Quantum circuit complexity and unsupervised machine learning of topological order

Domain: physics
Published: 2026-04-08
Editorial outcome: minor_revision

Abstract
Enabling the discovery of unknown quantum many-body phases of matter remains a fundamental challenge in machine learning for quantum physics. Here, inspired by the close relationship between Kolmogorov complexity and unsupervised machine learning, we explore quantum circuit complexity as a pivot to build intuitive and efficient unsupervised machine learning for topological order in quantum many-body systems. We argue that Nielsen’s quantum circuit complexity serves as an intrinsic informational distance between topological quantum states that results in interpretable manifold learning. To span a bridge from conceptual power to practical applicability, we present two theorems that connect Nielsen’s quantum circuit complexity of quantum path planning with quantum Fisher complexity (Bures distance) and entanglement generation, respectively. The resulting kernel functions demonstrate superior performance and enhanced interpretability in numerical multiqubit experiments. Our results establish connections between key concepts of quantum computation, quantum complexity, quantum metrology, and machine learning of topological quantum order.

Introduction
The discovery of topological phases of matter has opened a new chapter in modern physics, ranging from symmetry-protected band topologies with short-range quantum entanglement such as topological insulators and superconductors [1, 2], to topological quantum orders [3–10] featured by distributed non-local quantum entanglement such as in Kitaev’s toric code [7, 9]. Defying Landau’s paradigmatic approach, where phases and phase transitions are identified through local order parameters (that is, linear functionals of the density matrix of the phase and a local observable), topological phases encode the phase information in global –topological– system properties. This defining difference renders the detection and measurement of topological order challenging for both generic simulation and experimental demonstration.

Recent progress has shown the potential of machine learning for classifying topological phases of matter [11–34]. In particular, the supervised approach to quantum many-body physics problems, which can resort to well established theories of learnability and generalizability [35–37], has resulted in the formulation of rigorous guarantees [38–47]. However, in many practical scenarios, where a priori knowledge of the different phases of matter (and accordingly their labels) does not exist, only unsupervised machine learning can provide access to the desired information. Despite the practical relevance of unsupervised learning, a rigorous theory, tailored to topological quantum order and offering both interpretability and generalizability, has not yet been fully explored.

One prevalent approach to unsupervised learning is manifold learning [48–50], where the data is assumed to be distributed on a manifold characterized by an intrinsic distance metric. It is followed by a nonlinear mapping of the data manifold into a low-dimensional Euclidean space while preserving the original similarities and structures. Among many manifold-learning algorithms, diffusion map [51, 52], t-SNE [50], and kernel principal component analysis (PCA) [53, 54] require a kernel function constructed from the intrinsic distance. The kernel can be used either to build a probabilistic graph representation of the data manifold (e.g., in the diffusion map or the t-SNE), or for PCA in the nonlinear feature space (e.g., in the kernel PCA). Alternatively, the intrinsic distance metric itself can be applied directly to non-kernel manifold learning, such as the Isomap [48] and the metric-multidimensional scaling (metric-MDS) [55, 56]. In the context of topological quantum order, the basic task then is to formulate a distance metric that encodes the nonlinear and non-Euclidean features of topological quantum states and represents topological similarities or distances by exploiting the primitive notion of topological equivalence. Topologically equivalent quantum states, by definition, can be continuously transformed into each other via a smooth unitary transformation generated by a local Hermitian operator with bounded operator norm [4]. In this spirit, a similarity measure based on a path-finding algorithm has been proposed in [24] and has shown good performance in clustering symmetry-protected band topologies, which are characterized by short-range quantum entanglement. Other topological similarity measures, that focus on the closing of the spectral gap at topological transitions, have also been proposed and demonstrated [23, 57].

Despite the elegance of these methods in identifying band topologies, the increasing complexity and quantum entanglement in strongly interacting quantum many-body systems may undermine the efficiency and performance of these algorithms. For instance, the path-finding algorithm, when applied to interacting lattice models in order to identify a smooth unitary path generated by a local operator while keeping the parent Hamiltonian gapped, can be principally inconclusive due to the undecidability of the spectral gap for generic quantum many-body Hamiltonians [58].

The inherent problems of previous manifold-learning approaches call for a new similarity ansatz, an ansatz which is able to capture the topological distance and entanglement patterns at different scales while at the same time remaining interpretable and efficiently computable (see Fig. 1 for a schematic of the current state regarding the application of machine learning to the classification of topological phases of matter). To this end, we here adopt the alternative but equivalent perspective that two topologically equivalent multi-qubit states can be transformed into each other via a constant-depth quantum circuit composed of geometrically local quantum gates [3, 4], i.e., the smooth unitary transformation that connects them can be decomposed into a shallow quantum circuit. This implies that it is algorithmically trivial or cheap to generate one quantum state given the access to another in the same topological class by fully exploiting the shared structure and entanglement patterns between them.

The correspondence between topological equivalence of quantum states and shallow quantum circuit transformations indicates that, a complexity measure may be related to the topological distance. This is similar to the situation with Kolmogorov complexity, which represents an informational distance between two strings or objects [59], and may offer a basis for theoretically optimal solution to unsupervised machine learning (an idea, to the best of our knowledge, first formulated in a talk by Ilya Sutskever). We then assert that, the (conditional) generation and prediction of quantum states with the minimal quantum circuit cost may offer a theoretically optimal solution to the unsupervised learning of topological order. The complexity of the generating circuit can be interpreted as a quantum informational distance measure that captures the entanglement change at different scales. Specifically, we propose to use Nielsen’s quantum circuit complexity (QCC) of the unitary path connecting two quantum states as a suitable distance measure that can be used for the unsupervised manifold learning of topological order. While the QCC is already a central concept in quantum information theory [60–65], quantum field theory [66–69] and quantum gravity [70–74], its connection with, and relevance for, the unsupervised machine learning of topological phases of matter has remained mostly unexplored.

In this work, we use the QCC as a pivotal tool to understand and construct unsupervised machine learning of topological order for quantum many-body systems, in an interpretable and efficient manner. Although the exact QCC is generically intractable, we identify various quantities which are upper bounded by the QCC while preserving useful information and remaining implementable, and which thus can be efficiently utilized for classifying topological quantum many-body phases. Specifically, we establish entanglement-based and fidelity-based similarities / kernels, respectively. Numerical experiments with a bond-alternating XXZ qubit chain and Kitaev’s toric code model, respectively, demonstrate the excellent performance of these methods.

Results
2.1 Quantum path planning, topological order and quantum circuit complexity

As mentioned above, unsupervised machine learning of band topology has been extensively investigated. Band topologies, or the more general symmetry-protected topological (SPT) orders, such as those exhibited by topological insulators, are characterized by the absence of long-range quantum entanglement. This implies that, if the constraint of preserving the symmetry is dropped, SPT quantum states can be transformed smoothly into product states. In contrast, topological quantum order refers to quantum states with long-range quantum entanglement [4]; for instance, the ground state of Kitaev’s toric code exhibits topological quantum order that manifests in non-zero topological entanglement entropy [5, 6] and global logical operations [7], while SPT phases are confined to short-range entanglement. Thus, the task of identifying topological quantum order with unsupervised learning presupposes to construct similarity measures with feature maps that capture quantum entanglement at a variety of scales. For this purpose, we start by defining the quantum path planning between two multi-qubit quantum states, which lays the ground for our search for a distance measure of clustering topological orders.

