arXiv subjects

Leigh Lapworth

Publications and source records attributed to Leigh Lapworth.

8 recordsLinked to original sources

arXiv2026

Approximate Quantum Linear Solvers for Hybrid CFD: End-to-End Analysis with a Chebyshev-LCU Approach

Quantum linear solvers are well studied as standalone quantum algorithms; however, in hybrid classical-quantum routines, their practical value must be evaluated at the level of the full non-linear application. A central issue is whether the approximation error of the quantum linear solver remains controlled once embedded in a full iterative workflow. We study this question in the context of a hybrid computational fluid dynamics (CFD) scheme. Through numerical simulations, we analyze how an approximate quantum linear solver affects the convergence of the overall CFD iteration. We show that convergence can be preserved for a non-exact quantum solver with only a moderate overhead in iteration count, provided that the high-frequency components of the linear system are resolved with sufficient accuracy. In addition, we develop an approximate qubitization-based solver (Cheb-LCU) that can reduce quantum resource requirements relative to a Quantum Singular Value Transformation (QSVT)-based solver while inducing only a small loss in convergence performance. This claim is demonstrated through explicit implementation and compilation of the quantum algorithms and by examining their impact on the convergence of the full CFD scheme. We find that our approximate approach reduces the required number of single-qubit rotations by over an order of magnitude relative to the QSVT-based solver, while requiring only a modest increase in CFD iteration count.

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arXiv2025

Quantum compilation framework for data loading

Efficient encoding of classical data into quantum circuits is a critical challenge that directly impacts the scalability of quantum algorithms. In this work, we present an automated compilation framework for resource-aware quantum data loading tailored to a given input vector and target error tolerance. By explicitly exploiting the trade-off between exact and approximate state preparation, our approach systematically partitions the total error budget between precision and approximation errors, thereby minimizing quantum resource costs. The framework supports a comprehensive suite of state-of-the-art methods, including multiplexer-based loaders, quantum read-only memory (QROM) constructions, sparse encodings, matrix product states (MPS), Fourier series loaders (FSL), and Walsh transform-based diagonal operators. We demonstrate the effectiveness of our framework across several applications, where it consistently uncovers non-obvious, resource-efficient strategies enabled by controlled approximation. In particular, we analyze a computational fluid dynamics workflow where the automated selection of MPS state preparation and Walsh transform-based encoding, combined with a novel Walsh-based measurement technique, leads to resource reductions of over four orders of magnitude compared to previous approaches. We also introduce two independent advances developed through the framework: a more efficient circuit for d-diagonal matrices, and an optimized block encoding for kinetic energy operators. Our results underscore the indispensable role of automated, approximation-aware compilation in making large-scale quantum algorithms feasible on resource-constrained hardware.

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arXiv2025

Preconditioned Block Encodings for Quantum Linear Systems

Quantum linear system solvers like the Quantum Singular Value Transformation (QSVT) require a block encoding of the system matrix $A$ within a unitary operator $U_A$. 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 introduce preamplified quantum multiplication to reduce subnormalisation, which is of independent interest. 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 1024x1024.

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arXiv2025

Measurement Schemes for Quantum Linear Equation Solvers

Solving Computational Fluid Dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion arxiv:1806.01838. However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing. In this work we propose a scheme for measuring the output of QSVT matrix inversion algorithms specifically for the CFD use case. We use a Quantum Signal Processing (QSP) based amplitude estimation algorithm arxiv:2207.08628 and show how it can be combined with the QSVT matrix inversion algorithm. We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction. We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focusing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.

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arXiv2024

Evaluation of block encoding for sparse matrix inversion using QSVT

Three block encoding methods are evaluated for solving linear systems of equations using QSVT (Quantum Singular Value Transformation). These are ARCSIN, FABLE and PREPARE-SELECT. The performance of the encoders is evaluated using a suite of 30 test cases including 1D, 2D and 3D Laplacians and 2D CFD matrices. A subset of cases is used to characterise how the degree of the polynomial approximation to $1/x$ influences the performance of QSVT. The results are used to guide the evaluation of QSVT as the linear solver in hybrid non-linear pressure correction and coupled implicit CFD solvers. The performance of QSVT is shown to be resilient to polynomial approximation errors. For both CFD solvers, error tolerances of $10^{-2}$ are more than sufficient in most cases and in some cases $10^{-1}$ is sufficient. The pressure correction solver allows subnormalised condition numbers, $κ_s$, as low as half the theoretical values to be used, reducing the number of phase factors needed. PREPARE-SELECT encoding relies on a unitary decomposition, e.g. Pauli strings, that has significant classical preprocessing costs. Both ARCSIN and FABLE have much lower costs, particularly for coupled solvers. However, their subnormalisation factors, which are based on the rank of the matrix, can be many times higher than PREPARE-SELECT leading to more phase factors being needed. For both the pressure correction and coupled CFD calculations, QSVT is more stable than previous HHL results due to the polynomial approximation errors only affecting long wavelength CFD errors. Given that lowering $κ_s$ increases the success probability, optimising the performance of QSVT within a CFD code is a function of the number QSVT phase factors, the number of non-linear iterations and the number of shots. Although phase factor files can be reused, the time taken to generate them impedes scaling QSVT to larger test cases.

