The State Preparation of Multivariate Normal Distributions using Tree Tensor Network

TL;DR

This paper introduces a scalable quantum circuit method using tree tensor networks to efficiently prepare high-dimensional multivariate normal distributions, significantly reducing circuit complexity while maintaining fidelity.

Contribution

The paper presents a novel TTN-based approach for quantum state preparation of multivariate normal distributions with automatic structural optimization.

Findings

Reduces circuit depth and CNOT count compared to existing methods.

Efficiently represents distributions with 1D correlation structures using TTN.

Maintains high fidelity in state preparation for high-dimensional distributions.

Abstract

The quantum state preparation of probability distributions is an important subroutine for many quantum algorithms. When embedding $D$-dimensional multivariate probability distributions by discretizing each dimension into $2^n$ points, we need a state preparation circuit comprising a total of $nD$ qubits, which is often difficult to compile. In this study, we propose a scalable method to generate state preparation circuits for $D$-dimensional multivariate normal distributions, utilizing tree tensor networks (TTN). We establish theoretical guarantees that multivariate normal distributions with 1D correlation structures can be efficiently represented using TTN. Based on these analyses, we propose a compilation method that uses automatic structural optimization to find the most efficient network structure and compact circuit. We apply our method to state preparation circuits for various high-dimensional random multivariate normal distributions. The numerical results suggest that our method can dramatically reduce the circuit depth and CNOT count while maintaining fidelity compared to existing approaches.