Learning topological states from randomized measurements using variational tensor network tomography

TL;DR

This paper introduces a variational tensor network tomography method that uses randomized measurements to efficiently learn topologically ordered quantum states, demonstrated on surface code and quantum spin liquid states with theoretical and numerical analysis.

Contribution

It presents a novel heuristic approach combining tensor networks and randomized measurements for topological state tomography, with proven sample efficiency and measurement completeness.

Findings

Successfully learned ground states of surface code Hamiltonian

Achieved high fidelity with up to 48 qubits

Proved measurement scheme is tomographically complete for real pure states

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 ans\"atze 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.