Verifying random matrix product states with autoregressive local measurements

TL;DR

This paper introduces an efficient autoregressive sampling method for verifying matrix product states in quantum systems, significantly reducing classical overhead and enabling scalable validation of long 1D quantum states.

Contribution

It develops an autoregressive importance sampler and grouped measurement scheme to efficiently verify MPS and MPO with reduced classical and measurement overhead.

Findings

Reduces classical overhead from exponential to linear in qubit number.

Constructs commuting measurement settings via sorting strings, lowering estimator variance.

Extends verification methods to matrix product operators for long 1D systems.

Abstract

Matrix product states (MPS) are a central language for one-dimensional quantum matter and a practical target for near-term quantum simulators and variational algorithms. Yet, while substantial effort has focused on preparing MPS with shallow circuits, scalable methods to \emph{verify} that a many-body device has actually produced the intended state remain underdeveloped. Direct fidelity estimation (DFE) relies only on local Pauli measurements, but in many-body settings it suffers an exponential classical overhead from the preprocessing needed to sample Pauli strings. We eliminate this obstacle by introducing an \emph{autoregressive} importance sampler that draws Pauli strings sequentially from efficiently computable conditional distributions, reducing the per-shot classical overhead to linear scaling in the number of qubits. We further develop a grouped extension that constructs qubit-wise commuting measurement settings via a \emph{sorting string} and simultaneously estimates the entire commuting group from a single setting, significantly reducing estimator variance while preserving efficient postprocessing. Our approach extends naturally to matrix product operators (MPO), enabling scalable verification of tensor-network states and observables in long one-dimensional quantum systems. We utilize random MPS as a natural benchmark for generic 1D entangled states.