Practical Homodyne Shadow Estimation

TL;DR

This paper introduces a practical homodyne shadow estimation protocol for continuous-variable quantum systems, addressing real-world measurement limitations and providing scalable, unbiased state estimation with improved variance bounds.

Contribution

It develops a discretized homodyne shadow estimation method with finite phase settings, establishing conditions for informational completeness and analyzing variance scaling.

Findings

Shadow norm scales as O(n_max^4), better than previous bounds

Provides unbiased estimators for CV quantum states

Bridges theory and experiment for scalable quantum state characterization

Abstract

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 $n_{\mathrm{max}}$ photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as $\mathcal{O}(n_{\mathrm{max}}^4)$, improving upon previous $\mathcal{O}(n_{\mathrm{max}}^{13/3})$ bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.