Definition 1 [Quantum path planning (QPP).] The QPP problem is defined as: Given two arbitrary pure n-qubit quantum states ρ0 and ρ1, find continuous paths ρ(s)=U(s)ρ0U†(s) (s∈[0,1]) that connect ρ0 and ρ1, where the generator G(s) of U(s) is a hermitian operator, with the operator norm ∥∂sU(s)∥∞ =∥G(s)∥∞ <+∞(∀s∈[0,1]). In general, we have that U(s)∈SU(2n) and iG(s)∈su(2n), with the unitary given by
U(s) = P exp \left( -i \int_0^s G(\tau)d\tau \right), (1)
where U(s = 0) = Id is the identity and P denotes the path ordering operator. s can be the normalized time t (for instance, s = t/T, where T is the total evolution time for the state generation). In practical implementation, the QPP can involve finding the optimal path under various constraints.

Note that, for two topologically equivalent gapped quantum states, the generator G(s) in the above QPP must be additionally a geometrically local operator, which underlies why the two quantum states can be transformed into each other via a constant-depth quantum circuit [4] (See the following definition).

Definition 2 (Topological equivalence [3, 4].) We say that two gapped ground states ρ0 and ρ1 of local Hamiltonians H0 and H1, respectively, are in the same topological phase if there is a local unitary that maps the two states into each other without closing the energy gap. In other words, there is a smooth path of local Hamiltonians H(s) that connects the two parent Hamiltonians with H(s = 0) = H0 and H(s = 1) = H1 for s ∈ [0,1]. The Hamiltonian H(s) is gapped above its ground state ρ(s) for all s∈[0,1], where ρ(s) smoothly connects ρ0 and ρ1, which establishes a smooth path between the two quantum many-body states. In [75], it is show that such a path can be realized with a smoothly parametrized unitary U(s) whose generator G(s) is also a local operator, ρ(s)=U(s)ρ0U†(s), and naturally ρ1 = U(s = 1)ρ0U†(s = 1). This is further elaborated in Sec. II of the Supplementary Information.

The above Definition 2 is equivalent to the statement that the two topologically equivalent states can be deformed into each other by a constant-depth quantum circuit [4], each layer of which can be divided into geometrically local quantum gate operations. Therefore, the two quantum states differ from each other only in entanglement patterns at short-range scales, while they are equivalent in their long-range entanglement structure.

Intuitively, the depth of the quantum circuit quantifies the scale of the change of the quantum entanglement that is caused by applying the circuit on the reference quantum state ρ0. Then, the quantum circuit complexity (QCC) for realizing U(s = 1) can serve as a measure of the algorithmic information for generating the target state ρ1 from a reference state ρ0. As is explained above, this is similar to, and inspired by, the (conditional) Kolmogorov complexity K(x|y), which is the length of the shortest program on a universal computer that outputs a string or dataset x given the access of another string or dataset y, and as such represents an algorithmic information distance [59] or the shared structure between the two objects. We adopt and adapt this here, using a geometric lower bound of the quantum circuit cost, the Nielsen’s QCC CN(ρ0→ρ1), as a distance measure for clustering topological states with distinct entanglement patterns, whose definition is given in the following.

Definition 3 (Nielsen’s QCC.) The quantum circuit complexity of the unitary U(s = 1) can be evaluated through Nielsen’s geometric approach [76, 77]. For qubit systems, if we write G(s) = \sum_{\sigma} h\sigma(s) \sigma, where \sigma ∈ Pn = {I, X,Y,Z}⊗n and \sigma ̸= I⊗n is a n-qubit Pauli operator (I is the 2-dimensional identity operator and X,Y,Z are the three single-qubit Pauli operators, respectively), and h\sigma(s) denote real coefficients, the Nielsen’s quantum circuit complexity of the first order reads
CN(ρ0→ρ1)= inf \int_0^1 \sum_{\sigma} |h\sigma(s)| ds. (2)
The original Nielsen’s QCC can be generalized to systems other than qubits [67, 68, 78], where an orthonormal operator basis {Oi} replaces the n-qubit Pauli operators Pn, with the operator norm ∥Oi∥∞ ≤ 1 (∀i). Then the definition of Nielsen’s QCC above remains unchanged.

With the Nielsen’s QCC of geometrically local quantum circuit as a topological and informational distance measure, the proposed theoretically optimal similarity/ kernel function for the unsupervised manifold learning of topological order is given by
K(ρ0,ρ1)=exp[−β CN(ρ0→ρ1)], (3)
up to a normalization constant, where β >0 is a hyperparameter. The unitarity of the quantum circuit indicates that Nielsen’s QCC is symmetric under permutations of the two quantum states, i.e., CN(ρ0→ρ1) = CN(ρ1→ρ0), which leads to a symmetric kernel in (3). The similarity function in (3) can be used to construct conditional probabilities in the diffusion map [51, 52] and the t-SNE [50] manifold learning, or can be used for kernel PCA [53, 54] in the kernel feature space. Physically, Nielsen’s QCC can be viewed as a global order parameter of topological quantum orders, and also serves as an ideal topological distance in non-kernel manifold learnings such as the metric-MDS [55, 56].

2.2 Unsupervised machine learning of topological order derived from quantum circuit complexity

While it is in general hard to exactly compute CN(ρ0→ρ1) for the QPP, we now establish some approximate substitutes, which both capture essential aspects of Nielsen’s QCC and at the same time are easier to implement in practice. The two theorems that we present in the following provide rigorous guarantees for this purpose.

First, we define the quantum Fisher complexity (QFC) related to the QPP, denoted by CF(ρ0→ρ1), which coincides with the Bures distance between the two quantum states given by a fidelity measure and can be efficiently calculated in practice.

Definition 4 [Quantum Fisher complexity (QFC).] Defining the Uhlmann-Jozsa fidelity between two arbitrary quantum states (pure or mixed) as [79]
F(ρ,ρ˜)=tr \sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}, (4)
where tr denotes the trace operation, then the corresponding squared Bures distance reads [80–82]
DB2(ρ,ρ˜)=2[1−F(ρ,ρ˜)]. (5)
For a family of smoothly parametrized quantum states ρ(s) by s ∈ [0, 1], the quantum Fisher information (QFI) of the quantum state with respect to the parameters is given by [81–85] FQ(s) = tr [ρ(s)L2], with the symmetric logarithmic derivative (SLD) L determined by ∂sρ(s)= 1/2[ρ(s)L+Lρ(s)]. The QFI, which plays a central role in quantum parameter estimation and which is also used for characterizing multipartite entanglement and topological states [86], can be related to the infinitestmal Bures distance [80] with
1/4 FQ(s)d2s=DB2 [ρ(s),ρ(s+ds)], (6)
representing a susceptibility of the fidelity.

