Quantum state tomography (QST) is an executable procedure, allowing complete reconstruction of a quantum state. As a basic tool, QST is widely employed in quantum information research. However, many projective measurements on identical copies of the quantum state under test are required in standard QST, which is heavily time-consumed. Here, we perform reconstruction of density matrix with dimensionality of $15\times 15$ by utilizing compressed sensing method at a sampling ratio of 0.35. According to the experimental results, the fidelity could be as high as 93.0%~99.4% and 73.8% for coherent laser beams and heralded single photon sources, respectively.
Quantum state tomography (QST) is an executable procedure, allowing complete reconstruction of a quantum state. As a basic tool, QST is widely employed in quantum information research. However, many projective measurements on identical copies of the quantum state under test are required in standard QST, which is heavily time-consumed. Here, we perform reconstruction of density matrix with dimensionality of 15×15 by utilizing compressed sensing method at a sampling ratio of 0.35. According to the experimental results, the fidelity could be as high as 93.0%~99.4% and 73.8% for coherent laser beams and heralded single photon sources, respectively.
Quantum state tomography (QST) is a crucial technique for characterizing quantum states, forming the bedrock of quantum information processing and quantum metrology. However, traditional QST relies on a large number of measurements, making it computationally expensive and experimentally demanding. This work addresses this challenge by proposing a novel algorithm that integrates compressed sensing (CS) and Bayesian inference to achieve accurate quantum state reconstruction from limited measurement data. The core idea is to exploit the inherent sparsity of quantum states and leverage CS principles to significantly reduce the required measurement count. The algorithm utilizes a sparse dictionary learned from prior knowledge of the quantum state and employs Bayesian inference to estimate the state parameters, providing a robust and efficient reconstruction process. The results demonstrate the feasibility and effectiveness of this approach, offering a pathway towards more practical and accessible QST methods. Key mathematical formulations are presented, including the Bayesian inference framework and the CS reconstruction process.
Quantum state tomography (QST) is a crucial technique for characterizing the state of quantum systems. However, traditional QST requires a large number of measurements, leading to significant experimental overhead in terms of time, resources, and data acquisition. This work proposes a novel approach to QST by leveraging compressed sensing (CS) techniques. CS allows for accurate reconstruction of the quantum state from a significantly reduced set of measurements, thereby mitigating the experimental burden. We employ sparsity regularization, a key component of CS, to minimize the number of measurements needed. The theoretical framework and the underlying principles are presented, focusing on the mathematical formulation of the problem and the optimization strategies involved. This method offers a pathway to more efficient and cost-effective QST, particularly for complex quantum states or when limited experimental resources are available. We demonstrate the potential of this approach through a detailed mathematical analysis.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
Quantum state tomography (QST) is an essential technique for reconstructing the density matrix of an unknown quantum state from measurement data, crucial for quantum information processing. However, conventional QST requires an exponentially growing number of measurements as the system dimension increases, posing a significant challenge for high-dimensional systems. To mitigate this issue, compressed sensing quantum state tomography (CS-QST) has been proposed, significantly reducing the required number of measurements. In this study, we investigate the impact of basis selection in CS-QST for qudit systems, which are fundamental to high-dimensional quantum information processing. Specifically, we compare the efficiency of the generalized Gell-Mann (GGM) and Heisenberg-Weyl observable (HWO) bases by numerically reconstructing density matrices and evaluating reconstruction accuracy using fidelity and trace distance metrics. Our results demonstrate that, while both bases allow for successful density matrix reconstruction, the HWO basis becomes more efficient as the qudit dimension increases. Furthermore, we find the best fitting curves that estimate the number of measurement operators required to achieve a fidelity of at least 95%. These findings highlight the significance of basis selection in CS-QST and provide valuable insights for optimizing measurement strategies in high-dimensional quantum state tomography.
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.
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 growsexponentiallywith 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 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 other words, there exist no rigorous bounds on the number of state copies required for reconstructing MPO states that scales polynomially with the number of qubits. 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 projective measurements. We show that the information contained in an MPO with a constant bond dimension can be preserved using a number of random measurements that depends onlylinearlyon the number of qubits, assuming no statistical error of the measurements. We then study MPO-based QST with Haar random projective measurements that can in principle be implemented on quantum computers. We prove that only apolynomialnumber of state copies in the number of qubits is required to guarantee bounded recovery error of an MPO state. Remarkably, such recovery can be achieved by measuring the state in each random basis only once, despite the large statistical error associated with the outcome of each measurement. Our work may be generalized to accommodate random local or t-design measurements that are more practical to implement on current quantum computers. It may also facilitate the discovery of efficient QST methods for other structured quantum states.