Practical Homodyne Shadow Estimation

TL;DR

The paper addresses efficient estimation of observable expectations in continuous-variable quantum systems under realistic, discretized homodyne detection. It introduces a practical classical shadow framework based on a discretized POVM with phase settings $N$ and quadrature bins $M$, derives unbiased state estimators via $C_E^{-1}$, and proves both sufficient ($N \ge 2n_{\mathrm{max}}+1$, $M \ge n_{\mathrm{max}}+1$) and necessary (IC) conditions for complete state reconstruction within a truncated Fock space. A key result is the $O(n_{\mathrm{max}}^4)$ bound on the shadow norm, improving on prior $O(n_{\mathrm{max}}^{13/3})$ bounds and implying favorable sample complexity for estimating CV observables. The framework scales to multi-mode CV systems, with estimator construction and variance that factorize across modes and maintain tractable costs for local observables. Numerical simulations validate the theory, showing variance reduction with more phase settings, finer quadrature binning, and larger photon-number cutoffs, bridging theory and experimental feasibility for robust CV 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.

Figures (2)

• Figure 1: Overview of the experimental settings of our protocol. They are composed of a homodyne detector with a discrete-modulated local oscillator, an analog-to-digital converter discretizing the measurement output and a classical computer for the post-processing. LO: local oscillator; PM: phase modulator; PD: photo detector.
• Figure 2: Variance of the estimator of expectation $\langle \hat{n} \rangle = \langle a^\dag a \rangle$ versus system parameters $N$, $M$ and $n_{\mathrm{max}}$.

Theorems & Definitions (10)

• Theorem 4.1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 4.2
• proof
• Theorem 5.1
• proof