The QFC of the target quantum state ρ1 with respect to the reference state ρ0 is given by [66]
CF(ρ0→ρ1)= inf \int_0^1 \sqrt{1/2 FQ(s)} ds, (7)
which represents the optimal Bures distance between the two quantum states [82], reflecting the overall fidelity variation, as indicated by (6). Note that we have a symmetric QFC under permutations of its inputs, i.e., CF(ρ0→ρ1)=CF(ρ1→ρ0).

With the QFC and the Bures distance, we are ready to formulate the following theorem.

Theorem 1 (Quantum circuit complexity upper bounds quantum Fisher complexity and Bures distance.) The QCC for the QPP from ρ0 to ρ1 is lower bounded by the QFC and the Bures distance,
CN(ρ0→ρ1) ≥ CF(ρ0→ρ1)
≥ DB(ρ0,ρ1)/\sqrt{2}. (8)
Furthermore, the QCC of a geometrically local quantum circuit for the same QPP is approximately lower bounded by
CN(ρ0→ρ1) ≳ 1/\sqrt{2} \sum_{\Delta} DB [ρ0(∆),ρ1(∆)], (9)
where ρ(∆) is the reduced density matrix supported on the subsystem ∆ of constant size, and the summation goes over non-overlapping subsystems which, together with their neighboring environments, cover the whole system.

The proof of Theorem 1 is given in Sec. I of the Supplementary Information, and an asymptotic and tightness analysis of the bounds in Theorem 1 is provided in Sec. III. Note that, without the constraint of geometric locality of the quantum circuit, the overall Bures distance in (8) is equivalent to the L2-norm distance between the two density matrices, which has been shown not to be a suitable distance measure for topological phases [23].

Under the geometric locality of the quantum circuit, we exploit Theorem 1 to construct a fidelity-based kernel. The Uhlmann-Jozsa fidelity between the r-body reduced density matrices with r ={1,2,··· ,R} can be used for this purpose, where R is a scale of cutoff. For instance, for qubits hosted on a lattice Λ=[1, L]D of lattice size L and dimension D, the proposed kernel based on fidelity reads (up to some normalization constant)
KF(ρ,ρ˜)=exp \left\{ \beta \sum_{r=1}^R \omega_r \sum_{\Delta\in Pr(\Lambda)} F [ρ(∆), \tilde{\rho}(∆)] \right\}, (10)
where Pr(Λ) = {∆|∆ ⊂ Λ and |∆| = r}, with |∆| being the size (cardinality) of the sublattice ∆; ρ(∆)=tr(Λ−∆) ρ is the reduced density matrix supported on ∆; ωr is a weight function; and F(·,·) is the Uhlmann-Jozsa fidelity. While the dimension of the reduced density matrix depends exponentially on its size r, this fidelity-based kernel can be efficiently estimated in practical implementations for R=const. (For instance, a small r = 2 is already sufficient in many relevant applications. See Theorem 1 and the section on numerical experiments below.)

The second approximation of the Nielsen’s QCC that we propose focuses on the entanglement change. Denoting by ρ a pure density matrix of a n-qubit chain (rearranged from an arbitrary spatial geometry following a specified order), and Ak = {1,2,...,k} ⊂ [n] a subset of the system, which is obtained from a cut at k ∈[1, n−1], then the reduced density matrix supported on Ak reads ρ(Ak)=tr Bk ρ, where Bk ={k+1,k+2,...,n}. With the entanglement entropy over the cut k given by
Sk(ρ)=tr[ρ(Ak)lnρ(Ak)], (11)
we have the following theorem:

Theorem 2 [Quantum circuit complexity upper bounds entanglement-profile distance (A generalized version of Observation 1 in [78]).] The QCC for the QPP from ρ0 to ρ1 with a geometrically local quantum circuit (with respect to the n-qubit chain) is lower bounded by
CN(ρ0→ρ1)≥ c \sum_{k=1}^{n-1} |Sk(ρ1)−Sk(ρ0)|, (12)
for some constant c>0.

The proof of Theorem 2 is given in Sec. I of the Supplementary Information, and an asymptotic and tightness analysis of the bounds in Theorem 2 is provided in Sec. III. Theorem 2 can be readily generalized to situations where the qubits are arranged in higher spatial dimensions and other geometries, provided that the quantum circuit is applied in a geometrically local manner accordingly. Nielsen’s QCC is then lower bounded by the entanglement change distributed over suitable cuts covering non-trivial bonds of the system.

Theorem 2 inspires an entanglement-based kernel, given by
KE(ρ,ρ˜)=exp \left( -\beta \sum_{k=1}^{n-1} |Sk(ρ)−Sk(ρ˜)| \right), (13)
which is particularly useful when the entanglement profile over nontrivial bonds can be efficiently estimated. This is, for instance, the case in tensor-network simulations of quantum many-body states such as the density matrix renormalization group (DMRG)[87,88], or in experimentally prepared quantum states measured via classical shadow tomography [89].

Both lower bounds of Nielsen’s QCC in (9) and (12) potentially capture important information about the topological order. The fidelity-based kernel with reduced density matrices of constant size can be efficiently calculated and benchmarked (For instance, with classical shadow representations, see the following section). The entanglement-based kernel is more stringent and contains more information about quantum entanglement at multiple scales.

2.3 Relation to classical shadows

Classical shadow (CS) tomography is an experimentally friendly approach to access many properties of quantum many-body states with few-shot measurements [89]. In this formalism, the quantum state is approximated by its CS estimator,
ρ≈ST(ρ)= 1/T \sum_{t=1}^T \sigma(t)⊗\sigma(t)···⊗\sigma(t), (14)
in a total of T measurements, where \sigma(t)i is the local CS representation under the Pauli measurement for the i-th qubit in the t-th measurement. The shadow kernel suggested in [38], given by
KCS [ST(ρ),ST(ρ˜)]=exp \left\{ \beta \sum_{t, t′=1}^T \exp \left[ \nu \sum_{i=1}^n tr (\sigma(t)i \sigma˜(t)i) \right] \right\}, (15)
can be understood as a special case of the fidelity-based kernel in (10), if the quantum state ρ and the r-body reduced density matrices are represented by their classical shadows, if the weight function is taken to be \omega_r =\nu r/r!, and if we take the limit R→∞, where β >0 and \nu >0 are constant hyperparameters. In this case, density matrices with small values of r dominate in the summation, and randomized measurements reduce the complexity in calculating fidelity-based kernels.

2.4 Numerical experiments

In this section, we provide numerical demonstrations for the developed theory of unsupervised machine learning of topological order based on the QCC.

The bond-alternating XXZ spin chain. We begin with a benchmark model, the bond-alternating XXZ spin-1 chain [38, 90, 91], whose Hamiltonian is given by
H = \sum_{i=1}^{n-1} J_i (Xi Xi+1 +Yi Yi+1 +\delta Zi Zi+1)−h0 \sum_{i=1}^n Zi, (16)
where Ji = J1 if i is odd and Ji = J2 if i is even, respectively; \delta is a detuning parameter and h0 denotes the strength of the external field.

