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ACM Transactions on Quantum Computing

Acc-VQLS: Accelerated Variational Quantum Linear Solver for VSC Simulation

Shengzhe Chang, Wenliang Nan, Jiyuan Liu, Yongming Tang, He Li

Published Jun 18, 2026• 0 citations
Publisher

Abstract

Voltage source converters (VSCs) play a critical role in modern power systems, where their simulation requires solving linear system problems and remains computationally expensive. Quantum linear system solvers, particularly the variational quantum linear solver (VQLS) tailored for noisy intermediate-scale quantum devices, offer a competitive paradigm with the potential for developing linear solvers with logarithmically scaled computational complexity. We propose Acc-VQLS, a domain-specific quantum algorithm and circuit co-optimization method that accelerates traditional VQLS for VSC simulation. We adopt constant vector-based variational quantum circuits for fast convergence with efficient initial state preparation and employ multi-core processing to parallelize the cost function evaluation and gradient computation. To demonstrate the potential applicability of Acc-VQLS in VSC operating scenarios, we employ the commonly used fault behavior model for VQLS-based VSC simulation. Quantitative experimental evaluations and comparisons across the tested benchmarks show that the proposed Acc-VQLS reduces iterations required for cost function convergence by an average of 82.5%, and our multi-core parallel computing strategy achieves a 53.9 × speedup on the AMD EPYC 9554 CPU. By incorporating a short-circuit fault in the VSC simulation, the proposed Acc-VQLS with error compensation achieves less than 10 − 9 in both the mean absolute error and the root mean square error.

