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arXiv:2308.05660 · npj Unconventional Computing

Thermodynamic linear algebra

Maxwell Aifer, Kaelan Donatella, Max Hunter Gordon, Samuel Duffield, Thomas D. Ahle, Daniel Simpson, Gavin E. Crooks, Patrick J. Coles

Published Nov 5, 2024• 32 citations
arXivHTMLPDF

Abstract

Abstract Linear algebra is central to many algorithms in engineering, science, and machine learning; hence, accelerating it would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities. We consider an alternative physics-based computing paradigm based on classical thermodynamics, to provide a near-term approach to accelerating linear algebra. At first sight, thermodynamics and linear algebra seem to be unrelated fields. Here, we connect solving linear algebra problems to sampling from the thermodynamic equilibrium distribution of a system of coupled harmonic oscillators. We present simple thermodynamic algorithms for solving linear systems of equations, computing matrix inverses, and computing matrix determinants. Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension. Our algorithms exploit thermodynamic principles like ergodicity, entropy, and equilibration, highlighting the deep connection between these two seemingly distinct fields, and opening up algebraic applications for thermodynamic computers.

DOI
10.1038/s44335-024-00014-0

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Abstract Linear algebra is central to many algorithms in engineering, science, and machine learning; hence, accelerating it would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities. We consider an alternative physics-based computing paradigm based on classical thermodynamics, to provide a near-term approach to accelerating linear algebra. At first sight, thermodynamics and linear algebra seem to be unrelated fields. Here, we connect solving linear algebra problems to sampling from the thermodynamic equilibrium distribution of a system of coupled harmonic oscillators. We present simple thermodynamic algorithms for solving linear systems of equations, computing matrix inverses, and computing matrix determinants. Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension. Our algorithms exploit thermodynamic principles like ergodicity, entropy, and equilibration, highlighting the deep connection between these two seemingly distinct fields, and opening up algebraic applications for thermodynamic computers.
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Abstract Solving large-scale linear systems is an integral part of many scientific disciplines. Classical linear solvers have a polynomial time complexity and it seems impossible to further improve their performance. This fact has directed scientific research to the study of corresponding quantum algorithms, which promise even exponential acceleration compared to existing methods. In this paper, we present and analyze some of the most important related results. In particular, we present the HHL and WZP algorithms based on the eigen-decomposition of a matrix, the row and column iteration methods, as well as two hybrid algorithms that require the cooperation of a classical and a quantum computer, namely an algorithm that uses random walks and the VQLS algorithm. The latter is also examined from an experimental standpoint using the Qiskit open-source framework and a suitable quantum circuit is proposed which can enhance its performance.
This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.