In Fig. 2, we present a comparison between the fidelity- and entanglement-based kernels for this model, respectively, together with the two-dimensional representation in the diffusion space from manifold learning of the diffusion map algorithm [22–24, 51, 52]. The ground-state properties (fidelities and entanglements) are obtained from DMRG calculations of the model. Fixing h0 = 0, \delta = 3 and varying the value of J2/J1, we find that the unsupervised learning produces distinct clusters that can be associated with three phases (trivial, symmetry broken, and topological), in good agreement with independent calculations of topological invariants [38, 90, 91]. More details can be found in the caption of Fig. 2. While we are ultimately aiming at the unsupervised learning of topological order in the presence of non-local quantum entanglement, Fig. 2 shows that the fidelity- and entanglement-based kernels also perform well when clustering symmetry-broken phases and symmetry-protected topological orders [91] that are characterized by short-range entanglement.

Kitaev’s toric code. As a second demonstration, we cluster the ground state of Kitaev’s toric code with random product states, where the former has topological quantum order while the latter is topologically trivial. The toric code Hamiltonian defined on a 2D lattice of size Lx ×Ly is given by [7]
H =− \sum_{v} Av − \sum_{p} Bp, (17)
where v denotes a vertex and p is the face of the lattice, respectively; Av = \prod_{j∈star(v)} Xj and Bp = \prod_{j∈boundary(p)} Zj are the Kitaev’s star and face operators, respectively. The number of qubits for this model is n=2Lx Ly. The toric code ground state is gapped and exhibits area-law entanglement [5]. Therefore, for the purpose of a proof-of-principle demonstration and in order to obtain the entanglement profile for the kernel with a small system size, we use a MPS representation and DMRG to find a reasonably accurate solution for the ground state. For instance, numerical experiments show that, for a 4×4 lattice with periodic boundary conditions in both dimensions, a maximal MPS bond dimension of 64 is sufficient to produce accurate results.

In Fig. 3 we plot a comparison between the unsupervised machine learning of the toric code ground state (blue dots) and the random product states (RPS) (red dots) of qubits (i.e., random bit strings of spin ups and downs), with fidelity- and entanglement-based kernels, respectively. Both kernels perform well in clustering the topologically ordered quantum state and the trivial phase, while the entanglement-based one is more robust under the application of two-qubit Haar random unitaries to both the toric code and the RPS. This follows from the comparison between Fig. 3 (b) and (d), where the silhouette coefficient [56], which takes the value one for best clustering, is used for illustrating the performance degradation of the clustering results in Fig. 3 (a) and (c), respectively, under Haar random unitaries. Here we do not incorporate geometric locality for the two-body reduced density matrices in calculating the fidelity-based kernel (as not required in the shadow kernel learning). Moreover, with geometric locality, we find that the clustering performance of the fidelity kernel PCA will be boosted, but is still inferior to that of the entanglement-based one under Haar random unitaries. More details about the plot can be found in the caption of Fig. 3.

For more discussions on the effects of hardware noises and applied random quantum circuits, we refer the reader to Sec. V of the Supplementary Information. In addition, in Sec. VI of the Supplementary Information (see Fig. S1), we also use metric-MDS (which does not require a kernel formulation) based on the Bures and entanglement-profile distance metrics in (9) and (12), respectively, for unsupervised manifold clustering of the toric code and the product state. Similar conclusions can be obtained from Fig. S1.

The extended toric code. The original toric code is defined on a torus with periodic boundary conditions in both spatial dimensions. In contrast, the extended toric code (ETC) is defined on an infinite cylinder (e.g., the cylinder axis is along the x direction and the y direction is periodic with circumference Ly), and additionally features the Wilson and t’Hooft loop operators W and H, respectively. The corresponding Hamiltonian is given by [92]
H =H TC −JW W −JH H −h0 \sum_{i=1}^n Zi, (18)
where HTC is the original toric code Hamiltonian in (17), and h0 is a (uniform) local field strength. The loop operators are defined as products
W = \prod_{i∈Ev} Zi, H = \prod_{i∈Eh} Xi (19)
around the cylinder through vertical edges (Ev) and horizontal edges (Eh) of the lattice graph, respectively.

In Fig. 4, by fixing the values of JW = +1 and JH = −1, which stabilizes the ground state into one of the four degenerate sectors, and by varying the strength of the local perturbation field h, we plot a comparison between the unsupervised machine learning of the ETC ground state (blue dots) and the random product states (RPS) (red dots) of qubits (i.e., random bit strings of spin ups and downs), with fidelity- and entanglement-based kernels, respectively. Note that, for finite values of h, the ground states are no longer exactly solvable. The required maximal MPS bond dimension increases rapidly with larger values of h even for small system sizes (For the system size and the parameter range we adopted in Fig. 4, a maximal bond dimension of 600 is sufficient to produce accurate results). The topological clustering shows its stability against local perturbations. More details about the plot can be found in the caption of Fig. 4.

Methods
4.1 Theoretical framework and proof ideas of Theorems 1 and 2

The pivot of our method is an informational distance metric for topological dissimilarities, leveraging Nielsen’s geometric QCC of mapping two quantum many-body states as a primitive distance metric for manifold learning. This can result in both good interpretability and generalizability for unsupervised machine learning of topological quantum order. The theoretical toolbox includes differential geometry, quantum Fisher information, entanglement-growth rate, concept of topological equivalence, and quantum circuit computation, etc. By bridging conceptual power to practical implementation, Theorems 1 and 2 lead to approximate substitutes of the QCC, i.e., the Bures distance (fidelity change) and the entanglement-entropy distance, respectively, which preserve partial but important information of topological dissimilarity and which are intuitively more comprehensive and more accessible in simulations and measurements.

Now we describe the key ideas behind the proofs of Theorems 1 and 2. The insights for the the proof of Theorem 1 is that the integrand of the first-order Nielsen’s QCC (time complexity of the QPP) is lower bounded by the operator norm of the generator of the QPP (easy to obtain via the Pauli-string-operator basis), which also upper bounds the QFI (or the Fubini-Study metric) during the QPP. The constraint of geometrically local generators can be naturally incorporated via the Trotterization of the QPP into very small time steps [see details of the proof in Sec. I of the Supplementary Information (SI)]. In the proof of Theorem 2, one finds that the result for the Small Incremental Entangling (SIE) [111–115] is applicable to general QPPs, generalizing the initial product state [78] to an arbitrary quantum many-body state. This is useful in our manifold learning of arbitrary quantum states, the distance between which is a key ingredient in the method. Similarly, geometrically local generators and Trotterization are used in the proof of Theorem 2 (see details of the proof in Sec. I of the SI).

Note that Nielsen’s QCC, in general, does not require an adiabatic (gapped) QPP. However, assuming a gapped path (ground state) allows us to estimate the asymptotic scaling (with respect to the number of qubits n) of Nielsen’s QCC and the quantum Fisher complexity (see Secs. II and III in the SI for details; and also for a tightness analysis of the derived bounds in the two theorems of the main text).

