Quantum Box-Muller Transform

TL;DR

This paper introduces the Quantum Box-Muller transform, a quantum algorithm for efficiently generating multivariate Gaussian distributions and estimating expectations, with applications in quantum Monte Carlo methods.

Contribution

It presents a novel quantum algorithm that creates superpositions of Gaussian distributions using quantum arithmetic, improving state preparation for quantum Monte Carlo.

Findings

Quadratic gate complexity in the number of qubits

Efficient expectation value estimation with exponentially small error

Applicable to quantum Monte Carlo integration

Abstract

The Box-Muller transform is a widely used method to generate Gaussian samples from uniform samples. Quantum amplitude encoding methods encode the multi-variate normal distribution in the amplitudes of a quantum state. This work presents the Quantum Box-Muller transform which creates a superposition of binary-encoded grid points representing the multi-variate normal distribution. The gate complexity of our method depends on quantum arithmetic operations and, using a specific set of known implementations, the complexity is quadratic in the number of qubits. We apply our method to Monte-Carlo integration, in particular to the estimation of the expectation value of a function of Gaussian random variables. Our method implies that the state preparation circuit used multiple times in amplitude estimation requires only quantum arithmetic circuits for the grid points and the function, in addition to a single controlled rotation. We show how to provide the expectation value estimate with an error that is exponentially small in the number of qubits, similar to the amplitude-encoding setting with error-free encoding.