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arXiv:2309.05290

Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective

Alejandro Mata Ali, Iñigo Perez Delgado, Marina Ristol Roura, Aitor Moreno-Fernández-de-Leceta, Sebastián V. Romero

Published Sep 11, 2023• 0 citations
arXivHTMLPDF

Abstract

This work presents a new approach for simulating the HHL linear systems of equations solver algorithm with tensor networks. First, a novel HHL in the qudits formalism, the generalization of qubits, is developed, and then its operations are transformed into an equivalent classical HHL, taking advantage of the non-unitary operations that they can apply. The main novelty of this proposal is to perform a classical simulation of the HHL as efficiently as possible to benchmark the algorithm steps according to its input parameters and the input matrix. The algorithm is applied to three classical simple simulation problems, comparing it with an exact inversion algorithm, and its performance is compared against an implementation of the original HHL simulated in the Qiskit framework, providing both codes. It is also applied to study the sensitivity of the HHL algorithm with respect to its hyperparameter values, reporting the existence of saturation points and maximal performance values. The results show that this approach can achieve a promising performance in computational efficiency to simulate the HHL process without quantum noise, providing a higher bound for its performance.

DOI
10.48550/arxiv.2309.05290

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This work presents a new approach for simulating the HHL linear systems of equations solver algorithm with tensor networks. First, a novel HHL in the qudits formalism, the generalization of qubits, is developed, and then its operations are transformed into an equivalent classical HHL, taking advantage of the non-unitary operations that they can apply. The main novelty of this proposal is to perform a classical simulation of the HHL as efficiently as possible to benchmark the algorithm steps according to its input parameters and the input matrix. The algorithm is applied to three classical simple simulation problems, comparing it with an exact inversion algorithm, and its performance is compared against an implementation of the original HHL simulated in the Qiskit framework, providing both codes. It is also applied to study the sensitivity of the HHL algorithm with respect to its hyperparameter values, reporting the existence of saturation points and maximal performance values. The results show that this approach can achieve a promising performance in computational efficiency to simulate the HHL process without quantum noise, providing a higher bound for its performance.
Quantum computation offers a promising alternative to classical computing methods in many areas of numerical science, with algorithms that make use of the unique ways in which quantum computers store and manipulate data often achieving dramatic improvements in performance over their classical counterparts. The potential efficiency of quantum computers is particularly important for numerical simulations, where the capabilities of classical computing systems are often insufficient for the analysis of real-world problems. In this work, we study problems involving the solution of matrix equations, for which there is limited research on efficient, general, and fully implemented quantum procedures. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary, which we have adopted from research in the area of quantum walks, has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware for any well-conditioned system. The procedure is most useful for sparse matrix equations, as it allows for the inverse of a matrix to be applied with O(sN log(N)) complexity, where N is the number of unknowns, and s is an upper bound on the number of nonzero elements in any row of the system matrix. This efficiency is competitive with the most sophisticated available classical solvers, and is independent of the structure of the input matrix. Hence, this procedure represents an important step towards the achievement of quantum advantage over classical algorithms for practical problems in numerical science.
Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.
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Abstract This work proposes a qudit formulation of the Harrow-Hassidim-Lloyd algorithm and a classical tensor-network representation of its finite-resolution spectral filter. The tensor network evaluates the joint successful branch directly: it recovers the conditioned solution state on an exact non-aliased phase grid and otherwise applies fμ,τf_{\mu,\tau}fμ,τ​, with the success probability treated separately. The construction enables controlled studies of spectral discretization and aliasing. It is evaluated on model linear systems against direct classical references and, for small instances, against a Qiskit statevector implementation using matched spectral parameters. The framework is intended for benchmarking and methodological analysis of finite-resolution HHL.
Solving systems of linear equations is important both in classical and quantum computing. Harrow-Hassidim-Lloyd algorithm (HHL algorithm), a quantum algorithm for solving systems of linear equations, has achieved exponentially improved performance compared with corresponding classical algorithm. We will show in this article that the HHL algorithm is actually a duality quantum algorithm with non- unitary transformation from initial to the final state. We present a detailed realization of the HHL algorithm in a duality quantum computing formalism that allows the construction of non- unitary operations. The divider structure, combinor structure, and the total quantum circuit for the HHL algorithm in the duality quantum computing formalism are given explicitly.
The quantum algorithm for linear systems of equations for is one of the main fundamental algorithms expected to provide a speedup over their classical counterparts. HHL is going to be one of the most useful subroutines for any quantum machine learning algorithm because almost all machine learning uses some form of a linear system of equations. This chapter describes the HHL algorithm, gives an example step by step using qiskit command HamiltonianGate , and indicates the constrains of the algorithm. The challenge is due to the fact that the application of matrix exponentials are computationally expensive. Gauss elimination requires and approximate solution for a sparse matrix via conjugate gradient descent.