4.2 Machine learning algorithms and numerical experiments

The DMRG ground-state search in the numerical experiments was performed using the TeNPy Library [92], and some example codes there are reused for preparing the dataset of our machine learning model. In order to simulate effects of random circuits or noises on the clustering performance of the method (see Fig. 3 and Fig. S1), random unitary evolutions are applied to both the toric code and product states, via the time-evolving block decimation (TEBD) in the TeNPy Library [92]. This performs local two-site Haar random unitaries by a variable number of steps (two-qubit circuit depths). To implement the code in a reasonable time and memory, a maximum TEBD bond dimension can be set (e.g., to 500), and such random unitary evolutions of different circuit depths can be applied repeatedly to reach the maximum target bond dimension. More discussions on effects of noises and random circuits can be found in Sec. V of the SI.

Regarding specific manifold-learning algorithms for dimensionality reduction and phase clustering, we use the well-established framework with diffusion map [23, 51, 52], kernel PCA [53, 54], and metric-MDS [55, 56] for demonstrations in the current work. We believe that other manifold-learning algorithms such as the t-SNE [50] and the Isomap [48], equipped with our topologically informational distances, will give similar results.

In Sec. IV of the SI, We give an elaborated analysis on the sample and computational complexities of the method in terms of the number of qubits n and the sample size N, which shows that provably efficient unsupervised learning derived from Nielsen’s QCC is accessible in simulations and experiments. In addition, a (non-kernel) metric-MDS manifold learning [with a computational time of O(N2)] of the toric-code is also provided in Sec. VI of the SI, which is more efficient compared to kernel approaches [with a computational time of O(N3)] in case of large datasets.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, the author discussed the utility of quantum circuit complexity as a key concept in distinguishing topological phase and trivial phase of many-body ground states. The main contribution includes two lower bounds of quantum circuit complexity, which can be experimentally measured in practical quantum systems. Associated with these lower bounds, the authors proposed two kernel functions which can be used to classify topological phase and trivial phase in an unsupervised manner.\nI think the idea of connecting Nilsen's quantum circuit complexity with many-body phase classification is quite interesting. However, I have concerns regarding the novelty of the methods used to classify different phases. As part of the main results in this paper, the kernel methods used to classify topological phase and trivial phases are very similar to the approach developed in H-Y Huang et al. \"Provably efficient machine learning for quantum many-body problems\". Especially, the fidelity-based kernel is quite close to shadow kernel. Hence, the numerical results of successful classification are not surprised.\nTwo other comments are: 1. I am not sure how tight those two lower bounds of quantum circuit complexity are, especially for those many-body ground states discussed in the paper; 2. if I understand correctly, the entanglement-based kernel requires the estimation of subsystem entanglement entropy of all possible sizes, which is challenging to measure in scalable systems especially when it cannot be classically simulated by MPS.\nBased on the above comments, I believe this manuscript fails to satisfy the high criteria of Nature Communications.", "reviewer_2": "In the present paper the Authors focus on the task of identifying topological orders of many-body states with quantum algorithmic approach, namely building corresponding cirquit and evaluating its complexity. The paper proposes a new theoretical framework for unsupervised machine learning of topological quantum order by leveraging quantum circuit complexity (QCC), by providing a proof of several key theorems. The authors reinterpret the notion of topological equivalence between quantum states in terms of constant-depth quantum circuits. This connects machine learning kernels with a fundamental measure of quantum complexity for the two kernel constructions - fidelity-based kernel and entanglement-based kernel. Theorems establish lower bounds on QCC via Bures distance (fidelity changes) and entanglement changes. The Authors demonstrate that the proposed kernels can be efficiently computed in simulations like DMRG and potentially measured experimentally using classical shadow tomography. Examples contain bond-alternating XXZ spin chain and toric code. The paper is sound and represents an important step in building interpretable unsupervised machine learning techniques for identification of topological orders. The analytical proofs of theorems are clearly written. The presented figures explain well the main findings of numerical simulations and provide a clear comparison of two proposed kernels. I would recommend paper for publication after the following questions and comments are clarified:\n1. How well are presented kernels expected to scale for larger systems? In other words, whether the proposed kernel-based methods will be practically applicable for large condensed matter systems of interest?\n2. The entanglement-based kernel is shown to be more robust against two-qubit random noise. How do both kernels behave under realistic hardware noise models (e.g., correlated errors, decoherence)?\n3. In Fig.2, panels (c) and (d) the axes labels are indicated only in the caption. Could they be indicated also on the axes? In addition, both plots contain a single point of the yellow-light-green color, that stays in the middle between phases. This point corresponds approximately to J2/J1=1.5 . If more dense grid is taken for J2/J1 ratios, would it result in symmetry-broken phase being smoothly connected to one of the topo/trivial phases in the axes coordinates? This question requires clarification as one might expect the same problems of intersecting areas as one finds in principal component analysis classification of phases.\n(Remarks on code availability)\nThe run of code provides the same output as shown in figures. However, the code does not contain the readme file that would clearly indicate a structure and purpose of particular files / classes and functions. At the same time, file names and short readme files with instructions to run explain the general idea behind the code. In addition, not all the functions and classes implemented in the code have their clear descriptions in the form of docstrings or in readme file. My overall evaluation is that the code is good for reproducibility purposes for data shown in paper, but has limitations in re-use as it requires extra time to understand its internal structure.", "reviewer_3": "In this reviewed manuscript “Quantum circuit complexity and unsupervised machine learning of topological order”, the authors explore how to build interpretable and efficient unsupervised learning model for identifying topological order, with a focus on the quantum circuit complexity. The authors prove two theorems that link Nielsen’s geometric definition of quantum circuit complexity to fidelity variation and entanglement entropy differences between quantum states. From these relations, they derive two practical kernels (fidelity-based and entanglement-based kernels) to capture the topological similarity between states. They also carry out numerical experiments on classifying phases of XXZ spin chain, ground states of Kitaev’s toric code to support their theoretical findings. A connection to shadow kernel learning is discussed.\nUnsupervised learning topological phases of matter is a challenging task. This work makes a conceptual step by tying informational distances to quantum phase classification, providing an interpretable bridge between complexity theory and practical machine learning. I think this is an interesting paper and a timely contribution to both quantum information and condensed matter communities. Given that the manuscript is original, conceptually well-motivated, and technically sounds, I thus recommend the acceptance of this work in Nature Communications after the following comments/suggestions are carefully addressed:\n1)The toric code test with random unitaries suggests robustness of the entanglement kernel. It would be helpful to quantify how classification performance degrades with increasing noise levels. This would make the claims of robustness more concrete.