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

TL;DR

This work tackles the challenge of sample-efficient quantum state tomography for one-dimensional MPO states by leveraging informationally complete POVMs (IC-POVMs). It proves that SIC-POVMs and spherical $t$-designs yield linear-in-$n$ sample complexity to recover MPOs, with bounds $M = O(n d^2 \gamma(\boldsymbol{\rho}) \overline{r}^2 \log n / \epsilon^2)$ for $2$-designs (where $\gamma(\boldsymbol{\rho}) = K \max_k p_k$) and $M = O(n d^2 \overline{r}^2 \log n / \epsilon^2)$ for $t\ge 3$ designs; these results extend to rank-$s$ MPOs in trace distance. The authors also introduce a projected gradient descent algorithm with TT-SVD projection, proven to converge linearly to a neighborhood of the true MPO given a suitable initialization (e.g., spectral initialization). Numerical experiments with local IC-POVMs demonstrate practical recovery with polynomial scaling in system size, illustrating the potential for efficient tomography in realistic 1D quantum systems. Overall, the paper establishes a principled link between MPO structure, IC-POVM design, and provable, efficient state reconstruction.

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.

Figures (2)

• Figure 1: Illustration of (a) recovery error for different initialization methods, $\overline r$ and $n$ with $M = 3000$; (b) convergence rate of PGD for different $n$, spectral initialization (dotted lines), and random initialization (solid lines) with $M = 3000$ and $\overline r=4$; (c) recovery error with $n=5$ for different initialization methods, $\overline r$ and $M$; (d) $p_\text{max}$ for the MPDO.
• Figure 2: (a)Convergence comparison of PSGD with random initialization for various $n$; (b) Performance comparison between PGD and PSGD with random and spectral initializations.

Theorems & Definitions (26)

• Definition 1
• Lemma 1
• Definition 2
• Definition 3
• Lemma 2
• Lemma 3
• Theorem 1: Stable recovery with $2$-designs
• Theorem 2: Stable recovery with $t$-designs ($t\geq 3$)
• Lemma 4
• proof