Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for K times after the same unitary evolution, and repeats this procedure for M rounds of random sampled unitary. As a result, there are M K times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable O . We find that in addition to the shadow-norm ‖ O ‖ s h a d o w introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm ‖ O ‖ X s h a d o w . For both random Pauli and Clifford measurements, we analyze and show the upper bounds of ‖ O ‖ X s h a d o w . In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.
Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for K times after the same unitary evolution, and repeats this procedure for M rounds of random sampled unitary. As a result, there are M K times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable O . We find that in addition to the shadow-norm ‖ O ‖ s h a d o w introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm ‖ O ‖ X s h a d o w . For both random Pauli and Clifford measurements, we analyze and show the upper bounds of ‖ O ‖ X s h a d o w . In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.
We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, 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 consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.
Estimating many local expectation values over time is a central measurement bottleneck in quantum simulation and device characterization. We study the task of reconstructing the Pauli-signal matrix Sij=Tr(Oiρ(tj)) for a collection of M low-weight Pauli observables {Oi}i=1M over N timesteps {tj}j=1N, while minimizing the total number of device shots. We propose a Compressed Sensing Shadow Tomography (CSST) protocol that combines two complementary reductions. First, local classical shadows reduce the observable dimension by enabling many Pauli expectation values to be estimated from the same randomized snapshots at a fixed time. Second, compressed sensing reduces the time dimension by exploiting the fact that many expectation-value traces are spectrally sparse or compressible in a unitary (e.g., Fourier) transform basis. Operationally, CSST samples m≪N timesteps uniformly at random, collects shadows only at those times, and then reconstructs each length-N signal via standard ℓ1-based recovery in the unitary transform domain. We provide end-to-end guarantees that explicitly combine shadow estimation error with compressed sensing recovery bounds. For exactly s-sparse signals in a unitary transform basis, we show that m=O(slog2slogN) random timesteps suffice (with high probability), leading to total-shot savings scaling as Θ(N/s) (i.e., up to polylogarithmic factors) relative to collecting shadows at all N timesteps. For approximately sparse signals, the reconstruction error decomposes into a compressibility (tail) term plus a noise term. We present numerical experiments on noisy many-qubit dynamics that support strong Fourier compressibility of Pauli traces and demonstrate substantial shot reductions with accurate reconstruction.
Accurately estimating expectation values of quantum observables with as few measurements as possible is crucial to many quantum computing applications. We introduce a framework that covers many of existing measurement strategies and introduce heuristics that can be used to enhance randomized schemes, including those based on Pauli grouping with inverse probability weighting and variants of the classical shadow algorithm. We show how to maximize information gain from such schemes, while carefully optimizing the distribution of possible measurements, and show that simple grouping algorithms can get close to, and in some cases exceed, state-of-the-art accuracy for unbiased estimation of expectation values on a standard quantum chemistry benchmark. We show how these randomized methods may be compared to more recent measurement schemes, such as shadow grouping, derandomized shadow, and overlapped grouping measurement, we show how the same strategies can be used to augment these schemes, and we demonstrate that we can reduce measurement costs by up to a factor of two by allowing Clifford measurement circuits for otherwise Clifford-less methods.
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.
Shadow estimation provides an efficient framework for estimating observable expectation values using randomized measurements. While originally developed for discrete-variable systems, its recent extensions to continuous-variable (CV) quantum systems face practical limitations due to idealized assumptions of continuous phase modulation and infinite measurement resolution. In this work, we develop a practical shadow estimation protocol for CV systems using discretized homodyne detection with a finite number of phase settings and quadrature bins. We construct an unbiased estimator for the quantum state and establish both sufficient conditions and necessary conditions for informational completeness within a truncated Fock space up to nmax photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as O(nmax4), improving upon previous O(nmax13/3) bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.
We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown d-dimensional quantum state ρ and a known set of observables {Oi}i=1m, the goal is to estimate expectation values {tr(Oiρ)}i=1m to accuracy ε in Lp-norm, using possibly adaptive measurements that act on O(polylog(d)) number of copies of ρ at a time. We focus on the regime where ε is below an instance-dependent threshold.
Our main contribution is an instance-optimal characterization of the sample complexity as Θ~(Γp/ε2), where Γp is a function of {Oi}i=1m defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with L∞-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form ∑i=1mαiOi with ∥α∥q=1 (where q is dual to p) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of Θ(Γpob/ε2). We then show Θ~(Γp/ε2) is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, Γ∞ob=Γ∞. In both cases, allowing c-copy measurements improves the sample complexity by at most Ω(1/c).
Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.