Learning topological states from randomized measurements using variational tensor network tomography

TL;DR

The paper introduces a variational tensor-network tomography framework that combines randomized measurements with maximum likelihood estimation to reconstruct topologically ordered 2D states. Using an MPS prior along a snake path and random-$XZ$ measurements, the method achieves faithful learning of surface-code and Rydberg spin-liquid ground states with favorable sample complexity, and provides provable insights into MLE on tensor-network manifolds. It demonstrates the tomographic completeness of random-$XZ$ measurements for real pure states and leverages classical shadows for regularization, enabling efficient estimation of nonlocal observables and entanglement properties even at sizes up to 48 qubits. The work outlines extensions to higher-dimensional TNs (e.g., PEPS) and MPOs, discusses noise and practical considerations, and provides open-source code and data for community use, highlighting a pathway for practical tomography of exotic quantum phases on near-term devices.

Abstract

Learning faithful representations of quantum states is crucial to fully characterizing the variety of many-body states created on quantum processors. While various tomographic methods such as classical shadow and MPS tomography have shown promise in characterizing a wide class of quantum states, they face unique limitations in detecting topologically ordered two-dimensional states. To address this problem, we implement and study a heuristic tomographic method that combines variational optimization on tensor networks with randomized measurement techniques. Using this approach, we demonstrate its ability to learn the ground state of the surface code Hamiltonian as well as an experimentally realizable quantum spin liquid state. In particular, we perform numerical experiments using MPS ansätze and systematically investigate the sample complexity required to achieve high fidelities for systems of sizes up to $48$ qubits. In addition, we provide theoretical insights into the scaling of our learning algorithm by analyzing the statistical properties of maximum likelihood estimation. Notably, our method is sample-efficient and experimentally friendly, only requiring snapshots of the quantum state measured randomly in the $X$ or $Z$ bases. Using this subset of measurements, our approach can effectively learn any real pure states represented by tensor networks, and we rigorously prove that random-$XZ$ measurements are tomographically complete for such states.

Figures (8)

• Figure 1: Schematic illustration of variational tensor network tomography.Data collection: A target state vector $\hbox{$| \phi \rangle$}$ is measured in multiple bases for a collection of $N$ samples defined in Eq. \ref{['eq:data']}. For instance, the first panel shows measurements in a random $X$ or $Z$ basis for each qubit. Optimization: A tensor-network model is variationally optimized from $N$ samples using gradient descent, which outputs tensors $\hat{A}^{(N)}$ representing the state vector $\hbox{$| \psi(\hat{A}^{(N)}) \rangle$}$. The middle panel illustrates an MPS model prior, as numerically investigated in this work. The loss function given by Eq. \ref{['eq:loss_function_mps']} includes the negative log-likelihood (NLL), typically used in maximum likelihood estimation (MLE). It is optionally regularized by physical observables estimated from the dataset (e.g., using classical shadow tomography represented by the pink dashed box). Evaluation: The algorithm outputs a tensor network or, in the current work, an MPS representation $\hat{A}^{(N)}$ as the learned state. We sometimes omit the number of samples and abbreviate the output tensor as $\hat{A}$. In the right panel, we evaluate the performance of our learning protocol by computing the fidelity $F=\lvert\hbox{$\langle \phi | \psi(\hat{A}) \rangle$}\rvert$. From the reconstructed state, we can extract arbitrary physical observables, including the von Neumann entanglement entropy and highly nonlocal Pauli strings (the inset shows an example of the nonlocal $Z$ string operators considered in Fig. \ref{['fig:4_spin_liquid']}).
• Figure 1: Infidelity scaling for interpolation between a random and GHZ $3$-qubit state. The numerical results illustrate the polynomial scaling of the infidelity with the number of samples $N$ across a family of states, parametrized by $x$. Here, $x$ ranges from 0 (darkest color, representing the GHZ state) to 1 (lightest color, representing the random state). Dashed lines indicate polynomial fits of the form $1-F = c \times N^{-\alpha}$, with values for $c$ and $\alpha$ provided in the legend.
• Figure 2: Numerical results for the surface code.(a) Schematic representation of the surface code of dimension $(L_x,\, L_y)$ with $Z$ stabilizers (red patches) and $X$ stabilizers (blue patches). The MPS "snakes" through the system along the path shown (black arrow). (b) Scaling of the infidelity with the number of samples $N$ for a $3 \times 3$ surface code with global-$XZ$ (cross) or random-$XZ$ (circle) measurements. Shades of blue indicate the regularization strength $\beta$. (c) To investigate larger system sizes, we focus on the surface code in a strip geometry ($L_y=3$) with a perturbation ($h_z = 0.1$). The three lines indicate the infidelity for different numbers of qubits, $n$, with random-$XZ$ measurements. Dark blue indicates a regularization $\beta=5$. The dashed line marks the fidelity threshold for $n=9$. (d) Sample complexity plotted on a $log$-$log$ scale to reach a fixed local fidelity threshold $F_{\text{local}}=(0.99)^{n}$ extracted from (c). The number of samples required for achieving such a local fidelity is seen to scale at most polynomially with $n$. Colors and markers are the same as in (b).
• Figure 2: Rydberg atom array on a cylinder. The ruby lattice is placed on a cylinder, imposing periodic (open) boundary conditions along the $y$ ($x$)-axis. The numbers label the snake-like path for the tensors in the MPS representation of the ground state. An onsite field applied to the sites in the blue boxes compensates for the reduced coordination number of atoms situated along the open boundaries, thereby preventing them from getting pinned.
• Figure 3: Numerical results for learning quantum spin liquids.(a) The tuning parameters of the Rydberg Hamiltonian are the Rabi frequency $\Omega$, the detuning $\Delta$, and the blockade radius $R_b$. The gray arrow shows the mapping between a particular measurement bitstring on the ruby lattice and a dimer configuration on the kagome lattice: light (dark) blue circles correspond to an atom being in the excited $\hbox{$| r \rangle$}$ (ground $\hbox{$| g \rangle$}$) state and maps to the presence (absence) of a dimer. A careful choice of the Rydberg blockade radius $R_b=2a$, where $a$ is the lattice spacing, ensures that the dimers do not overlap. Bottom panel: The two phases as a function of the detuning ratio $\Delta/\Omega$. (b) The ground state belongs to the trivial disordered phase at $\Delta / \Omega = 0.5$. The infidelity decreases with an increasing number of samples $N$. For random-$XZ$ measurements with $N \leq 40000$, a regularization of $\beta=1$ (red circles) further improves the fidelity compared to the case without regularization $\beta=0$ (red squares). Global-$XZ$ measurements (gray crosses) perform worse than their random-$XZ$ counterparts (red). (c) Same as in (b) but for the spin-liquid state at $\Delta / \Omega = 1.7$. Relative to the case with no regularization $\beta=0$ (light blue), regularization $\beta=1$ (dark blue) does not improve the fidelity but reduces the number of nonconvergent outliers (above the black dashed line). Global-$XZ$ (gray cross) measurements perform similarly to random-$XZ$ ones (blue).

Theorems & Definitions (40)

• Theorem 1: Tomographic completeness
• Theorem 2: Probabilistic bound
• Definition 1: Projective Hilbert space haegeman_geometry_2014
• Definition 2: Projective MPS haegeman_geometry_2014 (Definition 13)
• Definition 3: Riemannian metric kuehnel_diffgeo_2008
• Definition 4: Riemannian manifold
• Definition 5: Riemannian connection kuehnel_diffgeo_2008
• Definition 6: Directional derivative
• Definition 7: Parallel transport
• Definition 8: Shortest connecting geodesic