online protocol (Monomi al. 2025) and with mid-cireuit measurements and reset operations (Murauer et al. 2025
Of all these approaches, the feedback protocol presents some particularly interesting features. First, the feedback protocol enables us to compute the expected values of observables from a single copy of the system by repeating one time step only. If only few copies of the system are available, this reduces the experimental time overhead for real-time applications compared to other approaches. Second, in contrast to previous QRC models, where an erasure mechanism added provide fun- damental properties such as the echo state property, simple unitary operations can provide these properties (Kobayashi et al. (2044). Finally, the dynamical equations of quantum reservoirs under the feedback protocol go beyond the standard state-affine system (SAS) paradigm of QRC models (Martínez-Panee Ortegal 2023). These properties make the feedback protocol a promising candi- date for exploring QRC applictioon.
B PROOFS FOR SECTION 4.
B. PROOF PROPOSITION 4.1
Proof. The proof a modification of the argument used to obtain GGoon & Jacquier 2025 sition 1). Recall that
Propo-
Fn.0 R.j
(x, (x,2) = : 2222ppp.00 (x, (x
+ +P (x, (z, E x Rd.
(18)
Fix (x, 2) E R. x Rd and (I...., N) and write Pm == pp.0' (x, 2) for m E {0, 1, 1,2,).. To prove the representation (7). let us first calculate Pm..
As first step, write
R-1
UV 10) pm
(0))
Vn Σ
0|41)
n-1
Vn ΣΣ
((++))
@ U (++)
147 + k).
141.1
0=
16

---

Published as a conference paper at ICLR 2026
Thus, for (0,1,2,3) we have
P
(1i + ml UV (0)
(40 ml
0
u (
141 +4+00 k)
vni
2-111
1=0 k=
1
U 1)
m+1.1
Next, we may calculate
Bi + +at (1),2)
10/] @ h11 h11 [u]]] 11, 1111 1 1111"11111111111 cos
2 cos
2
bi. + a (x, 2)
(u 1 2.1 juf") (0/-)1/,1 2,,1 sin
2
A
bid + atd (x, x)
( @ o 33.1 ju 12. [ug] 11,1 cos
sin
2
7
a
(0,
[u]d & u 111
11
(u] 22.1
2,1 = isin
sin
2
2
and thus
n2
2
bi + a
(ar, z)
2
Fo
COS
cos
2
2
N P1
2
bi a
(2x, 2)
sin
co8
2
n P2
a
(ax, 2)
COs
sin
n
2
2
a
(x,
2
P3
sin
Therefore, using cy) 2
cos( cos(2y) +1
obtain
2
Po P1
Co8
1i. a. (ar, z)
2
1
Σ
n
cos (bi a (x, z)),
n
2
2n
1=1
n
A
1
n
Pu P2
cos
2 +
cos (2
12
2n
Putting all together we obtain, for any given R > that
F.n (z,
(z, 2) R 2.( (x, 2) pn. (x, 2)]
R [1 4P. (Po + P1) (Po P2]
Reos (1) cos (bis + a. (x, z))
B.2 PROOF OF PROPOSITION 4.2
Proof. Let j E (I...., N) be fixed. As in the proof of Proposition in Gonon & Jacquier 2025 we may use the Fourier inversion theorem to represent
Fx = JRN x JR
{
(E) (2)
17

---

Published as a conference paper at ICLR 2026
which we may rewrite as, with (51, E2),
(x) -Je JRNNRd {cos (2T( 2) R)Re(F)(E)) cos x) 2 Im [E) dg
(19)
The hypothesis 0,F, E F implies that SRN XRU (E) < cc. Hence, applying differentiation under the integral sign yields
(2,,(2,2)) -2x { JRNxR
{Eesin (2T(x,) {)Rc(F + & sisin (2T(z, 2) - γ 2)) Im [F] [F) de.
(20)
Next, consider the random function
(x, 2) =
W. cos B, + (x,
(21)
1=1
for randomly selected weights W1,... W.. Bi...., Bn and A,,..., An valued in IR R and
R-V x IRé, respectively (for notational simplicity we leave the dependence on j implicit here). The distributions of these random variables are chosen as follows. First, we ZI...., Zn be iid. Bernoulli random variables with
P(Z. 1)= JRNNRR XR Re (E)]lE)
P(Z: 0) JRNXRA XAN [mm [m
(22)
JEN x (E)
JRN x (E)
and VRe and Pim be the probability measures on R x Red with densities
Re
Im
and
(23)
Je xRd Re [e (E)
JEN xR
Im (E)
respectively. In case Jg*xe x Re (E) = 0, instead we choose for Vhee an arbitrary probability measure and analogously for Vim in case JgN x [Im (E)](d = 0. Next, let UR U (resp. Ulm ..... U!m) be i.i.d. random variables with distribution PRee (resp. vim) and assume that U.m ..., Ulm U ...., URe Z Z....., Zn are independent. With these preparations, we are now ready to define the weights in 21
n
A; = 2m R
π B, = 2 2 Zz),
Re F (UR)
(U!m)
W, = [7-)
Z: +
(1 2z)
Re F (U)))
Im (U)
with the quotient set zero when the denominator is null.
Our goal now is to estimate
N+d
N+d
F, - 12 + Σ
|a,F - 1
E [IF,, - 2 2
+ Σ
[1a,F, - 0,4 (2
(24)
18

