Quantum state tomography (QST) is a central technique to fully characterize an unknown quantum state. However, standard QST requires an exponentially growing number of quantum measurements against the system size, which limits its application to smaller systems. Here, we explore the sparsity of underlying quantum state and propose a QST scheme that combines the matrix product states’ representation of the quantum state with a supervised machine learning algorithm. Our method could reconstruct the unknown sparse quantum states with very high precision using only a portion of the measurement data in a randomly selected basis set. In particular, we demonstrate that the Wolfgang states could be faithfully reconstructed using around 25% of the whole basis, and that the randomly generated quantum states, which could be efficiently represented as matrix product states, could be faithfully reconstructed using a number of bases that scales sub-exponentially against the system size.
Quantum state tomography (QST) is a central technique to fully characterize an unknown quantum state. However, standard QST requires an exponentially growing number of quantum measurements against the system size, which limits its application to smaller systems. Here, we explore the sparsity of underlying quantum state and propose a QST scheme that combines the matrix product states’ representation of the quantum state with a supervised machine learning algorithm. Our method could reconstruct the unknown sparse quantum states with very high precision using only a portion of the measurement data in a randomly selected basis set. In particular, we demonstrate that the Wolfgang states could be faithfully reconstructed using around 25% of the whole basis, and that the randomly generated quantum states, which could be efficiently represented as matrix product states, could be faithfully reconstructed using a number of bases that scales sub-exponentially against the system size.
Reconstructing quantum states from measurement data represents a formidable challenge in quantum information science, especially as system sizes grow beyond the reach of traditional tomography methods. While recent studies have explored quantum machine learning (QML) for quantum state tomography (QST), nearly all rely on idealized assumptions, such as direct access to the unknown quantum state as quantum data input, which are incompatible with current hardware constraints. In this work, we present a QML-based tomography protocol that operates entirely on classical measurement data and is fully executable on noisy intermediate-scale quantum (NISQ) devices. Our approach employs a variational quantum circuit trained to reconstruct quantum states based solely on measurement outcomes. We test the method in simulation, achieving high-fidelity reconstructions of diverse quantum states, including GHZ states, spin chain ground states, and states generated by random circuits. The protocol is then validated on quantum hardware from IBM and IonQ. Additionally, we demonstrate accurate tomography is possible using incomplete measurement bases, a crucial step towards scaling up our protocol. Our results in various scenarios illustrate successful state reconstruction with fidelity reaching 90% or higher. To our knowledge, this is the first QML-based tomography scheme that has been implemented on real quantum processors using exclusively classical measurements. This work establishes the feasibility of QML-based tomography on current quantum platforms and offers a scalable pathway for practical quantum state reconstruction.
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.
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.
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 is a key technique for quantum information processing but is challenging due to the exponential growth of its complexity with the system size. In this work we propose an algorithm which iteratively finds the best non-negative matrix product state approximation based on a set of measurement outcomes whose size does not necessarily grow exponentially. Compared to the tomography method based on neural network states, our scheme utilizes a so-called tensor train representation that allows straightforward recovery of the unknown density matrix in the matrix product operator form. As applications, the effectiveness of our algorithm is numerically demonstrated to reconstruct the ground state of the XXZ spin chain under depolarizing noise.
We study the problem of learning a sparse quantum state, an n-qubit quantum state whose density matrix has at most s nonzero matrix entries in an unknown product basis. While such states admit compact classical descriptions, they can carry long-range entanglement that prevents reconstruction from local reduced density matrices alone. Therefore, previous learning approaches addressed such long-range-entangled states using many entangling gates to extract the necessary information. In this work, we show that sparse states can nevertheless be efficiently learned using only single-qubit measurements. Specifically, when the sparsity s is constant, our algorithm can learn sparse states from single-qubit measurements with polynomial sample complexity and classical computational complexity. When s grows polynomially with n, sparse states can still be learned from single-qubit measurements with polynomial sample complexity, although efficient classical computation is not guaranteed in general. In this regime, however, the classical computational complexity becomes quasipolynomial when the state is sparse in an unknown basis that is a product of a known fixed finite set of single-qubit bases (e.g., eigenbases of Pauli operators). These results establish efficient learning of sparse states with long-range entanglement without entangling gates, and the single-qubit measurement requirements make our algorithms compatible with current quantum devices.