Quantum circuit complexity and unsupervised machine learning of topological order

TL;DR

The paper addresses unsupervised learning of topological order in quantum many-body systems by introducing Nielsen's quantum circuit complexity ($\mathcal{C}_{\mathcal{N}}$) as a topological distance that encodes multi-scale entanglement patterns. It derives two rigorous substitutes, the quantum Fisher complexity ($\mathcal{C}_{\mathcal{F}}$) and an entanglement-based bound, and builds two practical kernels, $\mathcal{K}_{\mathrm{F}}$ and $\mathcal{K}_{\mathrm{E}}$, for kernel-based and non-kernel manifold learning. Numerical experiments on the bond-alternating XXZ chain and Kitaev's toric code demonstrate effective clustering into distinct topological phases, with the entanglement-based kernel showing superior robustness to noise and better interpretability. The work forges connections between quantum circuit complexity, quantum metrology, and topology-aware machine learning, providing a pathway toward interpretable, scalable unsupervised classification of topological order and potential applicability to near-term quantum hardware via shadow tomography and entanglement measurements.

Abstract

Inspired by the close relationship between Kolmogorov complexity and unsupervised machine learning, we explore quantum circuit complexity, an important concept in quantum computation and quantum information science, as a pivot to understand and to build interpretable and efficient unsupervised machine learning for topological order in quantum many-body systems. We argue that Nielsen's quantum circuit complexity represents an intrinsic topological distance between topological quantum many-body phases of matter, and as such plays a central role in interpretable manifold learning of topological order. To span a bridge from conceptual power to practical applicability, we present two theorems that connect Nielsen's quantum circuit complexity for the quantum path planning between two arbitrary quantum many-body states with quantum Fisher complexity (Bures distance) and entanglement generation, respectively. Leveraging these connections, fidelity-based and entanglement-based similarity measures or kernels, which are more practical for implementation, are formulated. Using the two proposed distance measures, unsupervised manifold learning of quantum phases of the bond-alternating XXZ spin chain, the ground state of Kitaev's toric code and random product states, is conducted, demonstrating their superior performance. Moreover, we find that the entanglement-based approach, which captures the long-range structure of quantum entanglement of topological orders, is more robust to local Haar random noises. Relations with classical shadow tomography and shadow kernel learning are also discussed, where the latter can be naturally understood from our approach. Our results establish connections between key concepts and tools of quantum circuit computation, quantum complexity, quantum metrology, and machine learning of topological quantum order.

Figures (5)

