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Efficient learning of bosonic Gaussian unitaries

Marco Fanizza, Vishnu Iyer, Junseo Lee, Antonio A. Mele, Francesco A. Mele

TL;DR

This work presents the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis, and is believed to be the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.

Abstract

Bosonic Gaussian unitaries are fundamental building blocks of central continuous-variable quantum technologies such as quantum-optic interferometry and bosonic error-correction schemes. In this work, we present the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis. Our algorithm produces an estimate of the unknown unitary that is accurate to small worst-case error, measured by the physically motivated energy-constrained diamond distance. Its runtime and query complexity scale polynomially with the number of modes, the inverse target accuracy, and natural energy parameters quantifying the allowed input energy and the unitary's output-energy growth. The protocol uses only experimentally friendly photonic resources: coherent and squeezed probes, passive linear optics, and heterodyne/homodyne detection. We then employ an efficient classical post-processing routine that leverages a symplectic regularization step to project matrix estimates onto the symplectic group. In the limit of unbounded input energy, our procedure attains arbitrarily high precision using only $2m+2$ queries, where $m$ is the number of modes. To our knowledge, this is the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.

Efficient learning of bosonic Gaussian unitaries

TL;DR

This work presents the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis, and is believed to be the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.

Abstract

Bosonic Gaussian unitaries are fundamental building blocks of central continuous-variable quantum technologies such as quantum-optic interferometry and bosonic error-correction schemes. In this work, we present the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis. Our algorithm produces an estimate of the unknown unitary that is accurate to small worst-case error, measured by the physically motivated energy-constrained diamond distance. Its runtime and query complexity scale polynomially with the number of modes, the inverse target accuracy, and natural energy parameters quantifying the allowed input energy and the unitary's output-energy growth. The protocol uses only experimentally friendly photonic resources: coherent and squeezed probes, passive linear optics, and heterodyne/homodyne detection. We then employ an efficient classical post-processing routine that leverages a symplectic regularization step to project matrix estimates onto the symplectic group. In the limit of unbounded input energy, our procedure attains arbitrarily high precision using only queries, where is the number of modes. To our knowledge, this is the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.

Paper Structure

This paper contains 49 sections, 25 theorems, 170 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Technical contributions and significance.
  4. Technical overview
  5. Symplectic learning via coherent probing (\ref{['sec:vac-share', 'sec:sym-probe']}).
  6. Symplectic regularization (\ref{['sec:close-sym']}).
  7. Displacement learning with two-mode squeezed states (\ref{['sec:dis-entangle']}).
  8. Experimental feasibility (\ref{['sec:no-activesq']}).
  9. Displacement learning with single-mode squeezed states (\ref{['sec:algo-without-entanglement']}).
  10. End-to-end guarantee (\ref{['sec:end-to-end']}).
  11. Related work
  12. Local and global estimation.
  13. Learning the symplectic part of Gaussian unitaries.
  14. Process tomography with coherent probes.
  15. Connections to related learning tasks.
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $m\in\mathds{N}$, $z\ge 1$, $\bar{n}>0$, $\bar{n}_{\mathrm{in}}>0$, $\varepsilon\in(0,1)$, and $\delta\in(0,1)$ be known parameters. There exists a quantum algorithm that

Theorems & Definitions (44)

  • Theorem 1.1: (Informal, see \ref{['thm:end-to-end-diamond']} for details)
  • Lemma 2.1: (Euler decomposition)
  • Lemma 2.2: (Matrix $p$-th root representation and perturbation, see e.g. cardoso2012computation)
  • Lemma 2.3: (Tail bound for Gaussian operator norm; rescaled version of DavidsonSzarek2001)
  • proof
  • Lemma 2.4: (Estimation of first moments, see e.g. lugosi2017subgaussianestimatorsmeanrandom)
  • Proposition 2.1: (Bounds for displacement channels, see e.g. EC-diamond)
  • Proposition 2.2: (Bounds for symplectic Gaussian unitaries, see e.g. EC-diamond)
  • Lemma 4.1: (Query complexity for learning the symplectic part with vacuum-shared inputs)
  • proof
  • ...and 34 more