---

Published as a conference paper at ICLR 2026
by estimating the summands separately. To achieve this, first compute B[0 Z. 2)] and (z 2)}. Indeed, inserting the definitions, using independence and representation 19 yields
| 2)
E cos B1 + A1 x))]
Re[F,] [e)
Z1 Im [m))
1
(1 21)
[Ra[F]]( (UR))]
Im [m))
π 2
Z1) + 2 ( + + -.) (x) x))
cos
Re(E] | (U
=115
[P(Z) 1)
[UR()](UR()] cos (T URe (xx z))
Im (U(")
+P(Z1 = 0)
[mm| [B((!!)) cOs ( Um (x, (e(
Re Fe(E))E) cos (z, x)){ + /
Im (nnFF1(5) cos(
+ 2 (x, z))dE
JRN XR
JRNXR xBd
2
(F,
Analogously, using the representation 20 for the partial derivative F, instead, obtain
(20 2)] = -E[W, sin(B1 + A1 2)]]
Re [
[URe)
-T [P(Z) = 1)1E
Re sin (2 (xx
|Re| [Re[F]] (UR))
+P((Z, O)T
[m[P,] (Ulm)
U sin
π
2n U!m (a,, (11,2))
[m (Umm)
2
=-2 (fr JNN XRd
/ xRa
Re sin (ax, 2))dg
(.Im( sin( + 2 (xx, 2))
O (ar, 2).
(25)
Therefore, we may estimate the first expectation (24) as follows:
E
[1,
E
[f
[x, z) (x =))² dz)
1 NN xxa
V10 =)]u(dæ, dz)
RN XB*
1
V
W; cos B, + Ar (z, x)) uldz, dz)
n²
JRN xRd
1
V W cos B1 + A x)) u(de, dz)
JRN
1
8
E
(Wr cos( Bi + A (x. z))) u(dx, dz)
11.
RN X.d
1
- 1F
$
[W]]
n
(26)
19

---

Published as a conference paper at ICLR 2026
For the partial derivatives, obtain analogously
= /
E |10,13 - 0,
-0, (1(2(())
v10,4, 2)), dz)
RN XRd
1
{
V
WA AR, sin BB + A (z, 2)) a plde, dz)
n² JRN XR=
[
JRN JRNNRA XRa
I[WIA sin B1 + A1 (x, =))] pidx, dz)
(27)
Σ T.. JRN xB
(W241,1 (in B1 A1 (z, z)) 2))) / p(dx, dz)
1 11111
4m
$ E [W? A
[4
1Fyll
BN XB
where we used that E' [Ai
= 4 4
SEN x (E)
In particular,
26 and 27 imply that there exists scenario E Z such that (2,
E (.(w) cos( (:(w) Aw(w) (a, 2)) satisfies
[F 101122
|0,F 0,07 (z(N))
C
(28)
with C |15 + 4x Fyll2 SRN x EN+o < (E) Finally, (0) ......... ON) can then be
constructed by setting 0 (A,(w), Be(w), arccos( W.(w) F and so the proposition follows.
11= which guarantees that by
R
B.3 PROOF OF COROLLARY
The proof of this corollary requires the following lemma, which extends Lemma 4.10).
Gonon 20024
Lemma B.1. Let d, n, E N, let MI, M2 > O, let be non-negative random variable, and let Yi...., Yn be i..dd. IR -valuee random variables. Suppose B[U] I M1 and B||Y M2. Then
P U 3M,,
max I.....
| s (3nM2)
> 0.
Proof. The proof mimics that of in Gonon 2024 inequality for 1 by the more general version:
Lemma 4.10) by replacing the use of Markov's
Y 9
P([|
3nM2
3n
Proof of the corollary. The corollary follows by replacing the argument leading to 28 of Proposition 4.2 by Lemma B. and by noticing that
the proof
/
E
A 19
(E)
JRNKRd
JN xRd
B.4 PROOF OF PROPOSITION 4.4
Proof. It follows by combining the proof of Proposition 4.2 with the proof of Theorem Gonon & Jacquier 2025). More specifically, the same proof can be used for Proposition 4.2 except that
20

