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Classical simulation and quantum resource theory of non-Gaussian optics

Oliver Hahn, Ryuji Takagi, Giulia Ferrini, Hayata Yamasaki

TL;DR

This work addresses the classical simulation of continuous-variable systems with non-Gaussian inputs under Gaussian operations by decomposing non-Gaussian states into Gaussian superpositions and tracking relative phases with an extended covariance formalism. It introduces two simulators—an exact one costing $\mathcal{O}(\chi^2)$ and an approximate, sparsified one with linear scaling in $\|\mathbf{c}\|_1$—and defines the Gaussian rank $\chi$ and Gaussian extent $\xi$ as operational measures of non-Gaussianity. The authors connect these measures to a quantum-resource framework via robustness, prove monotonicity under Gaussian operations, and derive optimal decompositions for states relevant to CV quantum computing, including bounds for Gaussian boson sampling and cat-state breeding. The results provide both practical tools for classically bounding CV quantum advantages and fundamental insights into the resource cost of non-Gaussianity in continuous-variable computation.

Abstract

We propose efficient algorithms for classically simulating Gaussian unitaries and measurements applied to non-Gaussian initial states. The constructions are based on decomposing the non-Gaussian states into linear combinations of Gaussian states. We use an extension of the covariance matrix formalism to efficiently track relative phases in the superpositions of Gaussian states. We get an exact simulation algorithm, which costs quadratically with the number of Gaussian states required to represent the initial state, and an approximate simulation algorithm, which costs linearly with the $l_1$ norm of the coefficients associated with the superposition. We define measures of non-Gaussianity quantifying this simulation cost, which we call the Gaussian rank and the Gaussian extent. From the perspective of quantum resource theories, we investigate the properties of this type of non-Gaussianity measure and compute optimal decompositions for states relevant to continuous-variable quantum computing.

Classical simulation and quantum resource theory of non-Gaussian optics

TL;DR

This work addresses the classical simulation of continuous-variable systems with non-Gaussian inputs under Gaussian operations by decomposing non-Gaussian states into Gaussian superpositions and tracking relative phases with an extended covariance formalism. It introduces two simulators—an exact one costing and an approximate, sparsified one with linear scaling in —and defines the Gaussian rank and Gaussian extent as operational measures of non-Gaussianity. The authors connect these measures to a quantum-resource framework via robustness, prove monotonicity under Gaussian operations, and derive optimal decompositions for states relevant to CV quantum computing, including bounds for Gaussian boson sampling and cat-state breeding. The results provide both practical tools for classically bounding CV quantum advantages and fundamental insights into the resource cost of non-Gaussianity in continuous-variable computation.

Abstract

We propose efficient algorithms for classically simulating Gaussian unitaries and measurements applied to non-Gaussian initial states. The constructions are based on decomposing the non-Gaussian states into linear combinations of Gaussian states. We use an extension of the covariance matrix formalism to efficiently track relative phases in the superpositions of Gaussian states. We get an exact simulation algorithm, which costs quadratically with the number of Gaussian states required to represent the initial state, and an approximate simulation algorithm, which costs linearly with the norm of the coefficients associated with the superposition. We define measures of non-Gaussianity quantifying this simulation cost, which we call the Gaussian rank and the Gaussian extent. From the perspective of quantum resource theories, we investigate the properties of this type of non-Gaussianity measure and compute optimal decompositions for states relevant to continuous-variable quantum computing.
Paper Structure (33 sections, 219 equations, 1 table)

This paper contains 33 sections, 219 equations, 1 table.

Table of Contents

  1. Introduction
  2. Gaussian quantum optics
  3. Gaussian unitaries
  4. Tensor product and partial trace
  5. Gaussian completely positive and trace-preserving (CPTP) map
  6. Gaussian measurements
  7. Conditional Gaussian dynamics
  8. Phase-sensitive simulator for Gaussian states and operations
  9. Classical simulation algorithm for non-Gaussian optics
  10. Exact simulation
  11. Approximate simulation
  12. Comparison
  13. Measures of non-Gaussianity based on Gaussian decomposition
  14. Characterization of Gaussian extent by robustness measure
  15. Decompositions and Applications
  16. ...and 18 more sections