Classical shadow tomography has emerged as a powerful framework for predicting properties of quantum many-body systems with favorable sample complexity. Standard theoretical guarantees, however, rely on the assumption that experimental rounds are independent and identically distributed (i.i.d.). This idealization is often violated in practice, where parameter drift, environmental noise, and active feedback generate history-dependent sequences of states or channels. To address this, we introduce a robust classical shadow protocol based on a truncated mean estimator. We prove that its sample complexity for predicting properties of the time-averaged state or channel matches the standard i.i.d. scaling governed by the shadow norm, even when experimental rounds depend arbitrarily on the past. Our results establish the robustness of the shadow formalism beyond the i.i.d. regime.
Classical shadow tomography has emerged as a powerful framework for predicting properties of quantum many-body systems with favorable sample complexity. Standard theoretical guarantees, however, rely on the assumption that experimental rounds are independent and identically distributed (i.i.d.). This idealization is often violated in practice, where parameter drift, environmental noise, and active feedback generate history-dependent sequences of states or channels. To address this, we introduce a robust classical shadow protocol based on a truncated mean estimator. We prove that its sample complexity for predicting properties of the time-averaged state or channel matches the standard i.i.d. scaling governed by the shadow norm, even when experimental rounds depend arbitrarily on the past. Our results establish the robustness of the shadow formalism beyond the i.i.d. regime.
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.
Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory. While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates. Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050-1057 (2020)]. By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions. When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience. We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.
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.
Shadow tomography has appeared as a powerful tool for estimating observables on quantum computers from a small number of samples. We show that shadow-tomography-inspired ideas can offer similarly improved sample scaling for estimating observables on tensor network states on classical computers after proper adaptation. We develop strategies for both spin (bosonic) and fermionic systems, tailored to the contraction requirements of tensor networks, and generate scaling improvements of factors of O(N) to O(N3) (where N is system size), depending on the specific task and system type. For the important and difficult task of evaluating the expectation value of long-range interacting Hamiltonians, we achieve the optimal O(1) overall scaling (up to logarithmic factors) for an arbitrarily fixed relative Monte Carlo error in both spin and fermionic systems. Additionally, we show that shadow estimators offer more stable gradients of observables in variational optimization tasks than standard Monte Carlo estimators. We demonstrate practical advantage by simulating systems with long-range interactions, including the 2D long-range Heisenberg model and an ab-initio quantum chemistry Hamiltonian.
Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes. As part of our technical analysis we obtain ε-approximate relative-error k-designs over the n-qubit symplectic group in depth O(nk+log(1/ε)), which is optimal with respect to n, and may be of some independent interest.
Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory.While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates.Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050-1057(2020)].By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions.When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience.We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.