The Born Ultimatum: Conditions for Classical Surrogation of Quantum Generative Models with Correlators

TL;DR

This paper investigates the limitations of classical surrogates for quantum generative models, specifically QCBMs, by analyzing the discrepancies caused by deployment differences using Fourier correlators and tensor network methods.

Contribution

It introduces a framework to quantify classical surrogation discrepancies in QCBMs through Fourier decomposition and provides analytical and numerical insights into these effects.

Findings

Discrepancies between classical surrogates and quantum models are analytically quantified.

Numerical demonstrations show the impact of deployment differences on generative performance.

Closed-form expressions for Pauli propagation and Lie algebra are derived for specific circuits.

Abstract

Quantum Circuit Born Machines (QCBMs) are powerful quantum generative models that sample according to the Born rule, with complexity-theoretic evidence suggesting potential quantum advantages for generative tasks. Here, we identify QCBMs as a quantum Fourier model independently of the loss function. This allows us to apply known dequantization conditions when the optimal quantum distribution is available. However, realizing this distribution is hindered by trainability issues such as vanishing gradients on quantum hardware. Recent train-classical, deploy-quantum approaches propose training classical surrogates of QCBMs and using quantum devices only for inference. We analyze the limitations of these methods arising from deployment discrepancies between classically trained and quantumly deployed parameters. Using the Fourier decomposition of the Born rule in terms of correlators, we quantify this discrepancy analytically. Approximating the decomposition via distribution truncation and classical surrogation provides concrete examples of such discrepancies, which we demonstrate numerically. We study this effect using tensor-networks and Pauli-propagation-based classical surrogates. Our study examines the use of IQP circuits, matchcircuits, Heisenberg-chain circuits, and Haldane-chain circuits for the QCBM ansatz. In doing so, we derive closed-form expressions for Pauli propagation in IQP circuits and the dynamical Lie algebra of the Haldane chain, which may be of independent interest.