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Higher moment theory and learnability of bosonic states

Joseph T. Iosue, Yu-Xin Wang, Ishaun Datta, Soumik Ghosh, Changhun Oh, Bill Fefferman, Alexey V. Gorshkov

Abstract

We present a sample- and time-efficient algorithm to learn any bosonic Fock state acted upon by an arbitrary Gaussian unitary. As a special case, this algorithm efficiently learns states produced in Fock state BosonSampling, thus resolving an open question put forth by Aaronson and Grewal (Aaronson, Grewal 2023). We further study a hierarchy of classes of states beyond Gaussian states that are specified by a finite number of their higher moments. Using the higher moments, we find a full spectrum of invariants under Gaussian unitaries, thereby providing necessary conditions for two states to be related by an arbitrary (including active, e.g. beyond linear optics) Gaussian unitary.

Higher moment theory and learnability of bosonic states

Abstract

We present a sample- and time-efficient algorithm to learn any bosonic Fock state acted upon by an arbitrary Gaussian unitary. As a special case, this algorithm efficiently learns states produced in Fock state BosonSampling, thus resolving an open question put forth by Aaronson and Grewal (Aaronson, Grewal 2023). We further study a hierarchy of classes of states beyond Gaussian states that are specified by a finite number of their higher moments. Using the higher moments, we find a full spectrum of invariants under Gaussian unitaries, thereby providing necessary conditions for two states to be related by an arbitrary (including active, e.g. beyond linear optics) Gaussian unitary.

Paper Structure

This paper contains 1 theorem, 7 equations.

Key Result

Theorem 2

Let $\mathinner{\lvert\psi\rangle}= \mathcal{U}_S \mathinner{\lvert\bm f\rangle}$ for an unknown symplectic matrix $S \in \mathrm{Sp}(2n, \mathbb{R})$ specifying an arbitrary Gaussian unitary (modulo displacements) and an arbitrary Fock state $\mathinner{\lvert\bm f\rangle}$. If our measurements $\L where $f_{\rm max} = \max_i f_i$ and $s$ is the maximum magnitude of squeezing in $S$ (that is, $\m

Theorems & Definitions (2)

  • Definition 1
  • Theorem 2