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Universality of Classically Trainable, Quantum-Deployed Boson-Sampling Generative Models

Andrii Kurkin, Ulysse Chabaud, Zoltán Kolarovszki, Bence Bakó, Zoltán Zimborás, Vedran Dunjko

TL;DR

The Boson Sampling Born Machine is introduced and the role of constant-function postprocessing is introduced, generalizing the construction for IQP-QCBMs, which under suitable conditions can lead to universality while preserving the hardness of classically simulating the models.

Abstract

Recent work on the instantaneous quantum polynomial-time (IQP) quantum-circuit Born machine (QCBM) highlights a promising paradigm for generative modeling: train classically, deploy quantumly. In this setting, the training objective can be evaluated efficiently on a classical computer, while sampling from the resulting model may still be classically intractable. Furthermore, in the IQP-QCBM framework, extending the model family with ancillary qubits has been proven to yield universality. This paper asks whether similar results hold for linear-optical generative models. To this end, we introduce the Boson Sampling Born Machine (BSBM). Our analysis retraces analogous steps as were found for IQP-QCBMs with twists. Using recent results that enable classical approximation of broad classes of expectation values in linear optics, we show that BSBMs can be trained classically for wide families of loss functions. Next, we argue that "basic" BSBMs are not universal generative models, and that universality can be achieved by expanding the model while preserving efficient classical training and sampling hardness. In our approach, we introduce and analyze the role of constant-function postprocessing, generalizing the construction for IQP-QCBMs, which under suitable conditions can lead to universality while preserving the hardness of classically simulating the models. We showcase a family of BSBMs, characterized by a single hyperparameter, that allows for a monotonic increase in expressivity toward universality while retaining the capacity to represent ostensibly hard distributions. Furthermore, we discuss the possible modalities for the efficient classical training, in the sense of efficient estimation of gradients of the loss function.

Universality of Classically Trainable, Quantum-Deployed Boson-Sampling Generative Models

TL;DR

The Boson Sampling Born Machine is introduced and the role of constant-function postprocessing is introduced, generalizing the construction for IQP-QCBMs, which under suitable conditions can lead to universality while preserving the hardness of classically simulating the models.

Abstract

Recent work on the instantaneous quantum polynomial-time (IQP) quantum-circuit Born machine (QCBM) highlights a promising paradigm for generative modeling: train classically, deploy quantumly. In this setting, the training objective can be evaluated efficiently on a classical computer, while sampling from the resulting model may still be classically intractable. Furthermore, in the IQP-QCBM framework, extending the model family with ancillary qubits has been proven to yield universality. This paper asks whether similar results hold for linear-optical generative models. To this end, we introduce the Boson Sampling Born Machine (BSBM). Our analysis retraces analogous steps as were found for IQP-QCBMs with twists. Using recent results that enable classical approximation of broad classes of expectation values in linear optics, we show that BSBMs can be trained classically for wide families of loss functions. Next, we argue that "basic" BSBMs are not universal generative models, and that universality can be achieved by expanding the model while preserving efficient classical training and sampling hardness. In our approach, we introduce and analyze the role of constant-function postprocessing, generalizing the construction for IQP-QCBMs, which under suitable conditions can lead to universality while preserving the hardness of classically simulating the models. We showcase a family of BSBMs, characterized by a single hyperparameter, that allows for a monotonic increase in expressivity toward universality while retaining the capacity to represent ostensibly hard distributions. Furthermore, we discuss the possible modalities for the efficient classical training, in the sense of efficient estimation of gradients of the loss function.
Paper Structure (17 sections, 8 theorems, 19 equations, 1 algorithm)

This paper contains 17 sections, 8 theorems, 19 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear optics and boson sampling
  4. Boson Sampling Born Machine
  5. Classical training of BSBMs
  6. Extended mode: universality, expressivity, hardness and classical training
  7. Extended BSBM model
  8. Achieving monotone expressivity, universality and hardness inheritance
  9. Construction by direct parameter interpolation
  10. Efficient classical training
  11. Pushforward, non-identifiability, and lifting
  12. Deterministic lift.
  13. Interpreting training via a deterministic lift.
  14. Stochastic lift.
  15. Training procedure versus hardness.
  16. ...and 2 more sections

Key Result

Lemma 1

For any stationary and bounded kernel $k$ over $\{0,1\}^m$, let $G$ be the Fourier transform of $k$. The $\mathrm{MMD}$ loss of BSBM as specified in def:bsbm can be expressed as: where $\Pi_\alpha \coloneqq \bigotimes_{j=1}^m \Pi_j^{\alpha_j}$ and $\Pi_j$ is the single-mode parity operator $\Pi_j \coloneqq (-1)^{\hat{n}_j}$ with $\hat{n}_j$ being the number operator on mode $j$. Moreover, a parit

Theorems & Definitions (23)

  • Definition 1: Boson Sampling Born Machine (BSBM)
  • Remark 1: Specification of the model
  • Definition 2: Universality of a generative model
  • Definition 3: Maximum Mean Discrepancy (MMD)
  • Lemma 1: Generalized kernel MMD via parity expectation values
  • Lemma 2: Estimating parity word expectations in a BSBM
  • Theorem 1: Efficient estimation of MMD losses and gradients
  • proof : Sketch of proof
  • Remark 2: Trainability does not imply simulability
  • Remark 3: Unbiased estimator
  • ...and 13 more