Sample-Efficient Quantum State Tomography for Structured Quantum States in One Dimension

TL;DR

This paper demonstrates that for structured quantum states in one dimension, specifically MPO states, sample-efficient quantum state tomography is achievable with measurement schemes like SIC-POVMs and spherical t-designs, requiring only linear scaling in the number of parameters.

Contribution

The paper proves that MPO states in one dimension can be efficiently reconstructed using specific informationally complete measurements, establishing linear sample complexity in the number of parameters.

Findings

Sample complexity scales linearly with the number of parameters for SIC-POVMs and spherical 2-designs.

For spherical t-designs with t≥3, the number of copies needed is proportional to the MPO's parameter count.

The results affirm the existence of sample-efficient tomography protocols for structured quantum states.

Abstract

While quantum state tomography (QST) remains the gold standard for benchmarking and verifying quantum devices, it requires an exponentially large number of measurements and classical computational resources for generic quantum many-body systems, making it impractical even for intermediate-size quantum devices. Fortunately, many physical quantum states often exhibit certain low-dimensional structures that enable the development of efficient QST. A notable example is the class of states represented by matrix product operators (MPOs) with a finite matrix/bond dimension, which include most physical states in one dimension and where the number of independent parameters describing the states only grows linearly with the number of qubits. Whether a sample efficient quantum state tomography protocol, where the number of required state copies scales only linearly as the number of parameters describing the state, exists for a generic MPO state still remains an important open question. In this paper, we answer this fundamental question affirmatively by using a class of informationally complete positive operator-valued measures (IC-POVMs) -- including symmetric IC-POVMs (SIC-POVMs) and spherical $t$-designs -- focusing on sample complexity while not accounting for the implementation complexity of the measurement settings. For SIC-POVMs and (approximate) spherical 2-designs, we show that the number of state copies to guarantee bounded recovery error of an MPO state with a constrained least-squares estimator depends on the probability distribution of the MPO under the POVM but scales only linearly with $n$ when the distribution is approximately uniform. For spherical $t$-designs with $t\geq 3$, we prove that only a number of state copies proportional to the number of independent parameters in the MPO is sufficient for a guaranteed recovery of any state represented by an MPO.