Quantum Algorithm for Approximating Multivariate Traces

Conceptos Básicos

This paper presents a quantum algorithm for approximating multivariate traces, which are traces of products of matrices. The algorithm leverages a novel framework called Quantum Matrix State Linear Algebra (qMSLA) to efficiently construct quantum circuits that implement the multivariate trace formula.

Resumen

The paper 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.
The key contributions are:

Introduction of Quantum Matrix State Linear Algebra (qMSLA), a framework for performing matrix-level operations directly on quantum state preparation circuits. qMSLA provides a set of primitive operations that can be used to synthesize quantum circuits implementing various matrix computations.

Development of a quantum algorithm for approximating multivariate traces using qMSLA. The algorithm takes as input state preparation circuits for the input matrices and outputs two quantum state preparation circuits whose overlap encodes the multivariate trace. This overlap can then be estimated using existing quantum algorithms like Hadamard Test or Swap Test.

Discussion of how the proposed algorithm can be used to approximate spectral sums, which have wide-ranging applications in scientific computing and machine learning.

Analysis showing that the algorithm makes only mild assumptions on the input, requiring only access to state preparation circuits for the input matrices, without relying on more restrictive quantum input models like block encodings.

The paper demonstrates how the qMSLA framework can be leveraged to design quantum algorithms for fundamental matrix computations, showcasing its potential as a new paradigm for doing matrix computations in the quantum domain.

Estadísticas

The multivariate trace of matrices A1, A2, ..., Ak of compatible dimensions is defined as: MTrk(A1, ..., Ak) := Tr(A1A2 ... Ak).
Matrix moments Tr(Ak) = MTrk(A, ..., A) reveal useful spectral properties of A with applications in scientific computing and other fields.
Many machine learning techniques estimate spectral properties of various matrices by approximating appropriate spectral sums via polynomial spectral sums Tr(p(A)).

Citas

"Approximating multivariate traces is motivated as follows. For a square A ∈Rn×n, its matrix moments Tr(Ak) = MTrk(A, ..., A) for k = 1, 2, ..., which reveal useful spectral properties of A with applications in scientific computing [32] and other fields, are multivariate traces."
"Many machine learning techniques estimate spectral properties of various matrices by approximating appropriate spectral sums via polynomial spectral sums Tr(p(A)), e.g. Gaussian Processes [54], kernel learning [22], Bayesian learning [47], matrix completion [12], differential privacy problems [34], graph analysis [25], Hessian and neural network property analysis [53, 27] and many more [67, 14]."

Ideas clave extraídas de

Multivariate trace estimation using quantum state space linear algebra

by Liron Mor Yo... a las arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01098.pdf

Consultas más profundas

How can the proposed quantum algorithm for multivariate trace estimation be extended to handle matrices that are not necessarily square or Hermitian

The proposed quantum algorithm for multivariate trace estimation can be extended to handle matrices that are not necessarily square or Hermitian by introducing additional operations in the qMSLA framework. For non-square matrices, the algorithm can be modified to handle different dimensions by adjusting the circuit operations accordingly. This may involve padding the matrices with zeros or reshaping the circuits to accommodate the non-square dimensions.
To handle non-Hermitian matrices, the algorithm can incorporate operations for complex conjugation and transpose to manipulate the matrices appropriately. By including these operations in the qMSLA framework, the algorithm can effectively handle a wider range of matrix types, allowing for more versatile applications in quantum computing.

What are the limitations of the state preparation input model compared to the block encoding input model, and how can these limitations be addressed

The limitations of the state preparation input model compared to the block encoding input model primarily revolve around the complexity and efficiency of representing matrices in quantum circuits. The state preparation model requires circuits to encode matrices directly, which can be challenging for large or complex matrices. In contrast, the block encoding model offers a more structured approach by representing matrices as block encodings, which can be more efficient for certain operations.
To address these limitations, techniques such as circuit optimization, gate reduction, and qubit reordering can be employed to streamline the state preparation circuits and improve their efficiency. Additionally, advancements in quantum circuit design and compilation tools can help mitigate the complexity of handling large matrices in the state preparation model. By optimizing the circuit structures and implementing efficient quantum operations, the limitations of the state preparation input model can be minimized.

Can the qMSLA framework be further generalized to enable the design of quantum algorithms for a broader class of matrix computations beyond trace estimation

The qMSLA framework can be further generalized to enable the design of quantum algorithms for a broader class of matrix computations beyond trace estimation by expanding the set of primitive operations and composite operations. By introducing new operations that encompass a wider range of matrix algebra tasks, such as matrix multiplication, inversion, decomposition, and eigenvalue calculations, the qMSLA framework can be enhanced to support a more comprehensive set of quantum algorithms for various matrix computations.
Additionally, incorporating advanced quantum computing techniques, such as quantum phase estimation, quantum Fourier transforms, and quantum singular value transformations, can further extend the capabilities of the qMSLA framework for tackling complex matrix computations. By continuously refining and expanding the operations within the qMSLA framework, researchers can develop a versatile toolkit for quantum numerical linear algebra that addresses a diverse array of matrix-related problems in quantum computing.