Quantum-enhanced metrology with classical shadows

Abstract

Quantum metrology relies on observables that maximize information about the parameter of interest, but are often hard to access. We develop a protocol that combines Pauli moments estimated from a fixed ensemble of randomized local measurements into an optimized method-of-moments estimator. For local Pauli classical shadows, we derive the shadow information which determines estimator variance and sample-complexity guarantees. Identifying the separable-state sensitivity limit, we show that entanglement can lead to a constant-factor sensitivity enhancement, as illustrated with one-axis-twisted probes. This protocol allows for efficient circuit-based metrology without choosing an observable.

The paper asks how much metrological sensitivity can be extracted from fixed, uniformly randomized local Pauli measurements, without tailoring the readout observable to the probe. It combines classical-shadow estimates of all Pauli strings up to weight $K$ using generalized least squares, assuming known moment responses and covariance near a working point. An exact single-shot shadow covariance $\Gamma$ defines the “shadow information” $\mathcal I_K=\boldsymbol s_K^T\Gamma_K^{-1}\boldsymbol s_K$, giving local estimator variance $1/(n\mathcal I_K)$. Bernstein bounds yield small-error sample complexity $O([\eta^2\mathcal I_K]^{-1}\log(1/\delta))$. For collective single-qubit phase encoding, covariance and added-noise arguments establish $\mathcal I_K\le F_{\rm cs}\le2NF_Q/(2N+F_Q)\le2N$, where $F_{\rm cs}$ describes the complete measurement record. Thus this readout cannot retain Heisenberg scaling. Product-state calculations and convexity give the tight separable moment-estimator benchmark $\mathcal I_K\le N/3$; exceeding it witnesses entanglement. Quantum improvement over this benchmark is bounded by a factor below six, and weight-one moments obey the stronger ceiling $\mathcal I_1\le N/2$. State-vector calculations for one-axis-twisted probes with $4\le N\le12$ and $K\le3$ demonstrate entanglement-enhanced sensitivity, with about 15% improvement at $N=12$, mostly accessible through linear moments. Higher-weight moments help more for non-Gaussian probes. The guarantees are local and exclude calibration and covariance-learning costs. Numerics do not simulate finite-shot records or calibration errors. The separate-measurement comparison uses equal allocation rather than optimized readout, and even the full Pauli-moment estimator need not attain the record’s Fisher information.