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 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) 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 is a fundamental technique for characterizing quantum systems, but its scalability remains a significant challenge. This paper investigates an efficient alternative using classical shadows to reconstruct a Bell state with high fidelity. Classical shadows provide a compressed representation of a quantum state using randomized measurements, reducing the measurement complexity compared to full quantum tomography when full state reconstruction is not required. We employed a quantum circuit to generate a Bell state and utilize 1000 snapshots to construct its classical shadow. The reconstructed density matrix is evaluated using fidelity and norm difference metrics against the ideal Bell state. Our analysis demonstrates that as the number of snapshots increases, the fidelity of reconstruction stabilizes around 0.98-1.0, with the norm difference decreasing accordingly. The results also confirm the convergence of reconstructed states towards the ideal Bell state, highlighting the efficiency and accuracy of classical shadows in quantum state estimation. Shallow shadow tomography uses shallow random circuits to estimate global quantum state properties on noisy hardware. By combining low-depth entangling gates with randomized measurements and noise-aware postprocessing, it reduces sample complexity compared to Pauli shadows while remaining experimentally feasible. Recent experiments on superconducting processors confirmed up to fivefold measurement savings.
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.
Quantum shadow tomography based on the classical shadow representation provides an efficient way to estimate properties of an unknown quantum state without performing a full quantum state tomography. In scenarios where estimating the expectation values for only certain classes of observables is required, obtaining information about the entire density matrix is unnecessary. We propose a partial quantum shadow tomography protocol that estimates a subset of density matrix elements relevant to the expectation values of structured observables. Specifically, we identify specific subsets of the tomographically complete set Cl(2)⊗n and a simple pseudo-inverse of the associated channel, which can be used to estimate all elements of the density matrix with the same active order. By restricting the protocol to smaller subsets of single-qubit Pauli measurements, it becomes experimentally more efficient. We demonstrate the advantage over unitary designs, such as the Clifford, full Pauli basis, and methods utilizing mutually unbiased bases, by analytically deriving error bounds and numerically evaluating the protocol on structured operators. We experimentally demonstrate the partial shadow estimation scheme for a wide class of two-qubit states (pure, entangled, and mixed) in the nuclear magnetic resonance (NMR) platform. The full density matrix, reconstructed experimentally by combining different partial estimators, achieves fidelities around 99%.
Problem: Quantum state tomography (QST) is the bottleneck for characterizing intermediate-scale quantum devices, scaling exponentially in measurement complexity (4n) and reconstruction time. Standard iterative optimization methods become computationally intractable for systems beyond 5-6 qubits, limiting real-time characterization capabilities essential for quantum error correction and adaptive protocols. Solution: We introduce Geometric Shadow Inversion (GSI), a reconstruction framework that treats quan- tum state recovery as kernel regression on the density matrix manifold. GSI utilizes a Gaussian-weighted adjoint projection to triangulate the state geometry in a single non-iterative pass, eliminating the computational bottle- neck of iterative optimization entirely. Key Discovery: We identify a fundamental information threshold at measurement ratio M/P ≈1.17 where quantum state reconstruction undergoes a sharp performance transition. Systematic validation across M/P ∈[0.08, 7.84] for 4-qubit systems reveals three distinct regimes: (1) Underdetermined (M/P < 0.6): F < 0.997, performance sensitive to bandwidth; (2) Information Threshold (0.6 < M/P < 2): F crosses 0.9975 and becomes robust to bandwidth—variations of ±50% in σ (range 0.7–1.5) produce fidelity changes < 0.03; (3) Overdetermined (M/P > 2): F →0.999, saturated performance. This robustness at threshold is the signature of geometric regularization making theoretically-sufficient measurements practically sufficient. Our choice of σ = 0.78 operates within this robust plateau, explaining its consistent performance across system sizes n ∈[3, 6]. Below the information threshold, GSI extends into the strongly underdetermined regime (M/P= 0.73), achiev- ing F= 0.905 ±0.004 for 6-qubit states through geometric smoothness priors. Impact: With runtimes of ≈10ms (3 qubits) to ≈5min (6 qubits), GSI enables real-time calibration loops for drift monitoring in logical qubit blocks and tractable diagnostic verification for NISQ hardware. The information threshold phenomenon provides both a practical tool for continuous state characterization and a theoretical framework connecting kernel geometry to fundamental information-theoretic limits in quantum reconstruction.
Quantum state tomography (QST) via local measurements on reduced density matrices (LQST) is a promising approach but becomes impractical for large systems. To tackle this challenge, we developed an efficient quantum state tomography method inspired by quantum overlapping tomography [Phys. Rev. Lett. 124, 100401 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.100401], which utilizes parallel measurements (PQST). In contrast to LQST, PQST significantly reduces the number of measurements and offers more robustness against shot noise. Experimentally, we demonstrate the feasibility of PQST in a treelike superconducting qubit chip by designing high-efficiency circuits, preparing W states, ground states of Hamiltonians, and random states, and then reconstructing these density matrices using full quantum state tomography (FQST), LQST, and PQST. Our results show that PQST reduces measurement cost, achieving fidelities of 98.68% and 95.07% after measuring 75 and 99 observables for six-qubit and nine-qubit W states, respectively. Furthermore, the reconstruction of the largest density matrix of the 12-qubit W state is achieved with the similarity of 89.23% after just measuring 243 parallel observables, while 3^{12}=531 441 complete observables are needed for FQST. Consequently, PQST will be a useful tool for future tasks such as the reconstruction, characterization, benchmarking, and properties learning of states.