---

Published as a conference paper at ICLR 2026
need replace the (n) enror bounds in (26 and (27) by uniform bounds. For 26 we can
follow precisely the proof of Theorem 3 in Gonon & Jacquier (2025) to obtain
C
| - -Pa c0.M
S
(29)
Vi
with Cg 2( + + (N + d) F5lli (FRM XRU
EiSt (EI 1/2 Next, we
Also in this case, we may proceed as in the proof of Theorem 3 in Gonon & Jacquier 2025 and apply the same estimates to
turn to the derivatives, that is, we aim to estimate
1g F Or.P
Do.M
the random variables (x,), = W.A. sin( 13 + A; (x, 2 2)) Let é1.- En be i.i.d. Rademacher
random variables independent A = (AI.... An) and B =
B,). Symmetrisation and
independence then yield
1...R. R. D,F,
= E
sup (x)=)E|- M,MMM++
(Ul,(,,>) ((U
oc.M
1
$ 21E
sup
U (m,,)
(0x*) (M.M)) N+d
71
2 E
sup
a, sin(b; a, (2,
(0,)))| (w. M.A.NN+
(w.abb)((WAA.B)
Now fix a = (ai,..., an) E (R. Rdy,, b (bi,..., bn) E R", w = (wi....,,,). E IR" and denote
T = [(w;ar,((b; + a; (x, 2)))=,,,,,... (x,z) E [-M, M,M
2 (x) wiay, sin( w,a,k
R
for i 1... 1.....7. n. Then, using the definitions the first step, the comparison theorem (Ledoux
& Talagrand 2013 Theorem 4.) in the second step (note (:(0) 0 and @ is 1-Lipschitz), and standard Rademacher estimates (see, e.g.. Gonon (023)), ), we obtain
sup (0.=)E[- M
e,waai sin + a, (xc 2))
71 1=1
E
sup tET
eigi(li)
<
2E
sup tET
Eili
i=
2E
sup
E (waar (b, + a (æ, z))
1=1
S 2E
+ 2E
sup (m.))((-M.N))
(xt
a,
1=1
1/2
N+d
1/2
2
2M
+
w a
n
1=1
Putting everything together, we obtain
1/2
N+d
1/2
0 R.i
-3,13
5 21E
2
Σ W.² A. 13
2M
Σ
Σ
W?
W?A
00.M
1=1
i=1
N+d Σ
1/2
E
[[WAAA 1/2
+ +MI + d)//2
E [W?2APA443]
Vn
Vn
21

---

Published as a conference paper at ICLR 2026
with Cg = - (JEN (ATE(E))EE)
1/2
)
+ 4M + d2)
Here, the last estimate follows from the inequality
/ |RNX
E A3 I x 11
and
E [W,44,,44,1 16T 11 f
(E)
JRNRR XB=
Overall, we obtain (9) with Cj =k=0 Oge. chosen as
Cj = 2(x + 1)11 (8+ + 4Th d) 23
+N 1/2 1
B.5 PROOF OF COROLLARY 4.5
Proof. First, extending the proof of Corollary in Gonon Jacquier (2025), we show that F, can
be approximated on up to error 5 in C' by function in C. T Rd Indeed, first let
M 0 be such that C [-M, MMM, Mf Then, classical approximation results (see, e.g., Whitney
1934 Lemma 5) imply that there exists smooth function h: RV x R + R such that
sup (x, - h(x, x) + FV x) V Vh(x, 2) =) (m.²)EX'
(30)
Without loss of generality we may assume that h E C (IR x R.). Otherwise, we multiply h with cutoff function vi O (R- Rd) which equal in an open set with X c U (see, e.g., Hörmander 1990, Theorem 1.4. 1); thereby preserving (30).
In the next step, we now apply Proposition 4.4 to h. Since h is a Schwartz function, its Fourier transform h also a Schwartz function and thus is integrable and
J x
(1 + (
oc.
In particular, h E Fn for R > large enough and, as h is Schwartz function, also 0,h E F for all Thus, the hypotheses of Proposition are are satisfied and we obtain that there exist n E N and O such that
N+d
FR h
O.FRI R.j - 0,h
S e
c0.M
2
This estimate together with 330 then imply
sup (Z, 2) FR. (z, x)1 (F) - V (z. =||
(m,=)
sup |1 -) =) (x, Vh(x, =)]l
(m.))
+ sup
MMx, 2) -Fnl (2x 2)1 V R. (x. 2) vh(n, =)
(#,z)eX
< sup
[2, 2) - -h(x, 2) + V2 2) Vh(x, z)ll
(#.*)EX'
N+d
+ sup
FR. (x, 2) - h(x,, 2)1 + Σ
1 1 (m. - 0,h(2, 2)1
(m,=)cX
i=1
< sup
(2, - -h(x, 2)1 + | (x, - Vh(x. =)1l
(m,z)Ex'
N+d
+
Fn.0
R.j
h
+
0,F R.
1 10,N Se,
0,h
20.M
22

