Partition Function Estimation: Quantum and Quantum-Inspired Algorithms

TL;DR

This paper introduces quantum and classical algorithms for estimating partition functions of quantum spin Hamiltonians, highlighting the complexity and providing new computational insights.

Contribution

It presents the first DQC1 algorithm for complex temperatures and a classical algorithm for real temperatures with comparable performance, analyzing their computational hardness.

Findings

First DQC1 algorithm for complex temperatures

Classical algorithm for real temperatures with similar performance

Partition function estimation is DQC1-hard for Hamiltonian decomposition

Abstract

We present two algorithms, one quantum and one classical, for estimating partition functions of quantum spin Hamiltonians. The former is a DQC1 (Deterministic quantum computation with one clean qubit) algorithm, and the first such for complex temperatures. The latter, for real temperatures, achieves performance comparable to a state-of-the-art DQC1 algorithm [Chowdhury et al. Phys. Rev. A 103, 032422 (2021)]. Both our algorithms take as input the Hamiltonian decomposed as a linear combination Pauli operators. We show this decomposition to be DQC1-hard for a given Hamiltonian, providing new insight into the hardness of estimating partition functions.