Every Classical Sampling Circuit is a Quantum Sampling Circuit

TL;DR

This paper demonstrates that any classical sampling circuit can be transformed into a quantum state suitable for Quantum Monte Carlo Integration, potentially enabling quantum advantage if the preparation cost is comparable to classical sampling.

Contribution

It introduces Q-marginals, a method to encode classical probability distributions into quantum states directly from classical sampling circuits.

Findings

Q-marginals can be prepared from classical sampling circuits.

Quantum advantage depends on the cost of quantum state preparation.

Classical and quantum sampling circuits are fundamentally connected.

Abstract

This note introduces "Q-marginals", which are quantum states encoding some probability distribution in a manner suitable for use in Quantum Monte Carlo Integration (QMCI), and shows that these can be prepared directly from a classical circuit sampling for the probability distribution of interest. This result is important as the quantum advantage in Monte Carlo integration is in the form of a reduction in the number of uses of a quantum state encoding the probability distribution (in QMCI) relative to the number of samples that would be required in classical MCI -- hence it only translates into a computational advantage if the number of operations required to prepare this quantum state encoding the probability distribution is comparable to the number of operations required to generate a classical sample (as the Q-marginal construction achieves).