---

Published as a conference paper at ICLR 2026
where we used that
N+d
1/2
N+d
IVER R.j (a, z) Vh( 2| =
F0. Rg (at z) - 211(222212
0 F (a, z) -0, x)|,
since |//|| - | for all y E R2
C PROOFS FOR SECTION 4.2
C.I PROOF OF THEOREM 4.6
Proof. Choose M such that Bx Da [-M, M) M and [-R, Da |- M. MPN+ Firstly, our hypotheses on F guarantee that F satisfies the hypotheses of Proposition 4.4 there exists I E @ such that for any j E [I,...,N}.
Hence,
N+d
Cj vn
[RJ Fj
|.. RJ -d,F,
(31)
00.M
10.M
10.M
Then, for all x M, MI ,2 E Da
F
Fn.0 (zx
(x, 2)112 S F (x, 2) - V FeF(2,2)112 +
V +1 2)12
1/2
N
- (
+
(32)
S N max, N C.
+
Vn
Therefore, using that m M. V P (z,, 2)1l is the best Lipschitz-constant for F on [- M, M|* N for any given z E Da. we obtain for all x E [-M, MJN 2 E Da that
R
|| | (a: ,2) - F (a= =)12 S ||a+ 21
V Fm. max M.N-
R (ax, 2)11
N max-=1...... I.....N C
+ || 22122.
In particular, for n satisfying N2 (max, -...... 0
< n we obtain that Fu.o: BR x Da - Br
(1-Xy²
with BR- = far R. ||=|]] S RvN), J. contractive the first argument, hence the system 44 has the echo state property by Gonon et al. (2020, Proposition 1).
By the relation between the Lipschitz-consaan and the maximal norm of the Jacobian, the assump- tion [( z)1(2 X guarantees that F(,, z) is X-contractive for any E Da. Hence, we may estimate
[1)((2) - Uz))+1l ||=,- - = F (I 2,)
(iz-1; - zi>1
Frn0
< (1 21) - F (1)1-1 + F( 2x) - Fm (tx (0.- zx)
1/2
(33)
S ||| 2/-1/1+ +
pn.0
R.j
VN maxx== C
< / 11-111
nn
Iterating 33
we obtain
V maxj=1,.... Cg0
s X' 11 2.
x
Vn
(34)
J-1
VN max/=1,..... C3o
s <XVN((+RR)+
Σ
Vn
R=0
23

---

Published as a conference paper at ICLR 2026
Letting J - oc, thus arrive the bound 12
C.2 PROOF OF LEMMA 4.7
The proof of Lemma 4.7 is related to the approach introduced in Gonon & Ortega (2020) and sub-
sequently used, e.g., in Gonon al. (2023): Gonon & Ortega (2021).
Proof. We start by constructing a partition of &; as in the statement. If N = 1, we simply have i4 = [at] E R. Next, we define the reservoir vector F FRsg (F.....,FF) FFy.. Then, for (1.....41. and k = 2, 2..... K 1, E fla-1 1....,1... we have P,ã: Lt
k 1. (K) ........f 0] and P, 0 for j = 1K-1 + I. Inserting these choices
[
(k+1)
into 15 we may rewrite the dynamics as
(K) P/e 1411 (10 (k+() 11 0, 0, .I. z1), LEZ,,
(35)
for k = L...., -- and $ = Fix- (0, zz). In particular, 2. (K) Fix- l: (0, z.), which depends only on 2t, is explicitly given for all E Z_,, and for all k =I..... K we see
that a2(h) only depends on 2-11 (k+1)
..... 2.-1. Thus, (15) admits a unique solution which can be ex-
plicitly obtained from the recursion (3). that is, for all t E Z_,, we have 2; (K) -ix-r+tik (0, 2,), (K-1) Fix-a+ (1= O,...... 0). 22),
a(1) Fis, (1 (1 a(K) 0), 21). This
proves that F has the echo state property.
C.3 PROOF OF THEOREM 4.8
Proof. Without loss of generality we may assume e S 1, because proving for e S also
implies that (17) holds for e >>> 1.
Let Hy:: D2) 2- + Bm be the funetional associated to the filter U. Then, as in the proof of Gonon
& Ortega 2021 Theorem 2.1), there exists K E N and continuous function G: (Da)dK Bm
such that
sup =E(Da)*
Hz) G(z -(zzGG(zz..... :..,,..,
(36)
Moreover, e.g., by the argument in Gonon & Jacquier 2025 Ge C (RR)) Bm)) which satisfies
Corollary 4), there exists a function
sup =E(2
- |[((z)-((z))) <
(37)
Next, choose N = (K 1)d + and consider the recurrent QNN introduced in 4 Denote
05 Fn.0 R.j xz, 2) = 2R (x, 2) + Pn. (z, 2)],
(x,z) 2) E R R
(38)
= the update maps without preprocessing matrices. For 1 < i < j < N. write F
Fn.
R.j
(F .... and 4 = m + (k - 1)d for I,..., K. Define the constants
1/2
K K K-k+1
LG = max(Vd, sup =E(RdgN
[+
CG 4LG
Σ
Σ
(2LG)
(39)
Then, G E C ((R))K and the identity is smooth, Corollary guarantees that there exist nk, Rg and 0x E @d such that
4.5 (applied componentwise)
=eDa - (0, x) - x sup || V Fax. 0k Rx f 1d (0, x) - lall < Cg
sup Ox
(40)
xEDa
and (recursively). for all k = K -1,11... 1,.... , there exist nk. Rg and 0 E e such that
sup
1. R2dd2-14id, (x,x)= - æ1.ell + V / 0s 0s
Rad2-1 +1 (x, x)) - 1all
< 8
CG
(m.4)E[-R4 R1 M xD
(41)
24

