Improving shadow estimation with locally-optimal dual frames

TL;DR

This paper addresses the challenge of estimating quantum observables under finite measurement shots by introducing $k$-locally optimal ($k$-LO) dual frames, which construct correlated classical shadows from local measurements. The approach partitions qubits into highly correlated groups via mutual information, performs local tomography to obtain reduced states, and builds locally optimal duals for each group, yielding a global shadow as a tensor product of group shadows. Compared with standard shadows and Pauli-grouping methods, $k$-LO duals achieve unbiased estimators with lower variance across molecular energies, two-point correlations in Ising models, and large-scale molecular energies up to 40 qubits, using only single-qubit measurements and efficient post-processing. The results demonstrate substantial practical gains in estimation accuracy and highlight the importance of tailored, state-aware post-processing in quantum measurement protocols.

Abstract

Accurate estimation of observables in quantum systems is a central challenge in quantum information science, yet practical implementations are fundamentally constrained by the limited number of measurement shots. In this work we explore a variation of the classical shadows protocol in which the measurements are kept local while allowing the resulting classical shadows themselves to be correlated. By constructing locally optimal shadows, we obtain unbiased estimators that outperform state-of-the-art methods, achieving the same accuracy with substantially fewer measurements. We validate our approach through numerical experiments on molecular Hamiltonians with up to 40 qubits and a 50-qubit Ising model consistently observing significant reductions in estimation errors.

Figures (8)

• Figure 1: Summary of proposed methodology to construct state-aware correlated and locally-optimal shadows. (a) A quantum state $\rho$ is measured with an overcomplete (OC) POVM, e.g. single-qubit random Pauli measurements. (b) Using the observed frequencies, we build a graph of the pair-wise mutual information between the qubits, and partition it into disjoint groups of highest-correlated qubits, each of size at most $k$ ($k=4$ in the figure). (c) Reconstruct the local reduced density matrices for each of the groups from the measurement data using partial state tomography. (d) Construct optimal duals (shadows) for each of the groups. The corresponding global shadow is a tensor product of correlated $k$-qubit shadows which are locally-optimal for each the groups. Usual classical shadows are instead tensor products of single qubit operators and state-agnostic, in that they do not use available knowledge on the measured state.
• Figure 2: Comparison of $k$-locally optimal ($k$-LO) shadows against other methods (see Tab. \ref{['tab:comparison']}) for the task of molecular energy estimation on a benchmark set of molecules (4 to 16 qubits), obtained via the Jordan--Wigner transformation from their fermionic Hamiltonians hadfield2020github. $k$-LO duals allow for observable estimators which are more precise than other state-of-the-art approaches, while using same or fewer resources.
• Figure 3: The errors of the estimation of various TLD1433 ansatz energies for the ground and excited states (columns) in different system sizes in terms of qubits (rows) using CS-Pauli and $k$-LO duals. The absolute error is computed with respect to the ansatz energy, and the standard error is computed using the data and our estimator.
• Figure 4: The standard error $\sqrt{\text{Var}\qty[\bar{o}]/S}$ ($\sim 68\%$ CI) of the estimation of two-point correlation functions for 1D TFIM using $2^{19}$ shots from random Pauli measurements with canonical HuangShadows2020 (CS-Pauli) and k-LO duals. $\{0,i\}$-LO duals corresponds to the situation where we first trace out all but systems $0$ and $i$, and then compute the optimal duals for this subsystem.
• Figure 5: The estimation variance of the $ZZ$ observable \ref{['eq:zz_var']} and the state estimation MSE \ref{['eq:mse_state']} for two 2-qubit mixed and pure states, for various types of duals. For some values of $q$, the canonical estimator has a lower observable variance than the 1-LO estimator, even though the MSE error is consistently lower for the 1-LO duals. In these cases, 2-LO duals are equivalent to the globally optimal duals, and in fact proves to be the best for both observable and state estimation tasks.