We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$ -mode bosonic state $ρ$ , the goal is to output a pure Gaussian state whose infidelity with $ρ$ is at most $\mathrm{opt} + ε$ , 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 $n, 1/ε$ and $\log \log E$ , where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$ , our protocol uses $(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + ε$ or $\mathrm{opt} < c - ε$ , for any threshold $c\in(0,1)$ . We also prove $\mathrm{poly}(n,1/ε)$ 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.We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $ρ$, the goal is to output a pure Gaussian state whose infidelity with $ρ$ is at most $\mathrm{opt} + ε$, 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 $n, 1/ε$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + ε$ or $\mathrm{opt} < c - ε$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/ε)$ 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.

quant-pharXiv:2610.06810Oct 6, 20260 citations

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary

n

-mode bosonic state

ρ

, the goal is to output a pure Gaussian state whose infidelity with

ρ

is at most

\mathrm{opt} + ε

, 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/ε

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/ε)} \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 + ε

\mathrm{opt} < c - ε

, for any threshold

c\in(0,1)

. We also prove

\mathrm{poly}(n,1/ε)

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.