Learning Topological States from Randomized Measurements Using Variational Tensor-Network Tomography

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 matrix product state (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 with sizes of 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>X</a:mi></a:math> or <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>Z</c:mi></c:math> 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-<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>X</e:mi><e:mi>Z</e:mi></e:math> measurements are tomographically complete for such states.