DOI
10.1145/3821429

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Voltage source converters (VSCs) play a critical role in modern power systems, where their simulation requires solving linear system problems and remains computationally expensive. Quantum linear system solvers, particularly the variational quantum linear solver (VQLS) tailored for noisy intermediate-scale quantum devices, offer a competitive paradigm with the potential for developing linear solvers with logarithmically scaled computational complexity. We propose Acc-VQLS, a domain-specific quantum algorithm and circuit co-optimization method that accelerates traditional VQLS for VSC simulation. We adopt constant vector-based variational quantum circuits for fast convergence with efficient initial state preparation and employ multi-core processing to parallelize the cost function evaluation and gradient computation. To demonstrate the potential applicability of Acc-VQLS in VSC operating scenarios, we employ the commonly used fault behavior model for VQLS-based VSC simulation. Quantitative experimental evaluations and comparisons across the tested benchmarks show that the proposed Acc-VQLS reduces iterations required for cost function convergence by an average of 82.5%, and our multi-core parallel computing strategy achieves a 53.9 × speedup on the AMD EPYC 9554 CPU. By incorporating a short-circuit fault in the VSC simulation, the proposed Acc-VQLS with error compensation achieves less than 10 − 9 in both the mean absolute error and the root mean square error.
The Variational Quantum Linear Solver (VQLS), a hybrid quantum-classical algorithm for solving linear systems, faces a practical scalability bottleneck: the Linear Combination of Unitaries (LCU) decomposition requires O(L2)O(L^2)O(L2) circuit evaluations per optimizer iteration, where LLL can grow to 4n4^n4n in the worst case for an nnn-qubit system. We address this computational bottleneck through two complementary strategies. First, we present a distributed VQLS (D-VQLS) framework (https://code.ornl.gov/olcf-qcfd/DVQLS.git), built on NVIDIA CUDA-Q, that enables asynchronous, scalable distribution of the O(L2)O(L^2)O(L2) cost evaluations. Second, a fast Walsh--Hadamard transform (FWHT)-based Pauli decomposition with coefficient-amplitude pruning threshold τ=0.01τ=0.01τ=0.01 curbs LCU growth for the structured Toeplitz family, reducing LLL from O(2n)O(2^n)O(2n) to 64 for n>6n>6n>6 and compressing the circuit complexity per optimizer iteration from O(n4n)O(n4^n)O(n4n) to O(n)O(n)O(n). We derive the exact top-LLL Frobenius error and connect it to worst-case solution error. For a 10-qubit tridiagonal Toeplitz system, the L=64L=64L=64 pruning yields a 256×256\times256× reduction---from 23 million to 90k circuits per optimizer iteration. The D-VQLS framework is validated on the NERSC Perlmutter supercomputer using multi-node, multi-GPU ideal state-vector simulations, achieving over 99.99%99.99\%99.99% fidelity against classical solutions on tridiagonal Toeplitz and Hele--Shaw flow benchmarks, with near-ideal strong scaling up to 24 GPUs and 95.3%95.3\%95.3% weak scaling efficiency at 96 GPUs processing more than 360k circuits per optimizer iteration (from larger-LLL pruning) for the 10-qubit system. Systematic profiling identifies the optimal resource allocation for distributed quantum circuit workloads, yielding a 2.52×2.52\times2.52× speedup for the configurations studied.
Variational Quantum Algorithms (VQAs) have emerged as promising methods for tackling complex problems on near-term quantum devices. Among these algorithms, the Variational Quantum Linear Solver (VQLS) addresses linear systems of the form Ax=bAx=bAx=b, aiming to prepare a quantum state ∣x⟩|x\rangle∣x⟩ such that A∣x⟩A|x\rangleA∣x⟩ is proportional to the quantum state corresponding to bbb. A key advantage of VQLS is its use of amplitude encoding, which requires a number of qubits that scales logarithmically with the linear system size. However, the existing literature has primarily focused on linear systems of limited size or with a specific structure. In this study, we extend the application of VQLS to more general and larger problem instances, including problems where state preparation is non-trivial and problems within the real domain of fluid dynamics. Our investigation reveals some critical challenges inherent to VQLS, including the need for a sufficiently expressive ansatz, the large number of circuit executions required to estimate the cost function, and the high gate count in the circuits in the most general setting. Our analysis highlights the obstacles that need to be addressed for a broader application of VQLS and concludes that further research is necessary to fully leverage the algorithm's capabilities in addressing real-world problems.
This work develops simulation methods that enable the application of the variational quantum linear solver (VQLS) to simulate quantum transport in nanoscale semiconductor devices. Most previous work on VQLS applications in semiconductor device simulations focuses on solving the Poisson equation, where the coefficient matrix of the sparse linear system is real and symmetric. Solving the quantum transport equation, however, leads to coefficient matrices that are complex and non-symmetric. This work addresses the challenges of applying VQLS to quantum transport simulations. We propose new forms of cost functions to solve complex and non-symmetric linear systems with faster computing speed. We further develop efficient decomposition methods for cost function evaluation, which target reducing the quantum circuit complexity and improving noise robustness when solving the quantum transport equation using the nonequilibrium Green's function method. While classical computation faces the challenge of the “curse of dimensionality” as the spatial-energy numerical grid dimensions grow, the proposed quantum-computing-based method scales logarithmically with the grid size, which offers a promising opportunity for addressing the computational challenges of solving quantum transport in semiconductor devices.
Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare | x ⟩ such that A | x ⟩ ∝ | b ⟩ . We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that C ⩾ ϵ 2 / κ 2 , where C is the VQLS cost function and κ is the condition number of A . We present efficient quantum circuits to estimate C , while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of 1024 × 1024 . Finally, we numerically solve non-trivial problems of size up to 2 50 × 2 50 . For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in ϵ , κ , and the system size N .
As quantum computing technology continues to develop, quantum algorithms have the potential to outperform classical algorithms in future for certain computational tasks. One of these potential applications of quantum computing is numerically solving differential equations and integral equations. In this thesis the variational quantum linear solver algorithm (VQLS) proposed by Bravo-Prieto et al. is adapted to solve the Laplace and Poisson equations, ordinary differential equation initial value problems (IVPs) with both constant and function coefficients, and a Fredholm integral equation of the second type. As this thesis demonstrates, variational quantum algorithms for solving differential equations and integral equations such as VQLS are highly promising for use on Noisy Intermediate-Scale Quantum (NISQ) devices. The quantum circuit components of the VQLS algorithm were simulated using IBM Quantum's statevector simulator with the optimized solution circuits implemented on the IBM Quantum Guadalupe device (in most cases). A discussion of errors and convergence is also included. The application of VQLS to these differential equations and integral equations is novel and, to my knowledge, work using VQLS to solve the Poisson equation has only recently been published once by authors Liu et al. which was conducted concurrently and independently of this research.
The potential of quantum computing for scientific and industrial breakthroughs is immense, however, we are still in the Noisy Intermediate-Scale Quantum (NISQ) era, where the currently available quantum devices contain small numbers of qubits, are very sensitive to environmental conditions and prone to quantum decoherence. Even so, existing NISQ computers have already been shown to outperform conventional computers on specific problems and the key question is how to make use of today’s NISQ devices to achieve quantum advantage in the field of computational science and engineering (CSE). In this direction, this work proposes a hybrid computing formulation by combining quantum computing with machine learning for accelerating the solution of parameterized linear systems in NISQ devices. In particular, it focuses on the Variational Quantum Linear Solver (VQLS), which is a hybrid quantum- classical algorithm for solving linear systems of equations. VQLS employs a short-depth quantum circuit to efficiently evaluate a cost function related to the system solution. The algorithm’s circuit consists of a quantum gate sequence (unitary operators) that involves a set of tunable parameters. Then, well-established classical optimizers are utilized to tune the parameters of the sequence in order to minimize the cost function, which is equivalent to finding the system solution at an acceptable level of accuracy. In this work, it is demonstrated that machine learning tools such as feed-forward neural networks and nearest-neighbor interpolation techniques can be successfully employed to accelerate the convergence of the VQLS algorithm towards the optimal values for the circuit parameters when applied to parameterized linear systems that need to be solved for multiple parameter instances.