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Scientific Reports

A computationally efficient approach to quantum state reconstruction using robust classical shadows

Sanjay Sharma, Shyam Akashe, Govind Murari Upadhyay, Amanjot Kaur Lamba

Published Feb 2, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-026-35442-4

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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.
Extracting information efficiently from quantum systems is crucial for quantum information processing. Classical shadows enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Introducing a shallow random quantum circuit before measurements improves sample efficiency for high-weight Pauli observables and low-rank properties. However, in practice, these circuits can be noisy and bias the measurement results. Here, we propose the robust shallow shadows, which employs Bayesian inference to learn and mitigate noise in postprocessing. We analyze noise effects on sample complexity and the optimal circuit depth. We provide theoretical guarantees for the success of error mitigation under a wide class of noise processes. Experimental validation on a superconducting quantum processor confirms the advantage of our method, even in the presence of realistic noise, over single-qubit measurements for predicting diverse state properties, such as fidelity and entanglement entropy. Our protocol thus offers a scalable, robust, and sample-efficient method for quantum state characterization on near-term quantum devices. Fast and reliable characterisation of quantum systems is a key part of quantum technologies development. Here, the authors propose and demonstrate a way to embed noise characterisation in classical shadow estimation of nonlocal properties, enabling an efficient way to extract information from noisy quantum systems.
Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
We generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local-basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.
We provide practical and powerful schemes for learning properties of a quantum state using a small number of measurements. Specifically, we present a randomized measurement scheme modulated by the depth of a random quantum circuit in one spatial dimension. This scheme interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. We focus on the regime where depth scales logarithmically in the system size and provide evidence that this retains the desirable sample complexity properties of both extremal schemes while also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We achieve our findings by bringing together tools from shadow estimation, random circuits, and tensor networks.
Reliable and scalable readout strategies are essential for quantum technologies. As quantum processors grow, extracting useful information must remain feasible without measurement circuits becoming a dominant bottleneck. Randomized measurements and classical shadows provide a powerful route, but global estimation is conventionally associated with highly random ensembles that require increasing circuit depth and hence substantial experimental overhead. In this work, we show that substantially less randomness suffices when the readout is meaningfully adapted to the quantities being estimated. We introduce shallow phase shadows, based on a sparse Clifford-IQP ensemble, and prove efficient global estimation of stabilizer-state fidelities despite the ensemble not forming an approximate relative-error design. On all-to-all architectures, the protocol admits a constant-depth implementation using mid-circuit measurements and classical feedforward, or logarithmic depth without auxiliary systems. The protocol requires only controlled-phase entangling gates and offers a tunable trade-off between circuit resources and estimation accuracy, making it particularly amenable to experimentally relevant architectures with long-range connectivity. Our results show that scalable quantum readout need not reproduce generic randomness: task-adapted randomization can enable substantially shallower global characterization protocols.
Quantum state tomography is a fundamental task in quantum information science, enabling detailed characterization of correlations, entanglement, and electronic structure in quantum systems. However, its exponential measurement and computational demands limit scalability, motivating efficient alternatives such as classical shadows, which enable accurate prediction of many observables from randomized measurements. In this work, we introduce a bi-objective semidefinite programming approach for constrained shadow tomography, designed to reconstruct the two-particle reduced density matrix (2-RDM) from noisy or incomplete shadow data. By integrating NNN-representability constraints and nuclear-norm regularization into the optimization, the method builds an NNN-representable 2-RDM that balances fidelity to the shadow measurements with energy minimization. This unified framework mitigates noise and sampling errors while enforcing physical consistency in the reconstructed states. Numerical and hardware results demonstrate that the approach significantly improves accuracy, noise resilience, and scalability, providing a robust foundation for physically consistent fermionic state reconstruction in realistic quantum simulations.