We address the problem of solving a system of linear equations via the Quantum Singular Value Transformation (QSVT). One drawback of the QSVT algorithm is that it requires huge quantum resources if we want to achieve an acceptable accuracy. To reduce the quantum cost, we propose a hybrid quantum-classical algorithm that improves the accuracy and reduces the cost of the QSVT by adding iterative refinement in mixed-precision A first quantum solution is computed using the QSVT, in low precision, and then refined in higher precision until we get a satisfactory accuracy. For this solver, we present an error and complexity analysis, and first experiments using the quantum software stack myQLM.
We address the problem of solving a system of linear equations via the Quantum Singular Value Transformation (QSVT). One drawback of the QSVT algorithm is that it requires huge quantum resources if we want to achieve an acceptable accuracy. To reduce the quantum cost, we propose a hybrid quantum-classical algorithm that improves the accuracy and reduces the cost of the QSVT by adding iterative refinement in mixed-precision A first quantum solution is computed using the QSVT, in low precision, and then refined in higher precision until we get a satisfactory accuracy. For this solver, we present an error and complexity analysis, and first experiments using the quantum software stack myQLM.
Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).
Systems of linear equations are ubiquitous across science, engineering, machine learning, and even finance. While classical methods can be prohibitively slow for large-scale problems, quantum linear systems algorithms offer the potential for exponential speedup in certain parameter regimes. However, a significant gap persists between this theoretical promise and practical implementation, as the advantages are often obscured by the substantial quantum resources and high sensitivity to noise inherent in current quantum hardware. One way to bridge this gap is through the use of Iterative Refinement, a classical post-processing scheme that can exponentially improve the accuracy to which a linear system of equations can be solved using low-precision arithmetic. In the context of quantum linear systems algorithms, such as the HHL algorithm proposed by Harrow, Hassidim, and Loyd, Iterative Refinement can greatly reduce the quantum resources required to calculate an accurate solution in terms of tomography cost, circuit volume, and fault-tolerant overhead. Here, we compute and benchmark highly precise solutions to linear systems of equations of up to eight variables by running HHL with Iterative Refinement on NISQ quantum computers. We also present our open-source implementation, emphasizing that our circuit is not tailored to specific problem instances, as most available implementations are.
Solving linear systems of equations is central to many engineering and
scientific fields. Several quantum algorithms have been proposed for linear
systems, where the goal is to prepare ∣x⟩ such that A∣x⟩∝∣b⟩. While these algorithms are promising, the time horizon for their
implementation is long due to the required quantum circuit depth. In this work,
we propose a variational hybrid quantum-classical algorithm for solving linear
systems, with the aim of reducing the circuit depth and doing much of the
computation classically. We propose a cost function based on the overlap
between ∣b⟩ and A∣x⟩, and we derive an operational meaning for
this cost in terms of the solution precision ϵ. We also introduce a
quantum circuit to estimate this cost, while showing that this cost cannot be
efficiently estimated classically. Using Rigetti's quantum computer, we
successfully implement our algorithm up to a problem size of 32×32.
Furthermore, we numerically find that the complexity of our algorithm scales
efficiently in both 1/ϵ and κ, with κ the condition
number of A. Our algorithm provides a heuristic for quantum linear systems
that could make this application more near term.
High-Performance Computing (HPC) is today the main tool for large-scale scientific computing, but performance gains are increasingly limited by architectural and energy constraints. At the same time, quantum computing is an emerging paradigm, still under development, but already promising for accelerating certain computational kernels. A central challenge is therefore to understand how to efficiently integrate quantum processors within existing HPC infrastructures, rather than treating them as isolated accelerators. The objective of this thesis is to explore hybrid quantum-classical computing, both from the algorithmic perspective and in terms of programming models. We investigate how to coherently combine classical and quantum resources within hybrid workflows, leveraging the respective strengths of each paradigm. As a primary use case, we focus on solving linear systems of equations, a fundamental building block of scientific computing and HPC. First, we address the challenge of encoding classical data into quantum representations, which remains a major bottleneck for hybrid algorithms. We propose an improved preprocessing method for matrix encoding into unitaries using a Pauli decomposition, reducing the associated computational cost. Next, we introduce a hybrid solver based on the Quantum Singular Value Transformation (QSVT). Inspired by classical numerical linear algebra, we propose a mixed-precision iterative refinement approach, combining low-precision quantum solves with high-precision classical corrections. A detailed analysis of convergence, errors, and computational cost shows that hybridization can reduce the quantum resources required to reach a given precision.Finally, we address programming and implementation aspects through the design of a hybrid quantum-classical programming framework, Q-Pragma, aimed at facilitating the integration of quantum acceleration into existing HPC environments. Overall, this work contributes to the development of more practical and usable hybrid quantum-HPC codes, bridging the gap between theoretical algorithms and real-world high-performance applications.
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.
We propose an iterative improvement method for the Harrow-Hassidim-Lloyd (HHL) algorithm to solve a linear system of equations. This is a quantum-classical hybrid algorithm. The accuracy is essential to solve the linear system of equations. However, the accuracy of the HHL algorithm is limited by the number of quantum bits used to express the eigenvalues of the matrix. Our iterative method improves the accuracy of the HHL solutions, and gives higher accuracy which surpasses the accuracy limited by the number of quantum bits. In practical HHL algorithm, a huge number of measurements is required to obtain good accuracy, even if we provide a sufficient number of quantum bits for the eigenvalue expression, since the solution is statistically processed from the measurements. Our improved iterative method can reduce the number of measurements. Moreover, the sign information for each eigenstate of the solution is lost once the measurement is made, although the sign is significant. Therefore, the naïve iterative method of the HHL algorithm may slow down, especially, when the solution includes wrong signs. In this paper, we propose and evaluate an improved iterative method for the HHL algorithm that is robust against the sign information loss, in terms of the number of iterations and the computational accuracy.