\n2)The numerical demonstrations involve systems of 101 spins (XXZ chain) and 32 qubits (toric code). While this is reasonable for proof-of-principle, it would be helpful to comment more on the scalability.\n3)The authors may consider citing the following works, which report related advances and would enrich the discussion.\n1. npj Quantum Information 8, 116(2022);\n2.Fundamental Research 4, 1086(2024).\n4)Some typos:\nLine 72 : “a a suitable distance” --> “a suitable distance”.\nLine 158: “permmutations” --> “permutations”."}, "author_response": "In the following we provide a point-by-point response to the reviewers’ comments and suggestions:\n=======================================================\nResponse to Reviewer #1:\nReferee’s comment (general): In this manuscript, the author discussed\nthe utility of quantum circuit complexity as a key concept in distinguishing\ntopological phase and trivial phase of many-body ground states. The main\ncontribution includes two lower bounds of quantum circuit complexity,\nwhich can be experimentally measured in practical quantum systems.\nAssociated with these lower bounds, the authors proposed two kernel\nfunctions which can be used to classify topological phase and trivial phase\nin an unsupervised manner.\nI think the idea of connecting Nilsen's quantum circuit complexity with\nmany-body phase classification is quite interesting. However, I have\nconcerns regarding the novelty of the methods used to classify different\nphases. As part of the main results in this paper, the kernel methods used\nto classify topological phase and trivial phases are very similar to the\napproach developed in H-Y Huang et al. \"Provably efficient machine\nlearning for quantum many-body problems\". Especially, the fidelity-based\nkernel is quite close to shadow kernel. Hence, the numerical results of\nsuccessful classification are not surprised.\nReply:\nWe thank the reviewer for the positive comments on our main contribution\nin the manuscript, in particular, for acknowledging that \"the idea of\nconnecting Nielsen's quantum circuit complexity with many-body phase\nclassification is quite interesting\".\nWe agree with the reviewer that our usage of the kernel PCA to classify\ntoric code and trivial phases, in particular, the fidelity-based kernel, shares\nsome similarities with the approach developed in H-Y Huang et al. [1].\nHowever, let us clarify that, as Reviewer #2 and Reviewer #3 have also\npointed out, the focus of our manuscript is to make a conceptual step\ntoward solving the challenging task of interpretable machine learning of\nmany-body topological quantum order. The novelty is to link quantum\nalgorithms and informational distances (Nielsen's quantum circuit\ncomplexity) with topological quantum phase classification, building a bridge\nbetween complexity theory and interpretable machine learning, where\ninterpretability and generalizability are of central concern.\nIn our work, such an intrinsic distance metric (Nielsen’s quantum circuit\ncomplexity and its substitutes) is a core component for (unsupervised)\nmanifold learning of topological order, where the data is assumed to be\ndistributed on a manifold characterized by this distance metric. Once this\nintrinsic distance of data manifold is established, the following specific\nalgorithm of manifold learning can be diverse, including kernel or non-\nkernel methods. For instance, in the diffusion map (used in generating Fig.\n2 of our main text) and the t-SNE algorithms, a kernel built from a distance\nmetric is used to construct a probabilistic graph on the data manifold; In the\nkernel PCA (for generating Fig. 3 and is also adopted in [1]), data points\nare embedded into the space of leading eigen-components of the kernel\nmatrix. Regarding non-kernel methods, such as the Isomap and the\nmetric-multidimensional scaling (metric-MDS), a distance matrix of the data\nmanifold is already sufficient for the unsupervised clustering (Now we\nadded a metric-MDS classification of the toric code and product state in\nSec. VI of the Supplementary Information, as an alternatively non-kernel\napproach compared to the kernel PCA in Fig. 3).\nThe classical shadow kernel in [1] is formulated in a very elegant manner\nbut seems to lack explicit interpretability, in particular, when applied to\nclustering topological order (for instance, from the primitive notion of\ntopological equivalence). We agree with the reviewer that our fidelity-based\nkernel can be not new in the sense that it is similar to the shadow kernel [1]\n(which we already pointed out in the manuscript), while it is derived from a\nmanifold distance metric for topological order, i.e., the Nielsen’s QCC. This\nis an important new insight beyond the paper [1], and can be helpful in\nincreasing the interpretability of the classical shadow kernel. Moreover, an\nalternative distance / kernel approximated from the Nielsen’s QCC is\nentanglement-based, which arguably is the more generalizable and more\ninterpretable one for clustering topological phases. This has not yet been\nproposed in [1] or elsewhere.\nOther improved distance approximations (compared to the fidelity-based\none) could be discovered in the future. One possible choice can be the\nquantum Fisher complexity (QFC), because quantum Fisher information\nalso reflects the entanglement depth of quantum states, and we now show\nin the added Sec. III of the Supplementary Information that the QFC has\nthe same scaling behaviour as Nielsen's QCC.\nMoreover, our method is, in principle, not restricted to only clustering\ntopological and trivial phases. There can be multiple topologically distinct\nphases on the data manifold to be addressed. The method serves to\ndistinguish any two quantum many-body states that have different quantum\ncircuit complexities or long-range entanglement structures.\nIn the revised manuscript, we now added more elaborations to clarify these\npoints and the focus of our work (see the blue marked sentences in the\nAbstract, Introduction, Methods, etc.). Following the reviewer’s suggestion,\nwe also added tightness analyses of the bounds in Theorems 1 and 2; as\nwell as the scaling analysis of the method (see Sec. III and Sec. IV,\nrespectively, in the Supplementary Information, which are also mentioned /\nreferenced in the main text.). A non-kernel manifold learning of the toric\ncode is performed in Sec. VI of the Supplementary Information, which gives\nsimilar results as the kernel PCA in Fig. 3.\nWe hope that these revisions can address your comments and can remove\nyour concern satisfactorily.\nThank you very much for your constructive comments and suggestions\nwhich have been very helpful in improving the manuscript significantly.\nComment 1: I am not sure how tight those two lower bounds of quantum\ncircuit complexity are, especially for those many-body ground states\ndiscussed in the paper;\nReply: Many thanks for this constructive comment. Now in the revised\nversion, we have added “Sec. III: Asymptotic and tightness analysis of the\nbounds in Theorems 1 and 2” in the Supplementary Information to address\nthis comment.\nComment 2: If I understand correctly, the entanglement-based kernel\nrequires the estimation of subsystem entanglement entropy of all possible\nsizes, which is challenging to measure in scalable systems especially when\nit cannot be classically simulated by MPS.\nReply: We understand the reviewer’s concern. Indeed, the entanglement\nentropy profile requires the estimation of subsystem entanglement entropy\nof all possible sizes (at least half system size due to the symmetry). This is\na challenging task in the worst case, which indicates the hardness of\nlearning topological order in the worst case. Efficient tensor-network states\nand MPS, or quantum algorithms for certain quantum states can make the\nmeasurement of entanglement entropy more accessible. This is still an\nactive research area of quantum information and computation.\nNow we added elaborative discussions about this comment in “Sec. IV:\nComplexity and scaling analysis of the method” of the Supplementary\nInformation.\n=========\nReferences:\n[1]. H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill,\n“Provably efficient machine learning for quantum many-body problems,”\nScience 377, eabk3333 (2022).\n=======================================================\nResponse to Reviewer #2:\nReferee’s comment (general):\nIn the present paper the Authors focus on the task of identifying topological\norders of many-body states with quantum algorithmic approach, namely\nbuilding corresponding circuit and evaluating its complexity. The paper\nproposes a new theoretical framework for unsupervised machine learning\nof topological quantum order by leveraging quantum circuit complexity\n(QCC), by providing a proof of several key theorems. The authors\nreinterpret the notion of topological equivalence between quantum states in\nterms of constant-depth quantum circuits. This connects machine learning\nkernels with a fundamental measure of quantum complexity for the two\nkernel constructions - fidelity-based kernel and entanglement-based kernel.