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arXiv2024

L-QLES: Sparse Laplacian generator for evaluating Quantum Linear Equation Solvers

L-QLES is an open source python code for generating 1D, 2D and 3D Laplacian operators and associated Poisson equations and their classical solutions. Its goal is to provide quantum algorithm developers with a flexible test case framework where features of industrial applications can be incorporated without the need for end-user domain knowledge or reliance on inflexible one-off industry supplied matrix sets. A sample set of 1, 2, and 3 dimensional Laplacians are suggested and used to compare the performance of the Prepare-Select and FABLE block encoding techniques. Results show that large matrices are not needed to investigate industrial characteristics. A matrix with a condition number of 17,000 can be encoded using 13 qubits. L-QLES has also been produced to enable algorithm developers to investigate and optimise both the classical and quantum aspects of the inevitable hybrid nature of quantum linear equation solvers. Prepare-Select encoding that takes over an hour of classical preprocessing time to decompose a 4,096x4,096 matrix into Pauli strings can be can investigated using L-QLES matrices. Similarly, row-column query oracles that have success probabilities $\le 10^{-7}$ for the same matrix can be investigated.

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arXiv2022

Implicit Hybrid Quantum-Classical CFD Calculations using the HHL Algorithm

Implicit methods are attractive for hybrid quantum-classical CFD solvers as the flow equations are combined into a single coupled matrix that is solved on the quantum device, leaving only the CFD discretisation and matrix assembly on the classical device. In this paper, an implicit hybrid solver is investigated using emulated HHL circuits. The hybrid solutions are compared with classical solutions including full eigen-system decompositions. A thorough analysis is made of how the number of qubits in the HHL eigenvalue inversion circuit affect the CFD solver's convergence rates. Loss of precision in the minimum and maximum eigenvalues have different effects and are understood by relating the corresponding eigenvectors to error waves in the CFD solver. An iterative feed-forward mechanism is identified that allows loss of precision in the HHL circuit to amplify the associated error waves. These results will be relevant to early fault tolerant CFD applications where every (logical) qubit will count. The importance of good classical estimators for the minimum and maximum eigenvalues is also relevant to the calculation of condition number for Quantum Singular Value Transformation approaches to matrix inversion.

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arXiv2022

A Hybrid Quantum-Classical CFD Methodology with Benchmark HHL Solutions

There has been significant progress in the development of quantum algorithms for solving linear systems of equations with a growing body of applications to Computational Fluid Dynamics (CFD) and CFD-like problems. This work extends previous work by developing a non-linear hybrid quantum-classical CFD solver and using it to generate fully converged solutions. The hybrid solver uses the SIMPLE CFD algorithm, which is common in many industrial CFD codes, and applies it to the 2-dimensional lid driven cavity test case. A theme of this work is the classical processing time needed to prepare the quantum circuit with a focus on the decomposition of the CFD matrix into a linear combination of unitaries (LCU). CFD meshes with up to 65x65 nodes are considered with the largest producing a LCU containing 32,767 Pauli strings. A new method for rapidly re-computing the coefficients in a LCU is proposed, although this reduces, rather than eliminates, the classical scaling issues. The quantum linear equation solver uses the Harrow, Hassidim, Lloyd (HHL) algorithm via a state-vector emulator. Test matrices are sampled from the classical CFD solver to investigate the solution accuracy that can be achieved with HHL. For the smallest 5x5 and 9x9 CFD meshes, full non-linear hybrid CFD calculations are performed. The impacts of approximating the LCU and the varying the number of ancilla rotations in the eigenvalue inversion circuit are studied. Preliminary timing results indicate that the classical computer preparation time needed for a hybrid solver is just as important to the achievement of quantum advantage in CFD as the time on the quantum computer. The reported HHL solutions and LCU decompositions provide a benchmark for future research. The CFD test matrices used in this study are available upon request.

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