Matrix product states (MPS) are a central language for one-dimensional quantum matter and a practical target for near-term quantum simulators and variational algorithms. Yet, while substantial effort has focused on preparing MPS with shallow circuits, scalable methods to \emph{verify} that a many-body device has actually produced the intended state remain underdeveloped. Direct fidelity estimation (DFE) relies only on local Pauli measurements, but in many-body settings it suffers an exponential classical overhead from the preprocessing needed to sample Pauli strings. We eliminate this obstacle by introducing an \emph{autoregressive} importance sampler that draws Pauli strings sequentially from efficiently computable conditional distributions, reducing the per-shot classical overhead to linear scaling in the number of qubits. We further develop a grouped extension that constructs qubit-wise commuting measurement settings via a \emph{sorting string} and simultaneously estimates the entire commuting group from a single setting, significantly reducing estimator variance while preserving efficient postprocessing. Our approach extends naturally to matrix product operators (MPO), enabling scalable verification of tensor-network states and observables in long one-dimensional quantum systems. We utilize random MPS as a natural benchmark for generic 1D entangled states.
Matrix product states (MPS) are a central language for one-dimensional quantum matter and a practical target for near-term quantum simulators and variational algorithms. Yet, while substantial effort has focused on preparing MPS with shallow circuits, scalable methods to \emph{verify} that a many-body device has actually produced the intended state remain underdeveloped. Direct fidelity estimation (DFE) relies only on local Pauli measurements, but in many-body settings it suffers an exponential classical overhead from the preprocessing needed to sample Pauli strings. We eliminate this obstacle by introducing an \emph{autoregressive} importance sampler that draws Pauli strings sequentially from efficiently computable conditional distributions, reducing the per-shot classical overhead to linear scaling in the number of qubits. We further develop a grouped extension that constructs qubit-wise commuting measurement settings via a \emph{sorting string} and simultaneously estimates the entire commuting group from a single setting, significantly reducing estimator variance while preserving efficient postprocessing. Our approach extends naturally to matrix product operators (MPO), enabling scalable verification of tensor-network states and observables in long one-dimensional quantum systems. We utilize random MPS as a natural benchmark for generic 1D entangled 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.
Preparing arbitrary quantum states requires exponential resources. Matrix Product States (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50% and CNOT counts by 33%. We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.
We present and test a protocol to learn the matrix-product operator (MPO) representation of an experimentally prepared quantum state. The protocol takes as input classical shadows corresponding to local randomized measurements, and outputs the tensors of an MPO maximizing a suitably defined fidelity with the experimental state. The tensor optimization is carried out sequentially, similarly to the well-known density matrix renormalization group algorithm. Our approach is provably efficient under certain technical conditions expected to be met in short-range correlated states and in typical noisy experimental settings. Under the same conditions, we also provide an efficient scheme to estimate fidelities between the learned and the experimental states. We experimentally demonstrate our protocol by learning entangled quantum states of up to N=96 qubits in a superconducting quantum processor. Our method upgrades classical shadows to large-scale quantum computation and simulation experiments.
Abstract Preparing arbitrary quantum states requires exponential resources. Matrix product states (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50 % and CNOT counts by 33 % . We evaluate the full pipeline on datasets of varying complexity across systems of 19–50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.
Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A k=0 prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. Approach~A learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). Approach~B, our main proposal, is a \emph{fixed-protocol amortized} MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a χ-MPS, conditioning on an \emph{informative local} Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one (≈0.95, up to +0.59 over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives ≈90%-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity 0.90 at n=10, gain \emph{growing} in n; 0.88 at bond dimension χ=4), the parameterization is polynomial (native contraction to 20 qubits), and we close the loop on IBM hardware (5 states at 0.97 from hardware-measured Paulis).
We develop quantum algorithms for estimating properties of general matrix functions of Hermitian matrices, with applications to phase estimation, Green's function evaluation, and estimating measurement distributions of time-evolved states. The resulting methods exhibit commutator scaling in matrix parameters similar to that usually found for product formulae, lower circuit depth in other parameters, and require only a single ancillary qubit. Our central primitive consists of classically postprocessing randomly chosen product formulae circuits, which mathematically corresponds to an approximation of a Richardson extrapolation. Within our framework, we introduce a protocol for approximating the measurement distributions of quantum states, extending beyond standard observable estimation. We also provide tightened gate complexity bounds for practically relevant systems, including those with k-local interactions, long-tailed matrix ensembles, and conserved quantities. Finally, numerical experiments confirm that our method can achieve significantly shallower circuit depths than standard product formulae in certain parameter regimes, and highlight the potential of their heuristic application.