\nTheorems establish lower bounds on QCC via Bures distance (fidelity\nchanges) and entanglement changes. The Authors demonstrate that the\nproposed kernels can be efficiently computed in simulations like DMRG\nand potentially measured experimentally using classical shadow\ntomography. Examples contain bond-alternating XXZ spin chain and toric\ncode. The paper is sound and represents an important step in building\ninterpretable unsupervised machine learning techniques for identification of\ntopological orders. The analytical proofs of theorems are clearly written.\nThe presented figures explain well the main findings of numerical\nsimulations and provide a clear comparison of two proposed kernels. I\nwould recommend paper for publication after the following questions and\ncomments are clarified:\nReply:\nWe thank the reviewer for the positive comments, highlighting that “The\npaper proposes a new theoretical framework for unsupervised machine\nlearning of topological quantum order by leveraging quantum circuit\ncomplexity (QCC), by providing a proof of several key theorems”, and\n“represents an important step in building interpretable unsupervised\nmachine learning techniques for identification of topological orders”.\nWe thank the reviewer for the prospect of recommendation of our\nmanuscript for publication.\nBelow we provide a point-to-point response to the very constructive\ncomments from the reviewer, which have been very helpful in improving the\nmanuscript significantly.\nComment 1: How well are presented kernels expected to scale for larger\nsystems? In other words, whether the proposed kernel-based methods will\nbe practically applicable for large condensed matter systems of interest?\nReply: We thank the reviewer very much for pointing this scaling aspect\nout, which was missing in our original manuscript. Now we added an\nelaborative discussion on the scaling analysis (with respect to the system\nsize as well as the sample size) of estimating the kernels and of the\nmethod. Please see “Sec. IV: Complexity and scaling analysis of the\nmethod” in the Supplementary Information (which is also referenced /\nmentioned in the main text).\nComment 2: The entanglement-based kernel is shown to be more robust\nagainst two-qubit random noise. How do both kernels behave under\nrealistic hardware noise models (e.g., correlated errors, decoherence)?\nReply: We thank the reviewer for the constructive comment. Noises can\nhappen in the quantum state preparation or can be simulated by applying\nrandom quantum circuits. Now we added discussions on the noise effects\nconsidering potentially three scenarios. Please see “Sec. V: Effects of\nnoise and random quantum circuits” in the Supplementary Information\n(which is also referenced / mentioned in the main text).\nComment 3: In Fig.2, panels (c) and (d) the axes labels are indicated only\nin the caption. Could they be indicated also on the axes?\nIn addition, both plots contain a single point of the yellow-light-green color,\nthat stays in the middle between phases. This point corresponds\napproximately to J2/J1=1.5 . If more dense grid is taken for J2/J1 ratios,\nwould it result in symmetry-broken phase being smoothly connected to one\nof the topo/trivial phases in the axes coordinates? This question requires\nclarification as one might expect the same problems of intersecting areas\nas one finds in principal component analysis classification of phases.\nReply: We thank the reviewer for pointing it out. Now we added the axes\nlabels in panels (c) and (d). The outlier point corresponding to J2/J1=1.5 is\nthe (gapless) critical phase between the symmetry-broken and topological\nphases. This is due to the finite-size effect and the resulted finite-width of\nthe critical region. Now we replotted Figure 2 with a larger system size\n(number of qubits n = 151), where the clustering performance is improved\nand the size of each phase-cluster becomes smaller compared to the\norginal plots with n = 101.\nRemarks on code availability:\nThe run of code provides the same output as shown in figures. However,\nthe code does not contain the readme file that would clearly indicate a\nstructure and purpose of particular files / classes and functions. At the\nsame time, file names and short readme files with instructions to run\nexplain the general idea behind the code. In addition, not all the functions\nand classes implemented in the code have their clear descriptions in the\nform of docstrings or in readme file. My overall evaluation is that the code is\ngood for reproducibility purposes for data shown in paper, but has\nlimitations in re-use as it requires extra time to understand its internal\nstructure.\nReply: We thank the reviewer very much for kindly pointing this out. In a\nlater submission to the Code Ocean, we included a Readme text for all the\ncode files, describing the run steps to reproduce the figures in our\nmanuscript, which will be helpful in improving the readability and the re-\nusability of our code. Now we also add a Readme text for each code file in\nthis resubmission.\n====================================================\nResponse to Reviewer #3:\nReferee’s comment (general):\nIn this reviewed manuscript “Quantum circuit complexity and unsupervised\nmachine learning of topological order”, the authors explore how to build\ninterpretable and efficient unsupervised learning model for identifying\ntopological order, with a focus on the quantum circuit complexity. The\nauthors prove two theorems that link Nielsen’s geometric definition of\nquantum circuit complexity to fidelity variation and entanglement entropy\ndifferences between quantum states. From these relations, they derive two\npractical kernels (fidelity-based and entanglement-based kernels) to\ncapture the topological similarity between states. They also carry out\nnumerical experiments on classifying phases of XXZ spin chain, ground\nstates of Kitaev’s toric code to support their theoretical findings. A\nconnection to shadow kernel learning is discussed.\nUnsupervised learning topological phases of matter is a challenging task.\nThis work makes a conceptual step by tying informational distances to\nquantum phase classification, providing an interpretable bridge between\ncomplexity theory and practical machine learning. I think this is an\ninteresting paper and a timely contribution to both quantum information and\ncondensed matter communities. Given that the manuscript is original,\nconceptually well-motivated, and technically sounds, I thus recommend the\nacceptance of this work in Nature Communications after the following\ncomments/suggestions are carefully addressed:\nReply:\nWe thank the reviewer for the positive comments, in particular, for\nhighlighting our focus on quantum circuit complexity and on “providing an\ninterpretable bridge between complexity theory and practical machine\nlearning”, and for the (conditional) recommendation of acceptance of our\nwork.\nBelow we provide a point-to-point response to the very constructive\ncomments from the reviewer, which have been very helpful in improving the\nmanuscript significantly.\nComment 1: The toric code test with random unitaries suggests\nrobustness of the entanglement kernel. It would be helpful to quantify how\nclassification performance degrades with increasing noise levels. This\nwould make the claims of robustness more concrete.\nReply: We thank the reviewer very much for kindly pointing this out, which\nwas missing in the original manuscript. Now we added sub-figures which\nplot the silhouette coefficient / score, a measure of the clustering\nperformance. These plots show clearly that the performance degradation of\nthe entanglement-based approach is much slower as the noise level\nincreases. Please see Fig. 3 (b) and (d).\nComment 2: The numerical demonstrations involve systems of 101 spins\n(XXZ chain) and 32 qubits (toric code). While this is reasonable for proof-\nof-principle, it would be helpful to comment more on the scalability.\nReply: We understand the reviewer’s concern. Now we added more\ncomments and discussions on the scaling analysis of the method in “Sec.\nIV: Complexity and scaling analysis of the method” of the Supplementary\nInformation (which is also referenced / mentioned in the main text). Many\nthanks for pointing this out.