Multivariate trace estimation using quantum state space linear algebra

TL;DR

This paper introduces a quantum algorithm for efficiently approximating multivariate traces of matrices, utilizing a novel framework called quantum Matrix States Linear Algebra (qMSLA) that simplifies circuit construction without requiring QRAM.

Contribution

The paper presents a new quantum algorithm for multivariate trace estimation based on qMSLA, enabling direct circuit translation without complex encodings or specialized hardware.

Findings

Algorithm constructs state preparation circuits for multivariate traces

Operates independently of QRAM and complex encodings

Uses qMSLA for efficient matrix state operations

Abstract

In this paper, we present a quantum algorithm for approximating multivariate traces, i.e. the traces of matrix products. Our research is motivated by the extensive utility of multivariate traces in elucidating spectral characteristics of matrices, as well as by recent advancements in leveraging quantum computing for faster numerical linear algebra. Central to our approach is a direct translation of a multivariate trace formula into a quantum circuit, achieved through a sequence of low-level circuit construction operations. To facilitate this translation, we introduce \emph{quantum Matrix States Linear Algebra} (qMSLA), a framework tailored for the efficient generation of state preparation circuits via primitive matrix algebra operations. Our algorithm relies on sets of state preparation circuits for input matrices as its primary inputs and yields two state preparation circuits encoding the multivariate trace as output. These circuits are constructed utilizing qMSLA operations, which enact the aforementioned multivariate trace formula. We emphasize that our algorithm's inputs consist solely of state preparation circuits, eschewing harder to synthesize constructs such as Block Encodings. Furthermore, our approach operates independently of the availability of specialized hardware like QRAM, underscoring its versatility and practicality.