Quantum state tomography (QST) remains the gold standard for benchmarking and verification of near-term quantum devices. While QST for a generic quantum many-body state requires an exponentially large amount of resources, most physical quantum states are structured and can often be represented by a much smaller number of parameters, making efficient QST potentially possible. A prominent example is a matrix product state (MPS) or a matrix product density operator (MPDO), which is believed to represent most physical states generated by one-dimensional (1D) quantum devices. We study whether a general MPS/MPDO state can be recovered with bounded errors using only a number of state copies polynomial in the number of qubits, which is necessary for efficient QST. To make this question practically interesting, we assume only local measurements of qubits directly on the target state. By using a local symmetric informationally complete positive operator-valued measurement (SIC-POVM), we provide a positive answer to the above question for a variety of common many-body quantum states, including typical short-range entangled states, random MPS/MPDO states, and thermal states of one-dimensional Hamiltonians. In addition, we also provide an affirmative no answer for certain long-range entangled states such as a family of generalized GHZ states, but with the exception of target states that are known to have real-valued wavefunctions. Our answers are supported by a near-perfect agreement between an efficient calculation of the Cramer-Rao bound that rigorously bounds the sample complexity and numerical optimization results using a machine learning assisted maximal likelihood estimation (MLE) algorithm. This agreement also leads to an optimal QST protocol using local SIC-POVM that can be practically implemented on current quantum hardware and is highly efficient for most 1D physical states.
Quantum state tomography (QST) remains the gold standard for benchmarking and verification of near-term quantum devices. While QST for a generic quantum many-body state requires an exponentially large amount of resources, most physical quantum states are structured and can often be represented by a much smaller number of parameters, making efficient QST potentially possible. A prominent example is a matrix product state (MPS) or a matrix product density operator (MPDO), which is believed to represent most physical states generated by one-dimensional (1D) quantum devices. We study whether a general MPS/MPDO state can be recovered with bounded errors using only a number of state copies polynomial in the number of qubits, which is necessary for efficient QST. To make this question practically interesting, we assume only local measurements of qubits directly on the target state. By using a local symmetric informationally complete positive operator-valued measurement (SIC-POVM), we provide a positive answer to the above question for a variety of common many-body quantum states, including typical short-range entangled states, random MPS/MPDO states, and thermal states of one-dimensional Hamiltonians. In addition, we also provide an affirmative no answer for certain long-range entangled states such as a family of generalized GHZ states, but with the exception of target states that are known to have real-valued wavefunctions. Our answers are supported by a near-perfect agreement between an efficient calculation of the Cramer-Rao bound that rigorously bounds the sample complexity and numerical optimization results using a machine learning assisted maximal likelihood estimation (MLE) algorithm. This agreement also leads to an optimal QST protocol using local SIC-POVM that can be practically implemented on current quantum hardware and is highly efficient for most 1D physical states.
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≥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.
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.
Matrix product operators (MPOs) provide a scalable approach for quantum state tomography (QST) by offering a compact representation of many-body mixed states with limited entanglement, using only a number of parameters that scales polynomially with the system size. In this paper, we study QST for quantum density matrices that can be represented by MPOs. We first derive an equivalent characterization of Hermiticity in terms of the MPO core tensors and show that the coefficient tensor of an MPO under the Pauli or generalized Gell-Mann basis admits a real-valued low tensor-train (TT) rank structure. This establishes an explicit connection between MPO-based QST and noisy low-rank tensor completion. Motivated by this formulation, we develop an online Riemannian gradient descent (oRGD) algorithm that sequentially incorporates measurement data during the reconstruction process. With a proper initialization, we prove that oRGD converges linearly to the target MPO and succeeds with a number of distinct measurement settings that scales quadratically with the system size. As a byproduct, our analysis also yields a significantly improved sample complexity bound for the low TT rank tensor completion task. Furthermore, we propose a tailored spectral initialization method and establish its theoretical guarantee. Numerical experiments on several classes of quantum states validate the effectiveness and scalability of the proposed method.
Quantum computers solve ever more complex tasks using steadily growing system sizes.Characterizing these quantum systems is vital, yet becoming increasingly challenging.The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge.The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)-a daunting bottleneck given the scale of existing and nearterm quantum devices.Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs).We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion-without requiring ancillary ions to assist in measurements.More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot.Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination.SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods.The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state.Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running).We demonstrate this on an eight-qubit entangled state (which has 2 8 • 2 8 -1 = 65 535 degrees of freedom), as well as for fast state identification.All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.
Quantum state tomography (QST) is a technique used to reconstruct the density matrix of unknown quantum states based on experimentally obtained measurements. QST is a fundamental tool in the field of quantum information and quantum technology. It is commonly employed to assess the quality and limitations of experimental platforms. However, the density matrix reconstructed using the standard QST method often fails to guarantee semi-positive definiteness, which is physically unacceptable, due to limitations imposed by the randomness of quantum state measurements, noise in practical applications, and the number of measurements. To address this issue, a method is proposed that combines maximum likelihood (ML) estimation with population intelligence optimization algorithms. First, the issue of guaranteeing the semi-positive definiteness of the density matrix reconstructed using standard QST methods is analyzed. Subsequently, the ML method is introduced, and four commonly used population intelligence optimization algorithms are applied to find the density matrix that maximizes the likelihood of reproducing the experimental measurements. Finally, the superiority of the proposed method is demonstrated using an IBM Quantum (IBMQ) processor in scenarios involving separable and entangled states of multiple qubits, and compared with the standard QST method.
Quantum state tomography is a key technology for fully determining a quantum state. Unfortunately, standard quantum state tomography is intractable for general many-body quantum states, because the number of measurements and the post-processing time increase exponentially with the size of the system. However, for the matrix product states (MPSs), there exists an efficient method using linearly scaled local measurements and polynomially scaled post-processing times. In this study, we demonstrate the validity of the method in practice by reconstructing a four-photon MPS from its local two- or three-photon reduced-density matrices with the presence of statistical errors and systematical errors in experiment.