\nComment 3: The authors may consider citing the following works, which\nreport related advances and would enrich the discussion.\n1. npj Quantum Information 8, 116(2022);\n2.Fundamental Research 4, 1086(2024).\nReply: Thank you for pointing out these relevant and important works. Now\nwe cite these works in the updated manuscript.\nComment 4: Some typos:\nLine 72 : “a a suitable distance” --> “a suitable distance”.\nLine 158: “permmutations” --> “permutations”.\nReply: It is greatly appreciated. Now these typos have been corrected.\nPlease see the manuscript.\n=========================================================\nIn summary, we believe that we have addressed all the reviewers’\nconcerns and comments. We hope that now the paper is suitable for\npublication in Nature Communications."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thanks to the authors for their response to my comments. I appreciate that the main contribution of this work is to offer a new perspective and framework for the unsupervised learning of topological quantum phases, which enhances the interpretability of such classifications. Nevertheless, I remain unconvinced that these contributions justify publication in Nature Communications. My primary concern is that the link drawn between quantum circuit complexity and topological phases does not, in my view, yield fundamentally new insight. As noted in the authors’ response, fidelity-based methods here share similarities with earlier shadow-based approaches. Likewise, entanglement-based methods are not wholly novel, given the established understanding that subsystem entanglement entropy encodes topological features of quantum states.\nIn summary, while this work introduces a valuable perspective on unsupervised classification of topological phases, it lacks demonstrative results showing that the proposed approach surpasses prior methods in computational efficiency or experimental feasibility.", "reviewer_2": "In the revised manuscript, the Authors addressed all question and requests provided by the referees and significantly improved a manuscript in the number of subtle points. In particular, the discussion about the scaling of approach with sizes and the performance with the noise was significantly improved. Based on this evaluation and the timely appearance of these results, as was pointed out by the third Referee, I recommend the paper for publication in Nature Communications journal.\n(Remarks on code availability)\nThe Authors improved the clarity and reusability of the code by providing explanation files within each section, as was requested.", "reviewer_3": "The authors have addressed all of my concerns in the revised manuscript. I thus recommend the publication of this work in Nature Communications."}, "author_response": "Response to reviewers’ comments and recommendations in\nthe second round of review:\n========================================================================\nResponse to Reviewer #1:\nReferee’s comments: Thanks to the authors for their response to\nmy comments. I appreciate that the main contribution of this work\nis to offer a new perspective and framework for the unsupervised\nlearning of topological quantum phases, which enhances the\ninterpretability of such classifications. Nevertheless, I remain\nunconvinced that these contributions justify publication in Nature\nCommunications. My primary concern is that the link drawn\nbetween quantum circuit complexity and topological phases does\nnot, in my view, yield fundamentally new insight. As noted in the\nauthors’ response, fidelity-based methods here share similarities\nwith earlier shadow-based approaches. Likewise, entanglement-\nbased methods are not wholly novel, given the established\nunderstanding that subsystem entanglement entropy encodes\ntopological features of quantum states. In summary, while this\nwork introduces a valuable perspective on unsupervised\nclassification of topological phases, it lacks demonstrative results\nshowing that the proposed approach surpasses prior methods in\ncomputational efficiency or experimental feasibility.\nReply:\nWe thank the referee for appreciating our main contribution in this\npaper that “offer a new perspective and framework for the\nunsupervised learning of topological quantum phases, which\nenhances the interpretability of such classifications”; and that\n1\n“introduces a valuable perspective on unsupervised classification\nof topological phases”.\nThe new insight we provide here is a quantum algorithmic\ninformation paradigm to understand and, more importantly, to\ngeneralize unsupervised (classical and quantum) machine\nlearning of quantum many-body states. It is not restricted to\ninterpretable phase clustering, but also can be applied to\ngenerative modelling of (topological) quantum states via\noptimized / minimal quantum circuits (i.e., quantum program) from\na reference quantum state.\nThe Nielsen’s quantum circuit complexity itself has been widely\nstudied in quantum information and quantum field theories;\nSeparately, the same happened to quantum Fisher information\nand Bures distance (see the Introduction section of our paper).\nHowever, the bridge to these concepts in a provable manner\n(Theorem 1 in our paper), and thus, the connection to the shadow\nkernel learning in the literature, has not been established\nelsewhere.\nAdditionally, the geometric locality of the reduced density matrix is\na new ingredient in our fidelity-based model, while the shadow\nkernel does not require it explicitly. We find that the geometric\nlocality will boost the clustering performance, in particular, when\nthe system size becomes large (for instance, when the number of\nqubits is n = 151 in the bond-alternating XXZ spin model in our\npaper).\nRegarding the sample efficiency and the computational efficiency,\nwe have provided an elaborated analysis in Sec. IV of the\nwith respect to the number of qubits n, the entanglement-based\napproach is efficient for quantum states that are simulated by\nMPS or tensor networks, and is experimentally feasible under\nefficient quantum measurement and quantum estimation\nprotocols (e.g., the density matrix exponentiation or quantum\nalgorithms for entanglement entropy estimations; see our\ndiscussions in the SI). (2) For the scaling of the method with\nrespect to the number of samples N, the kernel-based methods\n(including the shadow kernel learning) are O(N^3), while manifold\nlearning directly based on the informational distances (without a\nkernelization) such as the metric-MDS (see Fig. S1 in the SI) is\nmore efficient, scaling with O(N^2). This is superior for large data\nset. We have provided such discussions in the SI (see Sev. IV)\nand the Methods section.\nThese are provable results that all demonstrative experiments\nshould obey.\nIn summary, we thank the referee for the appreciation and the\npositive comments on our main contribution in this work, and\nappreciate the reviewer’s significant contributions in the review\nprocess. We clarify new insights and efficiency of our method in\nresponse to other comments of the reviewer. In this paradigm of\nquantum algorithmic information, we hope our work would pave\nthe way toward an interpretable, generalizable, and efficient (both\nclassically and quantumly) unsupervised machine learning for\ntopological order.\n3\n========================================================================\nResponse to Reviewer #2:\nReferee’s comments: In the revised manuscript, the Authors\naddressed all question and requests provided by the referees and\nsignificantly improved a manuscript in the number of subtle points.\nIn particular, the discussion about the scaling of approach with\nsizes and the performance with the noise was significantly\nimproved. Based on this evaluation and the timely appearance of\nthese results, as was pointed out by the third Referee, I\nrecommend the paper for publication in Nature Communications\njournal.\n(Remarks on code availability): The Authors improved the clarity\nand reusability of the code by providing explanation files within\neach section, as was requested.\nReply:\nWe thank the reviewer for the positive comments and the\nrecommendation of our paper for publication. We appreciate the\nreviewer’s significant contributions in the review process.\n========================================================================\nResponse to Reviewer #3:\nReferee’s comments: The authors have addressed all of my\nconcerns in the revised manuscript. I thus recommend the\npublication of this work in Nature Communications.\nWe thank the reviewer for the positive comments and the\nrecommendation of our paper for publication. We appreciate the\nreviewer’s significant contributions in the review process.\n5"}]

Decision letter