Finding Gaussian Structure in Bosonic States

2026-10-05

AI Summary

This study addresses the agnostic tomography of pure bosonic Gaussian states, aiming to efficiently reconstruct an approximate Gaussian state from an arbitrary unknown input. To this end, it proposes a two-stage algorithm that first iteratively combines generic Gaussian measurements with robust statistics to obtain an initial estimate, and subsequently refines this result via non-Gaussian measurements and convex optimization. The key contributions include the first genuinely fault-tolerant protocol for testing Gaussianity, alongside a proof that non-Gaussian measurements are indispensable for strong agnostic guarantees and outperform classical robust estimation. Furthermore, the method achieves polynomial complexity in both high- and low-fidelity regimes, establishes computational lower bounds under the assumption NP⊆BQP, and significantly reduces sample and time overheads.

Abstract

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $\rho$, the goal is to output a pure Gaussian state whose infidelity with $\rho$ is at most $\mathrm{opt} + \epsilon$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/\epsilon$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/\epsilon)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt}>c + \epsilon$ or $\mathrm{opt}<c - \epsilon$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/\epsilon)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

Problem

• agnostic tomography
• bosonic Gaussian states
• infidelity
• Gaussianity testing
• quantum state estimation

Innovation

• agnostic tomography
• bosonic Gaussian states
• non-Gaussian measurements
• robust statistics
• Gaussianity testing