Quantum Box-Muller Transform

TL;DR

This work tackles the problem of efficiently generating Gaussian samples on quantum hardware for applications such as finance and Monte Carlo integration. It introduces the Quantum Box-Muller transform, which constructs two-dimensional normal samples from uniform quantum states using quantum arithmetic for the Box-Muller transforms, along with amplitude-estimation-based Monte Carlo evaluation. The authors provide a rigorous three-case framework (Case 1: $\theta\in[0,1]$, Case 2: $\theta\ge0$ with $\mathbb{E}[\theta^2]\le B^2$, Case 3: $\mathrm{Var}[\theta]\le\sigma^2$), deriving error bounds and scaling laws that yield $N=\mathcal{O}(\epsilon^{-3/2})$ and $t=\mathcal{O}(\epsilon^{-1})$ in favorable regimes. They also give detailed resource estimates for fixed-point arithmetic (multiplication, sine/cosine, logarithm, square root) and validate the approach via numerical experiments, including a Qiskit demonstration. Overall, the paper offers an arithmetic-only alternative to Gaussian-amplitude state preparation with clear implications for quantum Monte Carlo and financial modeling subroutines.

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.

Figures (3)

• Figure 1: Comparison of the standard method (state preparation) and our method (Box-Muller). Here, the random variable $\theta$ is given via $O_\theta:\ket{i}\ket{j}\ket{0}\to\ket{i}\ket{j}\ket{\theta(i,j)}$ and the controlled rotation is given via the operation $W_\theta:\ket{\theta}\ket{0}\to\ket{\theta}(\sqrt{\theta}\ket{0}+\sqrt{1-\theta}\ket{1})$.
• Figure 2: The approximation error of the quantum Box-Muller transform for different parameter values. In this experiment, $p$ is set to $10$ to avoid overflow, $n\in \lbrace 20,21,22,23,24,25 \rbrace$, $d\in \lbrace 1,2,3,4\rbrace$, $M\in \lbrace 4,8,16,32,64\rbrace$. In all tested cases, the error saturates at $M=32$ and $d=1$. The improvement from higher degrees is insignificant as it is offset by the limited precision of the fixed-point representation: $2^{p-n}$.
• Figure 3: The QQ plot and the histogram of 1024 samples from a Qiskit simulation of the quantum Box-Muller transform. This small experiment uses $5$ qubits to represent each uniform sample.

Theorems & Definitions (6)

• Theorem 1: Classical Box-Muller transform
• Lemma 1
• Lemma 2
• Theorem 2
• Theorem 3
• Theorem 4