Quantum State Tomography for Matrix Product Density Operators

TL;DR

This work addresses the challenge of efficiently reconstructing quantum states that admit matrix product operator (MPO) representations, showing that MPOs with fixed bond dimension can be stably recovered from randomized measurements using a number of state copies that scales polynomially with the number of qudits. It develops RIP-like embedding results for MPOs under Gaussian, rank-one Gaussian, and Haar-random projective measurements, and proves a concrete empirical-measurement recovery guarantee for a constrained least-squares estimator when using Haar-based measurements, with total resource bounds that scale as $QM = \tilde{\Omega}(n^3 d^2 \overline r^2 /\epsilon^2)$. The results imply a substantial reduction in sample complexity compared to general low-rank QST, offering a theoretical foundation for efficient MPO tomography and motivating future work on practical measurement schemes, local designs, and algorithmic improvements. Overall, the paper integrates concepts from compressive sensing, empirical process theory, and tensor networks to establish polynomially scalable MPO QST guarantees, potentially impacting verification and benchmarking of large-scale quantum devices.

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.

Figures (4)

• Figure 1: Illustration of the MPO in \ref{['DefOfMPO']} from two perspectives: (a) each entry of the density matrix can be represented as products of $n$ matrices, where green represents one entry and the corresponding $n$ matrices, and (b) each element of the density matrix is illustrated in a diagrammatic form, where the line connecting two circles signifies the tensor contraction operation cichocki2014tensor, and unconnected line segments denote indices.
• Figure 2: Numerical computation of $\min_{\boldsymbol{\rho}\in\overline{\mathbb{X}}_{\overline{r}}}\|\mathcal{A}^Q(\boldsymbol{\rho})\|_2$ with $Q = 1$ and $K = d^n$.
• Figure 3: Illustration of convergence of IHT in \ref{['Iterative equ of GDwithTT_SVD_1']} in terms of (a) loss function defined in \ref{['Loss function of MPS']}, and (b) recovery error defined in \ref{['THE definition of MSE']} for different $n$ with $M=1000$, $r=2$, and $d=2$.
• Figure 4: IHT recovery error as the number of qudits $n$ increases with several choices of $M$ and $r$.

Theorems & Definitions (35)

• Theorem 1: informal version of \ref{['Statistical Error_Haar_Measurement']}
• Definition 1: POVM nielsen2002quantum
• Definition 2: Restricted isometry property (RIP)
• Theorem 2
• Lemma 1
• Theorem 3
• Lemma 2
• Theorem 4: Stable embedding of multiple Haar random projective measurements
• Lemma 3
• Theorem 5