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

2024-10-03

AI Summary

Quantum state tomography (QST) suffers from exponential sample and computational overhead in many-body systems, rendering it infeasible for intermediate-scale quantum devices. This work addresses QST for one-dimensional physical states admitting a matrix product operator (MPO) representation—whose parameter count scales linearly in the number of qubits $n$—and proposes the first provably efficient protocol achieving sample complexity linear in the number of parameters ($O(r^2 n)$, where $r$ is the bond dimension). The method employs spherical $t$-design measurements with $t geq 3$, combined with constrained least-squares estimation and MPO tensor-network modeling. We rigorously establish that this approach yields stable, error-controlled reconstruction and prove that the attained sample efficiency is information-theoretically optimal. Furthermore, we show that symmetric informationally complete POVMs (SIC-POVMs) and spherical 2-designs also achieve linear sample complexity under uniform MPO distributions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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 $tge3$, we prove that only a number of state copies proportional to the number of independent parameters in the MPO is needed for a guaranteed recovery of emph{any} state represented by an MPO.

Problem

• Efficient tomography for structured quantum states
• Sample complexity for MPO state recovery
• Linear scaling with qubit number

Innovation

• Uses IC-POVMs for efficient quantum tomography
• Leverages MPO structure for linear parameter scaling
• Employs spherical t-designs for guaranteed state recovery