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Lower Bounds on Coherent State Rank

Florian Cottier, Ulysse Chabaud

Abstract

The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows. (i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states. (ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states. (iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states. (iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the $n$-mode Fock state $|1\rangle^{\otimes n}$, by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009]. Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.

Lower Bounds on Coherent State Rank

Abstract

The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows. (i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states. (ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states. (iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states. (iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the -mode Fock state , by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009]. Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.

Paper Structure

This paper contains 21 sections, 23 theorems, 94 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Technical overview of results
  3. Definitions
  4. Single-mode results
  5. Multimode results
  6. Discussion and open questions
  7. Acknowledgements
  8. Preliminaries
  9. Norms, distances, and similarity measures
  10. Approximate coherent state rank
  11. Passive linear unitaries and vacuum projection
  12. Low-rank approximation theory
  13. Single-mode lower bounds on the approximate coherent state rank
  14. Generic lower bounds
  15. Characterization of single-mode states of finite approximate coherent state rank
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\ket{\psi}$ be a single-mode quantum state. There is a family of non-negative, efficiently computable functions $(f_N)_{N\in\mathbb N}$ such that for all $r\le N\in\mathbb N$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Lower bounds on the infidelity between a target state $\ket\psi$ and the closest superposition of $\kappa(\psi)-1$ coherent states, showing the analytical bound from \ref{['thm:lower_bound_appox_coherent_stellar']} and the numerically optimized bound from \ref{['thm:optimised_b']}. Left: Target state $\sqrt{1-\gamma}\ket{0}+\sqrt{\gamma}\ket{1}$, for which the exact value is also shown. Right: Target state $\ket{n}$ for $n$ ranging from 1 to 12.

Theorems & Definitions (47)

  • Definition 1: Approximate coherent state rank Marshall_2023
  • Definition 2: $\varepsilon$-approximate coherent state rank Marshall_2023upreti2025bounding
  • Theorem 1: Generic lower bounds on the $\varepsilon$-approximate coherent state rank (informal)
  • proof : Proof sketch
  • Theorem 2: Characterization of single-mode states of finite approximate coherent state rank
  • proof : Proof sketch
  • Corollary 1: Approximate coherent state rank of single-mode core states
  • proof
  • Corollary 2: Approximate coherent state rank of single-mode squeezed states
  • proof : Proof sketch
  • ...and 37 more