---

Published as a conference paper at ICLR 2026
and there exist n1, R1 and 0₁ O such that
(1 [1 (1z K+1,......... (.+1,.....21,,)),22.)) z0) - -G(z ([z--+1,--.. x+1,,...,00) z0)1
sup (I=*++1mm 1. Ra
+ |VFRI. Ry.
- VG(Z-Kt1...., z0)l| < 4'
(42)
Without loss of generality we may choose At = Ri = Rg. since we can always replace Rg by max(Bg, Rx+1) (and hence ultimately replace Ri...., Rg by R) and absorb the change in an adjusted choice of parameters 7 (see representation 7 Moreover, by a similar reasoning we may assume without loss of generality that n = n2
ag. Indeed, otherwise we may again choose n to be the maximum of ,,,,...,. K'K, replace "I, by n and recover the same functions (7 by setting surplus terms > nig to by appropriate choice of Ths. The extra factor ". ".."
in turn, can be absorbed by modifying the choice of R.
Denote by Lg be the best Lipschitz constant for Fn.0e + Then 42 imply that La S vd + & S La for =K,... K..... 2 and Ln S suPzeRay [+V +
S Lg. In particular, La 2 max(Li,..... Lx) is a bound on the Lipschitz constant for all QNNs Fr.e. + and G. (K)
Partition d; = [ ,...,, tt E R x (R.))-11 Using the triangle inequality, we then obtain
G(z-K+1,,-.. z0) - 1
sup *++
-in. R. (10 -1...,... x (K) 0,, 2n)
sup =E(Da)(((1)
G(Z-K+1,.... 20)
G(z=KKI...., -[(( (2)
1 ................................ #
1, 200
(43)
| (2) [KK]] 20) Fu.
fm.1am (1 (2)
1:... x
(K)
0]. zo)
S LG [ (z-K+1, 20) (1t0
(K)
z0)
+ 4'
For the last norm, we write
[ (z z0) (1 (2) 1:
(K)
20)
||
2 =
K-2
2 2ー k-
x -1
(k-
2
K
2= K (10
k=
k=2
We proceed by backward induction over k to prove that for all k = K,... it holds
| 2t 1
K-k+1
8²
(2L |
Cτ
for arbitrary E Z_ Indeed, we have
2 ス-K+ - a:
(k) = 2 K-H#+ - 11.1- +11 (10 (k+1)
: .... #1,1,00 0, 0..),,,..,,0... z1)
and so for k = K it follows that
zーk+d+ 2 (k)
=1|20-
F Riix- 1t1d (0, )|
2²
=
< <
2LG
Cg
Cg
Assume that the bound holds for a fixed is E [K,...,3), 33, then for k- - 1 we estimate (with the notation fx-1 F 2
2 k k+ 1)+1 - 2
2(1-1))12
=
(k)
(K) 11-1 0,
0], 21)
2
K +h Sk- ([z K
(k+1)
(K)
0, 0), 01, 1|
+
([= (I_ (k++)
1-1
(K)
0
21) - - (k)
f1-1, 0, .0....... 0),2,)
(K)
2 5²
2²
=²
K-4+1
=²
72L
-
(k)
Σ
(2
$ Σ (2
Cg
C2
C5
C2
==1
==1
25

---

Published as a conference paper at ICLR 2026
which completes the induction. Therefore, we obtain
2-- -+.............................. 20) -1100
(K) --11
20)
Σ
2
K
(k)
k=2
K K-k+1
(2 6²
- -
$
1613
From 43
we thus obtain
G(z_K4t,...,.))) 2001
sup =E(D))K+1
(44)
I LG (E-K4I,... za) (100
x
(K)
2o)
+ 8 e
5 44" 2
Setting W to be the projection onto the first block 2 Wi,1 for m.) and putting together (36)
(that is, W has zero entries except for and (44) yields
(37
sup
sup [ - How (z)]l -
sup =E(Da)²
|u( -
=E(Da)² 1E
+ sup
[|((z) - -((z)]++
sup =E(D))(+t
G(z=k+I..... =0) -Hüw (=)
(45)
=E(R")
8 8
=c
4
4
remains to be shown that (4) has the echo state property. Recall that we partition a; [ (1)
..... 1 (K)
Rm x K-1 For k 1, (1.....41). and k 2,..., K 1.
j E (4-1 + 1,...,141 1.), select P3 as the matrix with zero entries, except for (Pz)1,44 = for
I = 1, .,. d(K k) and let P3
for j lK-1 +.... N. Then, for k 1, E {1, (
41
and k = 2.....
........ K 1,j E (4-11 + +,.....,44 J.). we have Pj = [=(+++1 ,...,..,," 2. (K) 0,................. 0] and
pj for j 1--1- + I,..., N. Then, echo state property follows by calling Lemma 4.7
Therefore, the approximation bound for the functional (45) immediately implies the corresponding
bound for the filter (17). which completes the proof of the theorem.
D CONSTRUCTION OF V
In this appendix we provide further details on the choice of V appearing in the quantum circuit. Our presentation follows Gonon & Jacquier (2025).
Generally, the matrix E Coexou can be any unitary matrix mapping (0) @n
the state (+))
vn ΣE= |4i) which, for n 2 also explicitly given (+)) v Σ (t) 100).
As V [0] 00 (b) is the only property required in the proof, many alternative choices of are possible and one may thus select the one that is most suitable from the perspective of hardware requirements or limitations.
Example One explicit example for is given by V = 2 (g) (مe - I, with
10) + (V)
(p) = /2 (1 + (0)+))
where we write [0) in place of [0) @n for brevity here. One easily checks that v - 2 (vp) -I = v
and thus vVi vfv I. Furthermore, a straightforward computation yields that
v (0) = (2 (p) (م) - ())))))0))))
00 (1 ()) + (V) (1 + (1)00 10) = =1b)
1+ (0/1)
26

