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arXiv:2502.20908 · Quantum Science and Technology

Preconditioned block encodings for quantum linear systems

Leigh B. Lapworth, Christoph Sünderhauf

Published Oct 3, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/ae0f4b

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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.
Unstructured meshes are widely used in modelling and simulation in Computational Fluid Dynamics (CFD) due to their geometric flexibility. However, they require an efficient block encoding circuit describing the system. Here, we present a new block encoding methodology for unstructured mesh solvers based on decomposing the system matrix into a sum of 1-sparse matrices. This block encoding is engineered to reduce subnormalisation even in the absence of structure by hiding the contribution of large matrix elements behind others. We investigate three methods of 1-sparse decomposition. Our circuit constructions can optionally take advantage of limited matrix bandwidth and repeated values. For example we consider digitisation (reduced precision), where the number of multiplexed rotation gates is equal to the number of non-unique matrix entries. We evaluate our methodologies using 2D and 3D CFD matrices for unstructured aerofoil meshes, structured cavity, and duct meshes, with and without matrix preconditioning. Our results show that we are able to achieve a block encoding with a near constant subnormalisation factor as the number of matrix preconditioning infill levels increases. This significantly reduces the subnormalised condition number and, in one case, reduces the degree of the QSP polynomial approximation to 1/x1/x1/x by 100,000. Reduced precision digitisation is shown to become more effective as the number of preconditioning infill levels is increased and, in one case, reduces the number of multiplexed rotations from 2,785,576 to 546 whilst maintaining an infidelity of 2.6×10−52.6\times10^{-5}2.6×10−5 relative to the classical solution. The effect of digitisation on QSVT solutions is demonstrated using 2D and 3D unstructured mesh aerofoil cases. Due to its flexibility and automatic applicability, we believe our 1-sparse decomposition block encoding can offer benefits to other quantum applications.
Preconditioning is a fundamental technique for accelerating classical linear system solvers, and understanding when its benefits persist in quantum linear system (QLS) solvers is important for assessing the practical resource requirements of quantum linear algebra. In QLS algorithms, however, the potential advantage of preconditioning may be offset by the normalization overhead incurred by composing separate block-encodings of the system matrix and the preconditioner, as observed in recent work. This limitation leaves open whether additional algebraic structure can make preconditioning effective in quantum access models. Motivated by this question, we show that Pauli-structured representations of both the system matrix and the preconditioner allow the preconditioned operator to be accessed through regrouped Pauli expansions. In this setting, algebraic regrouping of Pauli products can reduce the Pauli coefficient weight of the preconditioned operator, thereby altering the normalization parameters relevant to quantum algorithms. We derive explicit size and coefficient-weight bounds for the regrouped Pauli representations, and we trace their consequences for both direct block-encoding constructions and randomized Pauli linear system solvers. These results identify when Pauli-structured preconditioning can reduce the effective complexity parameters of QLS algorithms, rather than merely improving the classical condition number. Numerical experiments on a finite-dimensional synthetic benchmark show reductions in norm-aware direct block-encoding diagnostics and in the randomized QLS per-sample depth proxy.
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.
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.
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.
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.