• Figure 1: Landscape of machine learning topological phases of matter. Supervised learning requires prior information or labels, whose efficiency guarantees have been extensively investigated HuangScience2022ManybodyLewisNC2024ChePRR2024RouzeNC2024ChoNC2024ZhaoPRXQ2024OnoratiArxiv2024DuArxiv2024WannerArxiv2024arxiv:2312.17019. Under the constraint of geometric locality LewisNC2024RouzeNC2024 or with a bounded number (No.) of parameters (param.) ChePRR2024, provably efficient machine learning of quantum many-body systems has been established, e.g., in the framework of probably approximately correct (PAC)-learnable Valiant1984ChePRR2024. Supervised learning is conceptually more straightforward due to the existence of well-established statistical learning theories. In contrast, unsupervised learning, whose theory is more complicated and less addressed, becomes practically relevant when there is no access to prior labels. The symmetry-protected topological (SPT) order features short-range entanglement, while the topological quantum order exhibits long-range entanglement. Unsupervised learning with the path-finding algorithm ScheurerPRL2020 and kernels focusing on closing gaps at topological transitions ChePRB2020LongPRL2023, respectively, have shown good performance and interpretability for SPT orders or band topologies. However, the direct generalization of these methods to more complex quantum systems with increased complexity and quantum entanglement faces challenges, due to the intrinsic hardness of identifying (or even the undecidability CubittNature2015 of) the spectral gap for generic quantum many-body Hamiltonians (See more elaborations and references in the introduction of the main text). Note that reinforcement learning for quantum systems are not included in this landscape.
• Figure 2: Unsupervised manifold learning of the bond-alternating XXZ qubit chain with fidelity- and entanglement-based informational distances. (a) and (b) show the fidelity- and entanglement-based kernels, $\mathcal{K}_{\mathrm{F}} \left( \rho, \tilde{\rho}\right)$ and $\mathcal{K}_{\mathrm{E}} \left( \rho, \tilde{\rho}\right)$, respectively. One sees clearly three clusters in the heat map, corresponding to three distinct quantum phases of the model (trivial, symmetry broken, and topological) in a range of the model parameter $J_2/J_1 \in (0,\ 2.5]$, which is represented by the two axes of (a) and (b), respectively. The colorbar encodes the value of the normalized kernel. (c) and (d) are the corresponding two-dimensional ($\mathrm{2D}$) nonlinear representations of the samples, through the diffusion map algorithm [plotted are the second and the third dimensions in the diffusion space (Components 1 and 2, respectively), after normalization by the standard deviation]. The colorbar encodes the value of $J_2/J_1$, and the cross in each plot indicates the centre of the cluster found by a $k$-means clustering algorithm. The outlier in (c) with the value of $J_1 / J_2 \approx 1.5$ is the critical phase due to the finite-size effect and the resulted finite-width of the critical region. We use $n = 151$ qubits, and the DMRG [with the singular value decomposition (SVD) cutoff $10^{-10}$ and the maximal energy error $10^{-10}$] to determine the ground state. We set $h_0 = 0$ and $\delta = 3$, while $N = 30$ samples are drawn uniformly in the parameter range of $J_2/J_1 \in (0,\ 2.5]$ in an ordered manner. The hyperparameter of the kernel is $\beta = 50.0$ for (a) and (c); and $\beta = 10.0$ for (b) and (d). The entanglement profile is accessible in the DMRG calculation with the matrix product state representation. In the fidelity-based kernel, we use $n$ geometrically local (nearest-neighbour) two-body reduced density matrices to cover the system, i.e., we only use terms with $r = 2$ and $\omega_{r=2} = 1/n$ in (\ref{['eq:K_F']}).
• Figure 3: Unsupervised clustering of the ground state of Kitaev's toric code (TC) [blue dots in sub-figures (a) and (c)] and random product states (RPS) [red dots in sub-figures (a) and (c)] without or with applying two-qubit random unitaries, via fidelity- and entanglement-based kernels, respectively. (a) and (c) show a one-dimensional ($\mathrm{1D}$) representation of the data set in terms of the kernel's principal components (i.e., via the algorithm of kernel PCA with shifting the center to zero and normalizing by the standard deviation), by applying repeatedly different depths of random two-qubit unitaries with respect to the Haar measure tenpy to both the toric code and the RPS. (a) and (c) utilize the fidelity- and entanglement-based kernels, $\mathcal{K}_{\mathrm{F}} \left( \rho, \tilde{\rho}\right)$ and $\mathcal{K}_{\mathrm{E}} \left( \rho, \tilde{\rho}\right)$, respectively. (b) and (d) plot the degradation of clustering performance under Haar random noises for results in (a) and (c), respectively, with the silhouette coefficient Sklearn_Silhouette (which takes the value one for best clustering) as a benchmark metric. The four subplots share the same horizontal axis. We use $N = 20$ samples in total, with 10 samples of the toric code (blue dots) and 10 samples of the RPS (random spin ups and downs of $n$ qubits) (red dots). We use $n = 32$ qubits (on a lattice of size $4 \times 4$ with toric boundary condtions; i.e., with a small code distance of $4$) for this proof-of-principle demonstration, and the DMRG (with the SVD cutoff $10^{-10}$ and the maximal energy error $10^{-10}$) to solve for the ground state of the toric code and, at the same time, to extract the entanglement profile. The hyperparameter of the kernel is $\beta = 0.1$ for (a) and $\beta = 2.0$ for (c). With the fidelity-based kernel, we randomly sample $n$ two-body reduced density matrices Note_RDM, i.e., we only use terms with $r = 2$ and $\omega_{r=2} = 1/n$ in (\ref{['eq:K_F']}). The result shows that, in the absence of noise (i.e. with circuit depth zero), both kernels types perform well in clustering the toric code and the RPS, while the entanglement-based kernel in (c) is found to be more robust to Haar random two-qubit unitaries [see (b) and (d)].
• Figure 4: Unsupervised clustering of the ground state of the extended toric code (ETC) (blue dots) and random product states (RPS) (red dots) without or with applying local perturbations, via fidelity- and entanglement-based kernels, respectively. (a) and (b) show a one-dimensional ($\mathrm{1D}$) representation of the data set in terms of the kernel's principal components (i.e., via the algorithm of kernel PCA with shifting the center to zero and normalizing by the standard deviation), by applying different field strengths of local perturbations ($h$) to the ETC. (a) and (b) utilize the fidelity- and entanglement-based kernels, $\mathcal{K}_{\mathrm{F}} \left( \rho, \tilde{\rho}\right)$ and $\mathcal{K}_{\mathrm{E}} \left( \rho, \tilde{\rho}\right)$, respectively. The two subplots share the same horizontal axis. We use $N = 20$ samples in total, with 10 samples of the ETC (blue dots) and 10 samples of the RPS (random spin ups and downs of $n$ qubits) (red dots). The boundary conditions are given by an infinite cylinder geometry, where it is periodic in the $y$ direction with a circumference of $L_y = 4$; In the $x$-direction it is infinite, and we choose a unit cell of width $L_x = 2$. The MPS-based DMRG (with the SVD cutoff $10^{-10}$ and the maximal energy error $10^{-10}$) is used to solve for the ground state of the ETC and, at the same time, to extract the entanglement profile for both zero and nonzero values of $h$. The hyperparameter of the kernel is $\beta = 0.2$ for (a) and $\beta = 2.0$ for (b). With the fidelity-based kernel, we randomly sample $n$ two-body reduced density matrices, i.e., we only use terms with $r = 2$ and $\omega_{r=2} = 1/n$ in (\ref{['eq:K_F']}). The result shows that the topological clustering is robust to local perturbations.
• Figure S1: Unsupervised manifold clustering of the ground state of Kitaev's toric code (blue dots in all sub-figures) and product states (red dots in all sub-figures) via the Bures distance (four sub-figures on the top) and entanglement distance (four sub-figures in the bottom) based metric-MDS, respectively. Two-qubit Haar random gates tenpy of different circuit depth (Circuit depth $= 0, 1, 2, 3$) are applied repeatedly to both the toric code and the product state $| 0\rangle ^{\otimes n}$. We use $N = 20$ samples in total, with $10$ samples of the toric code (blue dots) and $10$ samples of the product state (red dots). We use $n = 32$ qubits (on a lattice of size $4 \times 4$ with toric boundary conditions; i.e., with a code distance of $4$), and the DMRG (with the SVD cutoff $10^{-10}$ and the maximal energy error $10^{-10}$) to solve for the ground state of the toric code and, at the same time, to extract the entanglement profile. We use $n$ geometrically local $2$-body reduced density matrices to cover the system and to calculate the local Bures distance in Eq. (9) of Theorem 1 in the main text. The result shows that, in the absence of noise (i.e. with circuit depth zero), both the Bures- and the entanglement-metrics perform well in clustering the toric code and the product states, while the entanglement-MDS is more robust to Haar random noises, indicating that the entanglement metric is a better choice in both the clustering performance and the interpretability.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Theorem 2