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Computer Science Research Notes

Variational Quantum Eigen-Decomposition Preconditioning Method for Solving Computational Mechanics Problems

Eunsik Choi, Jungin E. Kim, Yan Wang

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Recently, quantum computing emerged as a paradigm for solving systems of linear equations.However, large condition numbers significantly increase the computational complexity of quantum linear equation solvers.In this work, a new quantum preconditioning approach called variational quantum eigen-decomposition (VQED) is proposed, where the preconditioner is defined as a weighted projector onto the subspace spanned by the eigenvectors of a matrix.A variational quantum algorithm with projection-based deflation is performed to calculate eigenvalueeigenvector pairs for constructing the preconditioner.The proposed VQED method is used with quantum singular value transformation to solve linear systems for mechanics examples.It is demonstrated that the VQED can reduce the condition numbers to as low as 1.03, which is a significant improvement over the original condition number.

DOI
10.24132/csrn.2025-a63

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•Jan 6, 2025•1 citation
Recently, quantum computing emerged as a paradigm for solving systems of linear equations.However, large condition numbers significantly increase the computational complexity of quantum linear equation solvers.In this work, a new quantum preconditioning approach called variational quantum eigen-decomposition (VQED) is proposed, where the preconditioner is defined as a weighted projector onto the subspace spanned by the eigenvectors of a matrix.A variational quantum algorithm with projection-based deflation is performed to calculate eigenvalueeigenvector pairs for constructing the preconditioner.The proposed VQED method is used with quantum singular value transformation to solve linear systems for mechanics examples.It is demonstrated that the VQED can reduce the condition numbers to as low as 1.03, which is a significant improvement over the original condition number.
Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be inverted is a decisive parameter. A well known classical, and now quantum, remedy is to precondition the linear system Ax=bA x = bAx=b by premultiplying it by a matrix HHH in such a way that the condition number of HAHAHA is significantly smaller than the condition number of AAA. In this work, we focus on a family of preconditioners called domain decomposition. First, we prove that it is feasible to apply quantum domain decomposition. We provide upper bounds for the block-encoding parameters of the Poisson problem discretized by the finite element method and preconditioned by the two-level Additive Schwarz preconditioner (one of the most fundamental domain decomposition techniques). From these bounds, we deduce the complexity of the quantum linear system solver. Second, we focus on a particular choice of local solver within the domain decomposition preconditioner by applying recent work by [Deiml and Peterseim, \textit{Math. Comput.}, 2025] on the Bramble--Pasciak--Xu (BPX) preconditioner. Finally, we provide details on how the operators are implemented.
Abstract Quantum linear system solvers like the quantum singular value transformation (QSVT) require a block encoding of the system matrix A within a unitary operator UA . Unfortunately, block encoding often results in significant subnormalisation and increase in the matrix’s effective condition number κ, affecting the efficiency of solvers. Matrix preconditioning is a well-established classical technique to reduce κ by multiplying A by a preconditioner P. Here, we study quantum preconditioning for block encodings. We consider four preconditioners and two encoding approaches: (a) separately encoding A and its preconditioner P, followed by quantum multiplication, and (b) classically multiplying A and P before encoding the product in U PA . Their impact on subnormalisation factors and condition number κ are analysed using practical matrices from computational fluid dynamics (CFD). Our results show that (a) quantum multiplication introduces excessive subnormalisation factors, negating improvements in κ. We study preamplified quantum multiplication to reduce subnormalisation. Conversely, we see that (b) encoding of the classical product can significantly improve the effective condition number using the sparse approximate inverse preconditioner with infill. Further, we introduce a new matrix filtering technique that reduces the circuit depth without adversely affecting the matrix solution. We apply these methods to reduce the number of QSVT phase factors by a factor of 25 for an example CFD matrix of size 1024 × 1024.
Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
We apply preconditioning, which is widely used in classical solvers for linear systems Ax=bA\textbf{x}=\textbf{b}Ax=b, to the variational quantum linear solver. By utilizing incomplete LU factorization as a preconditioner for linear equations formed by 128×128128\times128128×128 random sparse matrices, we numerically demonstrate a notable reduction in the required ansatz depth, demonstrating that preconditioning is useful for quantum algorithms. This reduction in circuit depth is crucial to improving the efficiency and accuracy of Noisy Intermediate-Scale Quantum (NISQ) algorithms. Our findings suggest that combining classical computing techniques, such as preconditioning, with quantum algorithms can significantly enhance the performance of NISQ algorithms.
We consider the quantum linear solver for Ax=bAx=bAx=b with the circulant preconditioner CCC. The main technique is the singular value estimation (SVE) introduced in [Kerenidis and Prakash, Quantum recommendation system, in ITCS (2017)]. However, the SVE should be modified to solve the preconditioned linear system ${C}^{\ensuremath{-}1}Ax={C}^{\ensuremath{-}1}b$. Moreover, different from the preconditioned linear system considered in [Phys. Rev. Lett. 110, 250504 (2013)], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of CCC and ${C}^{\ensuremath{-}1}A$, as well as the Frobenius norm ${\ensuremath{\parallel}A\ensuremath{\parallel}}_{F}$.
This research paper explores the application of singular value decomposition (SVD) in quantum image processing (QIP), specifically focusing on the computation of eigenvalues using variational quantum algorithms. SVD is a powerful mathematical tool in image processing, used for tasks such as image compression, noise reduction, and feature extraction. In this study, we propose quantum variational singular value decomposition (QVSVD) based on the variational quantum deflation (VQD) algorithm to determine the eigenvalues that contributes in calculating the singular values of the image matrix and eventually doing the SVD. This facilitates the extraction of eigenvalues with exponential speedup on real hardware compared to classical methods. We detail the implementation of this quantum algorithm within the framework of QIP, highlighting its advantages in terms of computational efficiency and accuracy on a Fault-Tolerant Quantum Computing (FTQC). Furthermore, we present a comparative analysis of the quantum and the classical method of SVD, demonstrating the accuracy of the image data. The simulation results validate the theoretical advantages of using quantum algorithms for SVD in image processing. This work not only underscores the potential of quantum computing in enhancing image processing techniques but also sets the stage for future research in the field, exploring more complex image processing tasks and other quantum algorithms. Our findings suggest that quantum image processing can offer unprecedented capabilities, paving the way for advancements in various applications such as medical imaging, remote sensing, and multimedia processing.