Multivariate trace estimation using quantum state space linear algebra

TL;DR

This work introduces qMSLA, a BLAS-like framework for quantum linear algebra that operates on matrices encoded as quantum state preparations. It defines a formal method to encode multivariate traces ${\bf MTr}_{k}({\bm{A}}_{1},\dots,{\bm{A}}_{2k})$ as overlaps between two state-preparation circuits produced by MVTracePrep, enabling estimation via Hadamard/Swap tests or phase estimation. The authors demonstrate concrete Level-1 and Level-2 primitives for matrix operations, along with warm-ups for k=2,3 and a general k>2 MVTr formula, and discuss end-to-end complexity under a lax input model that avoids block encodings or QRAM. The approach supports non-Hermitian/non-square matrices and can be extended to spectral-sum approximations, offering a versatile pathway for quantum-assisted spectral analysis and learning tasks. Overall, qMSLA provides a conceptual and practical toolkit for executing matrix algebra directly in the quantum state space, facilitating new quantum algorithms for numerical linear algebra and machine learning tasks.

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.

Figures (9)

• Figure 1: Circuit Implementing of Hadamard test
• Figure 2: Circuit Implementing of Swap Test
• Figure 3: Graphical description of $\text{qmsla.identity}$ (Input: $n$, Output: ${\cal U}_{{\bm{\mathrm{I}}}_{n}}$).
• Figure 4: Graphical description of $\text{qmsla.matrix}$ (Input: ${\cal U}$, Output: ${\cal U}_{{\bm{\mathrm{M}}}\left({\cal U}\right)}$).
• Figure 5: Graphical description of $\text{qmsla.transpose}$ (Input: ${\cal U}_{{\bm{\mathrm{A}}}}$, Output: ${\cal U}_{{\bm{\mathrm{A}}}^{\textsc{T}}}$).

Theorems & Definitions (25)

• Proposition 1: Hadamard Test
• Remark 2
• Proposition 3: Swap Test
• Definition 4: State preparation Circuit
• Definition 5: Block encoding of a matrix gilyen2019quantum
• Definition 6: Matrix state preparation circuit
• Remark 7
• Remark 8
• Remark 9
• Lemma 10