[Paper Review] Quantum State Tomography for Matrix Product Density Operators

Zhen Qin, Casey Jameson+3 more|arXiv (Cornell University)|Jun 15, 2023

Quantum Information and Cryptography4 citations

TL;DR

This paper establishes theoretical guarantees for efficient quantum state tomography (QST) of one-dimensional quantum states approximated by matrix product operators (MPOs) with constant bond dimension. By leveraging compressive sensing and empirical process theory, it proves that only a polynomial number of state copies—linear in the number of qubits—are required for stable MPO reconstruction using Haar random projective measurements, even with statistical noise from single-shot measurements.

ABSTRACT

The reconstruction of quantum states from experimental measurements, often achieved using quantum state tomography (QST), is crucial for the verification and benchmarking of quantum devices. However, performing QST for a generic unstructured quantum state requires an enormous number of state copies that grows \emph{exponentially} with the number of individual quanta in the system, even for the most optimal measurement settings. Fortunately, many physical quantum states, such as states generated by noisy, intermediate-scale quantum computers, are usually structured. In one dimension, such states are expected to be well approximated by matrix product operators (MPOs) with a finite matrix/bond dimension independent of the number of qubits, therefore enabling efficient state representation. Nevertheless, it is still unclear whether efficient QST can be performed for these states in general. In this paper, we attempt to bridge this gap and establish theoretical guarantees for the stable recovery of MPOs using tools from compressive sensing and the theory of empirical processes. We begin by studying two types of random measurement settings: Gaussian measurements and Haar random rank-one Positive Operator Valued Measures (POVMs). We show that the information contained in an MPO with a finite bond dimension can be preserved using a number of random measurements that depends only \emph{linearly} on the number of qubits, assuming no statistical error of the measurements. We then study MPO-based QST with physical quantum measurements through Haar random rank-one POVMs that can be implemented on quantum computers. We prove that only a \emph{polynomial} number of state copies in the number of qubits is required to guarantee bounded recovery error of an MPO state.

Motivation & Objective

• To address the exponential resource cost of standard quantum state tomography (QST) for large-scale quantum systems.
• To establish theoretical bounds on the number of state copies required for reconstructing matrix product operator (MPO)-structured quantum states.
• To demonstrate that MPO states with constant bond dimension can be stably recovered using only polynomially many state copies.
• To show that Haar random projective measurements enable efficient QST with single-shot measurements despite statistical noise.
• To generalize the framework to practical measurement settings like t-designs for near-term quantum devices.

Proposed method

• Uses tools from compressive sensing and empirical process theory to analyze the stability of MPO recovery from random measurements.
• Analyzes two measurement models: Gaussian measurements and Haar random projective measurements, showing linear dependence on qubit count for information preservation.
• Applies Bernstein-type inequalities and multinomial concentration bounds to control statistical error in empirical measurements.
• Derives probabilistic bounds on recovery error using the trace norm minimization framework under random measurement ensembles.
• Establishes that a single measurement per random basis suffices for bounded error recovery, even with noisy outcomes.
• Generalizes results to local random or t-design measurements, which are more feasible on current quantum hardware.

Experimental results

Research questions

• RQ1 Can quantum state tomography be performed efficiently for MPO-approximated states with a constant bond dimension?
• RQ2 What is the minimum number of state copies required to stably recover an MPO state using random measurements?
• RQ3 Can Haar random projective measurements enable QST with only a polynomial number of copies, despite statistical noise?
• RQ4 Is it possible to achieve bounded recovery error using single-shot measurements per basis?
• RQ5 Can the framework be extended to more practical measurement settings like t-designs for current NISQ devices?

Key findings

• The number of random measurements required to preserve MPO information scales linearly with the number of qubits, not exponentially.
• Only a polynomial number of state copies in the number of qubits is needed to guarantee bounded recovery error for MPO states under Haar random projective measurements.
• Stable recovery is achievable even when each measurement basis is used only once, due to the concentration of empirical estimates.
• Theoretical bounds on recovery error are derived using multinomial concentration inequalities and empirical process techniques.
• The framework generalizes to t-design and local random measurements, making it applicable to near-term quantum devices.
• The results provide the first rigorous polynomial sample complexity bound for MPO state tomography, filling a key theoretical gap.