---

Published as a conference paper at ICLR 2026
Construction of ( In the no 0, there is explicit construction of v) in terms of Hadamard gates acting on the control qubits. Indeed, for n 2 2. (Gonon Jacquier 2025 Lemma A.2) shows that
n
H (0)
100)
E MONTE CARLO ERROR
In practice, the empirical sampling error leads to an additional error component of order 1/V5 for S independent shots, see, e.g.. Qi et al. 2023); Liu et al. (2025). Here, outline how this Monte Carlo error could be taken into account in the present setting.
More specifically, QNNs in and 44 are defined using probabilities, rather than their Monte
Carlo estimates
Tan.0
Σ
A(n (i(+))
with (()) the measured state in the i-th shot. To obtain refined bounds incorporating the sampling er- ror, one would proceed as follows. Denote by FR F..0.. the RQNN state map with output probabilities estimated by shots, by & the associated state and by Us the associated filter.
For the state map itself, the L2-error can be directly controlled (as in Gonon & Jacquier 2025 by
1/2
E
Fr. (x,, 2)
Fn... (xr, 2 judda, dz)
xR
(fr XRd
|pmn (I., -pp. (xx, -p,.0(00, 2)
27 u(dx, dz)
1/2
2R
Σ
E
(46)
4R
Vg¹
using that [X,]
1 s
x,
Var(Xi) for i.i.d. random variables XI...., Xs. $
For the associated filter, one may proceed as follows. Firstly, (33) in the proof of Theorem 4.6 can be adapted to
Us(2); - U(z)e|l = | - æzll =
Fin p.. (2.-11 F (x xn)
< (F(x(-1,2)) - F( z1))|
F zi) - -Fm.0.
(a. ( 2z)
1/2
1/2
N
N
2
2
S > x - a
Σ
po.0.5
níj
R2j
lo.M
+
Σ
RT.j
Fj
(47)
1/2
VN max i....,, Oge
x1-1 SA
<A
Σ
F..0.S
Fu.0l
Vn
R.j
R.j
The last error term can be bounded as
1/27
1/2
E
Σ
- FR.J D0.M
<
E
e.S
Fn.0
C
co.M
VS
for suitable constant C using techniques from statistical learning theory, provided that fe is Lipschitz continuous as function of (æ, 2). Inserting this into (47) and proceeding precisely as in the proof of Theorem t.6 yields bound that incorporates also the sampling error.
27

---

Published as a conference paper at ICLR 2026
Alternatively, as the Lipschitz continuity may be hard to verify, may obtain L touud analo- gously to Theorem 4.6 as follows. First, using that the shots are independent across evaluations, we may apply (46) to estimate
1/2
E
Σ
pn... (z (
RR.j (á) zi)
1/2
N
Fm. Hi (t. 2x) 119)
<
E
Fm.0..
R.j ( (
4R
VV
VS'
where the expectations taken with respect to sampling the probabilities evaluate
T (+)-
Next, by proceeding as in may estimate
E| Us(() - U(=)e||] & X 1 - -2 11 VN max,=1..... Oj
VT.
1/2
N
+ E
F ( zt) - F.n (t) ( Zz)
(48)
$ N maaxj-1..... C3. + VN
4R
$ $>|1
vn
VS
with the expectations again taken with respect sampling the probabilities evaluate Fn.0.S
R.j (2.{1,22), 2t). In particular, taking expectations also with respect to random process (taking values in (Da)²-> and sampling at each evaluation, the estimate (88 and the same arguments as in the proof of Theorem yield the bound
1 7->
VN maxi ..... C
4R
sup ElIlU*(() (2))ll S 122
VN
(49)
Vn
VS
28


