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Native linear-optical protocol for efficient multivariate trace estimation

Leonardo Novo, Marco Robbio, Ernesto F. Galvão, Nicolas J. Cerf

Abstract

The Hong-Ou-Mandel test estimates the overlap between spectral functions characterizing the internal degrees of freedom of two single photons. It can be viewed as a photon-native protocol that implements the well-known quantum SWAP test. Here, we propose a native linear-optical protocol that efficiently estimates multivariate traces of quantum states called Bargmann invariants, which are ubiquitous in quantum mechanics. Our protocol may be understood as a photon-native version of the cycle test in the circuit model, which encompasses many-photon multimode quantum states. We show the protocol is sample-efficient and discuss applications, such as generalized suppression laws, efficient quantum kernel estimation for quantum machine learning, eigenspectrum estimation, and the characterization of multiphoton indistinguishability.

Native linear-optical protocol for efficient multivariate trace estimation

Abstract

The Hong-Ou-Mandel test estimates the overlap between spectral functions characterizing the internal degrees of freedom of two single photons. It can be viewed as a photon-native protocol that implements the well-known quantum SWAP test. Here, we propose a native linear-optical protocol that efficiently estimates multivariate traces of quantum states called Bargmann invariants, which are ubiquitous in quantum mechanics. Our protocol may be understood as a photon-native version of the cycle test in the circuit model, which encompasses many-photon multimode quantum states. We show the protocol is sample-efficient and discuss applications, such as generalized suppression laws, efficient quantum kernel estimation for quantum machine learning, eigenspectrum estimation, and the characterization of multiphoton indistinguishability.
Paper Structure (11 sections, 7 theorems, 29 equations, 1 figure)

This paper contains 11 sections, 7 theorems, 29 equations, 1 figure.

Table of Contents

  1. Framework --
  2. Multivariate trace estimation ---
  3. Generalized suppression laws ---
  4. Quantum kernel methods --
  5. Eigenvalue estimation --
  6. Partial photonic distinguishability --
  7. Discussion --
  8. Acknowledgments --
  9. Appendix
  10. Quasiprobability distributions -
  11. Proofs

Key Result

Proposition 1

Let $\Omega= \bigotimes_{j} \rho_j$ be a bosonic quantum state where each $\rho_j$ belongs to a multimode bosonic Fock space $\mathcal{H}$. The multivariate trace $\mathrm{Tr}\left[ \rho_1\rho_2...\rho_M \right]$ can be estimated with probability $1-\delta$ and precision $\epsilon$ with $O(\epsilon^

Figures (1)

  • Figure 1: Generalized HOM test for estimation of overlaps of generic input states $\{\rho_1,\rho_2\}$ involving multiple photons occupying several spatial modes, as well as further internal d.o.f. The $i$th spatial mode of $\rho_1$ interferes via a 50:50 beam splitter with the $i$th spatial mode of $\rho_2$. We assume the beam-splitter leaves invariant other internal modes (e.g. frequency, polarization). Total mode occupations $S_j$ are measured by photon counting and outcomes are binned according to $f(\vec{S})$ to estimate probabilities $P_j$ (see Eqs. \ref{['eq:P_j']}), for an estimation of overlap $\mathrm{Tr}\left[ \rho_1 \rho_2 \right]$. A generalized HOM suppression is observed if $\rho_1$ and $\rho_2$ are identical pure states for outcomes $\vec{S}$ such that $f(\vec{S})=1$.

Theorems & Definitions (11)

  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 1 more