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Classical algorithms for measurement-adaptive Gaussian circuits

Changhun Oh, Youngrong Lim

TL;DR

Bosonic circuits with adaptivity are analyzed and it is proved that when the number of adaptive measurements is small, the mean-value problem admits efficient classical algorithms even if a large amount of non-Gaussian resources are present in the input state, whereas less constrained regimes are computationally hard.

Abstract

Gaussian building blocks are essential for photonic quantum information processing, and universality can be practically achieved by equipping Gaussian circuits with adaptive measurement and feedforward. The number of adaptive steps then provides a natural parameter for computational power. Rather than assessing power only through sampling problems -- the usual benchmark -- we follow the ongoing shift toward tasks of practical relevance and study the quantum mean-value problem, i.e., estimating observable expectation values that underpin simulation and variational algorithms. More specifically, we analyze bosonic circuits with adaptivity and prove that when the number of adaptive measurements is small, the mean-value problem admits efficient classical algorithms even if a large amount of non-Gaussian resources are present in the input state, whereas less constrained regimes are computationally hard. This yields a task-level contrast with sampling, where non-Gaussian ingredients alone often induce hardness, and provides a clean complexity boundary parameterized by the number of adaptive measurement-and-feedforward steps between classical simulability and quantum advantage. Beyond the main result, we introduce classical techniques -- including a generalization of Gurvits' second algorithm to arbitrary product inputs and Gaussian circuits -- for computing the marginal quantities needed by our estimators, which may be of independent interest.

Classical algorithms for measurement-adaptive Gaussian circuits

TL;DR

Bosonic circuits with adaptivity are analyzed and it is proved that when the number of adaptive measurements is small, the mean-value problem admits efficient classical algorithms even if a large amount of non-Gaussian resources are present in the input state, whereas less constrained regimes are computationally hard.

Abstract

Gaussian building blocks are essential for photonic quantum information processing, and universality can be practically achieved by equipping Gaussian circuits with adaptive measurement and feedforward. The number of adaptive steps then provides a natural parameter for computational power. Rather than assessing power only through sampling problems -- the usual benchmark -- we follow the ongoing shift toward tasks of practical relevance and study the quantum mean-value problem, i.e., estimating observable expectation values that underpin simulation and variational algorithms. More specifically, we analyze bosonic circuits with adaptivity and prove that when the number of adaptive measurements is small, the mean-value problem admits efficient classical algorithms even if a large amount of non-Gaussian resources are present in the input state, whereas less constrained regimes are computationally hard. This yields a task-level contrast with sampling, where non-Gaussian ingredients alone often induce hardness, and provides a clean complexity boundary parameterized by the number of adaptive measurement-and-feedforward steps between classical simulability and quantum advantage. Beyond the main result, we introduce classical techniques -- including a generalization of Gurvits' second algorithm to arbitrary product inputs and Gaussian circuits -- for computing the marginal quantities needed by our estimators, which may be of independent interest.

Paper Structure

This paper contains 26 sections, 17 theorems, 170 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related previous studies
  3. Preliminary
  4. Problem setup
  5. Classical algorithms
  6. Gaussian circuits without feedforward
  7. Photon-number-measurement-adaptive Gaussian circuits
  8. Gaussian-measurement-adaptive Gaussian circuits
  9. Classical algorithms using low-rank structures
  10. Computing the marginal probabilities of Gaussian circuits on photon-number basis
  11. Generalized Gurvits' second algorithm
  12. Warm-up example: Permanent
  13. Amplitude of low-mode states
  14. Low-rank structure
  15. Proof sketch of Lemma \ref{['lemma:MVS']}
  16. ...and 11 more sections

Key Result

Theorem 1

Consider an $M$-mode Gaussian circuit $\hat{G}$ and an arbitrary product input state $|\psi\rangle$, and an $M$-mode product operator $\hat{O}=\hat{O}_A\otimes \hat{\mathbb{1}}_B$. The expectation value $\langle \psi|\hat{G}^\dagger \hat{O}\hat{G}|\psi\rangle$ can be approximated within additive err where $\hat{\rho}_A\equiv \Tr_B[\hat{G}|\psi\rangle\langle \psi|\hat{G}^\dagger]$ is the reduced de

Figures (2)

  • Figure 1: Schematics of quantum circuits of interest in this work. The input state is a product state, and the quantum circuits are composed of Gaussian circuits, measurements, and feedforward operations. More specifically, we consider photon number measurements (Sec. \ref{['sec:adaptive_photon']}) and Gaussian measurements (Sec. \ref{['sec:adaptive_gaussian']}) for feedforward operations. The final measurement on the computing register is always Gaussian for sampling and implicit for the quantum mean-value problem, depending on the observable. Note that the final measurement can also be adaptive through the final circuit layer $\hat{G}_{L+1}$.
  • Figure 2: Schematics of Lemma \ref{['lemma:low-rank']}. The lemma shows that for a Gaussian unitary circuit $\hat{G}_0$ and a phase shifter $\hat{P}(\bm{\varphi})$ where $\bm{\varphi}\equiv (\varphi_1,\dots,\varphi_L,0,\dots,0)$, $\hat{G}_0^\dagger \hat{P}(\bm{\varphi})\hat{G}_0$ is equivalent to $\hat{U}\hat{S}(\bm{r})\hat{V}$, where $\bm{r}=(r_1,\dots,r_{2L},0,\dots,0)$.

Theorems & Definitions (30)

  • Definition 1: Sampling problem
  • Definition 2: Quantum mean-value problem
  • Theorem 1: Quantum mean-value problem in Gaussian circuits without feedforward
  • proof : Proof Sketch
  • Theorem 2: Sampling in Gaussian circuits with photon-number measurement and Gaussian feedforward
  • proof : Proof
  • Theorem 3: Quantum mean-value problem in Gaussian circuits with photon-number measurement and Gaussian feedforward
  • proof : Proof sketch
  • Lemma 1: Computing photon-number marginal probabilities in Gaussian circuits
  • Theorem 4: Quantum mean-value problem in Gaussian circuit augmented by Gaussian measurement and Gaussian feedforward
  • ...and 20 more