Reviews:
[{"review_id": "0PCPiIU9Zv", "reviewer": "Reviewer_A7sM", "summary": "This paper investigates the expressive power of recurrent quantum neural networks (RQNNs) in processing temporal data. The authors address a significant gap in quantum machine learning theory – namely, whether quantum recurrent models (a form of quantum reservoir computing with feedback) can universally approximate sequences and dynamical systems, and if so, under what resource requirements. They develop a rigorous theoretical framework combining quantum neural network function approximation results with classical reservoir computing theory. The main contributions include quantitative approximation error bounds and universality theorems for RQNNs with simple linear output layers.", "questions": "1. The paper asserts that RQNNs have approximation capabilities “as competitive as” classical reservoir families like echo state networks or state-affine systems. However, it doesn’t provide a direct comparison or quantification of any potential advantage. Hmm.. IMHO, most ppl in QML would ask about the potential for quantum advantage here, for instance, do there exist learning instances such that RQNNs and their classical counterparts exhibit a learning separation in terms of either time or sample complexity?\n2. Have the authors considered the practical side of how one would train or set the parameters $\\theta$ for an RQNN to approximate a given system? Would one use gradient-based variational quantum circuits training to fit observed data from the target system?", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "1. IMHO, this work is the first to establish quantitative universal approximation bounds for recurrent quantum neural networks. Prior to this, the literature lacked error guarantees for quantum recurrent models. The paper fills that gap by providing rigorous theorems (with proofs) that demonstrate RQNNs’ ability to approximate a broad class of time-dependent functions to arbitrary accuracy. \n2. Also, it shows that RQNNs can achieve universality without the need for high-degree polynomial readout functions. \n3. to reach a desired approximation error, the required number of qubits grows only logarithmically with $1/\\varepsilon$. In other words, exponentially increasing the accuracy only adds a linear number of additional qubits. This is a remarkable claim as it suggests no curse of dimensionality in qubit resources.", "weaknesses": "1. Thm 4.6 relies on the requirement that the state transition function lies in the Barron function class and has bounded first derivatives (plus contractivity $\\lambda<1$), which means the results apply primarily to “well-behaved” systems (smooth, band-limited, and not too chaotic). Real-world temporal processes might violate these conditions (e.g., non-smooth or highly non-contractive dynamics). \n2. Minor weakness: the paper does not include any experimental or numerical simulation results to complement the theory. All results are analytical. This is a weakness in the context of a machine learning conference. Still, what I said is merely a comment, rather than a criticism.\n3. The paper assumes the existence of optimal parameters $\\theta$ for the RQNN (since it’s a universal approximation argument), but does not discuss how one might find these parameters in practice. Training a quantum model with many parameters is non-trivial, e.g., you might face barren plateaus, circuit noises, etc. You can argue that this is beyond the scope of this paper. Still, I think it is quite an important aspect to at least discuss them in the conclusion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "euyUQpYTKX", "reviewer": "Reviewer_7EQ4", "summary": "Summary\n\nThe manuscript rigorously derives approximation error bounds and a\nproof for universality of recurrent quantum neural networks (RQNNs).\nThe main novelty is the inclusion of recurrency/feedback, as similar\nresults have been recently presented for feedforward quantum neural\nnetworks (Gonon & Jacquier 2025). Universal approximation theorems are\nan important topic and well researched for classical neural networks,\nas they provide important insights on the power of different neural\nnetwork architectures. Given the importance of recurrent networks in\nprocessing time series, the results of this study are highly relevant.\n\n\nSoundness\n\nThe study is purely theoretical. The obtained results, however, look\nsound as detailed proofs are presented in the appendices. I admit that\nI did not dive deeply into all aspects of the proofs, but many aspects\nseem to rely on closely related previous studies on feedforward QNNs.\n\n\nPresentation\n\nThe general structure of the study is very good. The main text\nfocuses on the main results, and detailed proofs are being relegated\nto appendix sections. Also the appendix on previous techniques in\nquantum neural networks is very helpful.\n\nThe authors further attempt to provide simple introductions to the\nindividual sections to guide the reader through the logic of the\nconstruction of the quantum circuits and the proofs for their\napproximation capabilities. While this helped me to a certain extent,\nin my opinion, the manuscript is still hard to follow for non-experts\non quantum neural networks. Given that the ICLR community has very\nbroad backgrounds, publication at ICLR therefore requires some\nimprovements in terms of presentation. The main issue I have is that\nthe manuscript is not self-contained enough. See detailed points in\nweaknesses below.\n\n\nContribution\n\nThe authors present a systematic way to construct unitary operators\nfrom a suitable combination of rotation and Hadamard gates that upon\nmeasurement after application to initial states define a set of\nfunctions that can approximate target activation functions of neurons\nin recurrent networks to arbitrary accuracy. This is done by tuning\nthe parameters of the gates. Mathematically rigorous proofs are\npresented for these construction steps and the resulting approximation\npower. The analysis largely relies on a recently published study on\nsimilar error bounds and universality properties of quantum\nfeedforward neural networks. For non-expert readers, the novelty, i.e.