Extracting classical information from quantum systems is of fundamental importance, and classical shadows allow us to extract a large amount of information using relatively few measurements. Conventional shadow estimators are unbiased and thus agree with the true mean in expectation. In this Letter, we consider a biased scheme, intentionally introducing a bias in the expectation value by rescaling the conventional classical-shadow estimators to reduce the error in the finite-sample regime. The approach is straightforward to implement and requires no quantum resources. We analytically prove average-case as well as worst- and best-case scenarios, and rigorously prove that it is, in principle, always worth biasing the estimators. We illustrate our approach in a quantum simulation task of a 12-qubit spin-ring problem and demonstrate how estimating expected values of nonlocal perturbations can be significantly more efficient using our biased scheme.
Extracting classical information from quantum systems is of fundamental importance, and classical shadows allow us to extract a large amount of information using relatively few measurements. Conventional shadow estimators are unbiased and thus agree with the true mean in expectation. In this Letter, we consider a biased scheme, intentionally introducing a bias in the expectation value by rescaling the conventional classical-shadow estimators to reduce the error in the finite-sample regime. The approach is straightforward to implement and requires no quantum resources. We analytically prove average-case as well as worst- and best-case scenarios, and rigorously prove that it is, in principle, always worth biasing the estimators. We illustrate our approach in a quantum simulation task of a 12-qubit spin-ring problem and demonstrate how estimating expected values of nonlocal perturbations can be significantly more efficient using our biased scheme.
Expectation values of observables are routinely estimated using so-called classical shadows-the outcomes of randomized bases measurements on a repeatedly prepared quantum state. In order to trust the accuracy of shadow estimation in practice, it is crucial to understand the behavior of the estimators under realistic noise. In this Letter, we prove that any shadow estimation protocol involving Clifford unitaries is stable under gate-dependent noise for observables with bounded stabilizer norm-originally introduced in the context of simulating Clifford circuits. In contrast, we demonstrate with concrete examples that estimation of "magic" observables can lead to highly misleading results in the presence of miscalibration errors and a worst case bias scaling exponentially in the system size. We further find that so-called robust shadows, aiming at mitigating noise, can introduce a large bias in the presence of gate-dependent noise compared to unmitigated classical shadows. Nevertheless, we guarantee the functioning of robust shadows for a more general noise setting than in previous works. On a technical level, we identify average noise channels that affect shadow estimators and allow for a more fine-grained control of noise-induced biases.
The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.
A crucial subroutine for various quantum computing and communication algorithms is to efficiently extract different classical properties of quantum states. In a notable recent theoretical work by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)NPAHAX1745-247310.1038/s41567-020-0932-7], a thrifty scheme showed how to project the quantum state into classical shadows and simultaneously predict M different functions of a state with only O(log_{2}M) measurements, independent of the system size and saturating the information-theoretical limit. Here, we experimentally explore the feasibility of the scheme in the realistic scenario with a finite number of measurements and noisy operations. We prepare a four-qubit GHZ state and show how to estimate expectation values of multiple observables and Hamiltonians. We compare the measurement strategies with uniform, biased, and derandomized classical shadows to conventional ones that sequentially measure each state function exploiting either importance sampling or observable grouping. We next demonstrate the estimation of nonlinear functions using classical shadows and analyze the entanglement of the prepared quantum state. Our experiment verifies the efficacy of exploiting (derandomized) classical shadows and sheds light on efficient quantum computing with noisy intermediate-scale quantum hardware.
Randomized protocols are procedures that incorporate probabilistic choices during their execution and they play a central role in quantum algorithms, spanning Hamiltonian simulation, noise mitigation, and measurement tasks. In practical implementations, the dominant cost of such protocols typically arises from circuit execution and measurement, and depends on hardware-specific resources such as gate counts, circuit depth, runtime, or dissipated energy. We introduce a general framework for applying classical importance sampling to randomized quantum protocols. Given a cost function for running quantum circuits, the proposed approach minimizes a net-cost figure of merit that jointly captures the computational expense per circuit and the estimator variance. We further extend the framework to scenarios where the quantum computation is subject to errors arising either from algorithmic approximations or from physical noise, proving that importance sampling preserves estimator bias despite altering the sampling distribution, and to settings with error-detection schemes, where we characterize the resulting changes in the optimal sampling strategy and achievable net-cost reduction. Representative applications include the Qdrift protocol, dephasing channels, mixed-states simulation, composite observables estimation, classical shadows, and probabilistic error cancellation. Overall, our results establish a principled approach for reducing the computational resources required by randomized quantum protocols through classical sampling optimization.
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U. In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ``real'' classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
Abstract Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k -local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in- k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.