\nthe specific differences in the derivations that feedback and recurrency\nintroduce, are hard to detect and not clearly presented enough.", "questions": "Reservoir computing typically optimizes only the readout layer. The\nauthors mention in Section 1.2. that their results are for networks\nwhere all parameters are trainable, but they claim that they are also\ngeneralizable to random parameters in the recurrent layer. This\ngeneralization does not become clear from the current presentation.\nThis point is only mentioned again in the conclusion section, but it\nis not clear how the analyses of the current study support this claim.\nThis point needs to be elaborated much further.\n\n\nMinor points:\n- Clarify why it is necessary to extend previous results on functions to the first derivatives.\n- The abbreviation SAS is not defined.\n- Please spell out Barron-type conditions somewhere. \n- Please elaborate more on the function of control and target qubits for non-expert readers.\n- Please make clearer around line 266 that one needs N parallel circuits to have one for approximating each component of F in eq. (1)?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "The derivations in the study seem mathematically rigorous. The authors\nalso provide useful background on filters and functionals, and a\nhelpful review of existing literature in the related works and\nappendix sections.", "weaknesses": "The study is not self-contained enough:\nreferences in many places to previous work Gonon and Jacquier (2025).\nThis makes it harder to follow. In particular, this issue appears when\nthe authors aim to introduce the unitary matrix V, which seems to be\nimportant for the recurrent quantum neural network architecture. But\nall details are relayed to the previous publication.\n\nThe study needs to better work out the novelty:\nAs often mentioned many times throughout the manuscript, the work\nlargely follows the techniques in Gonon & Jacquier 2025 and others on\nfeedforward QNNs. The differences between the feedforward and the\nrecurrent case of QNNs, however, needs to be highlighted more strongly\nso that the novelty of the results also becomes more apparent to\nreaders that are not experts in the field and familar with the\nprevious studies. In particular concerning the constructions in\nSection 3: how do they differ explicitly from the case of feedforward\nnetworks? This is not obvious, but crucial to judge the advances\ncompared to Gonon & Jacquier 2025.\n\nProposition 4.1:\nThis proposition seems to be central for the understanding of the\nprocedure. The proof of proposition 4.1 is only relayed to Gonon &\nJacquier 2025. For a more self-contained presentation that targets the\nbroader community of ICLR, it would be better to also present the\nproof of proposition 4.1 in the appendix of the current study.\n\nThe \"curse of dimensionality\" is emphasized in the abstract,\nintroduction and the summary of contributions, but never mentioned in\nthe results sections. It would be helpful to point to the results on\ndimensionality more specifically throughout the manuscript and\nemphasize the importance on the log scaling there.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C2CesCi3fl", "reviewer": "Reviewer_mUbU", "summary": "The paper derives approximation bounds and universality statements for recurrent quantum neural networks. The proposed approach is based on a uniformly controlled quantum gate to apply multicontrolled rotations to a set of control and target qubits, and it has been recently shown that it can be efficiently implemented.\n\nThe authors first prove that RQNNs are able to uniformly approximate the filters induced by any contracting Barrontype state-space system.\nSecond, they extend this universality property to the much larger category of arbitrary fading memory, causal, and time-invariant filters. In this last result, neither Barron-type integrability nor contractivity conditions are needed for the target filter.\n\nThe paper is a strong theoretical contribution to the field. One of its major limitations for ICLR is that it is only theoretical.", "questions": "Could you discuss the fact that error rates do not suffer from the curse of dimensionality? I think you refer to d, but  I'm not sure.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- Relevant theoretetical contribution\n- Excellent technical depth and rigor\n- Clear positioning in literature and in particular w.r.t recent literature", "weaknesses": "- Lack of empirical validation. Theoretical findings are strong. However, no numerical or experimental results are a limitation for this paper.\n- Some assumption (e.g., Barron-type integrability) may restrict practical applicability\n- A comparison between the proposed approach and classical RNNs or RC models for large n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GeheFLrQWh", "reviewer": "Reviewer_WKBd", "summary": "This theoretical paper discusses the universal approximation bound for RQNNs. The authors extend approximation error bounds from previous QNN work and introduce a time factor as a state-space system to derive the approximation bound for RQNNs.", "questions": "1. In equation (12), where N = 2^{qubit count} relates to qubit size and n relates to parameter count, the role of λ (the gradient bound) in the denominator appears extremely important. How would λ be determined in practice? For example, does it relate to quantum noise? \n2. Can the measurement shot requirement for extracting quantum information from the QRNN play an important role in the approximation bound? For example, the empirical estimation of expectation value typically has a distance error of the form O(1/√N_{shot}) (ref. eq. (19) in [1] and also eq. (19) in [2]).\n\n[1] Qi, J., Yang, CH.H., Chen, PY. et al. Theoretical error performance analysis for variational quantum circuit based functional regression. npj Quantum Inf 9, 4 (2023)\n\n[2] Liu, CY., Kuo, EJ., Abraham Lin, CH. et al. Quantum-Train: rethinking hybrid quantum-classical machine learning in the model compression perspective. Quantum Mach. Intell. 7, 80 (2025).", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and logically structured. \n2. The subject is important for the quantum reservoir computing community, providing theoretical foundations for a field dominated by empirical studies.", "weaknesses": "The presentation could be more reader-friendly with graphical illustrations of the problem statement and derivation direction (nice to have but minor).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)