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NISQ-Compatible Variational Quantum Solver for Lyapunov Equations

Amir Hossein Salehi Shayegan

Published Jun 12, 2026• 0 citations
Publisher

Abstract

Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation $A^T X + X A = -Q$. Using the Kronecker product formulation, the matrix equation is transformed into the linear system $H z = b$, where $H = I_n \otimes A^T + A^T \otimes I_n$ and $z = \mathrm{vec}(X)$. VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to $25 \times 25$ demonstrate residual norms as low as $10^{-9}$, with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.

DOI
10.21203/rs.3.rs-9849163/v1

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Abstract \noindent This paper applies the Variational Quantum Linear Solver (VQLS) to the continuous-time Lyapunov equation ATX+XA=−QA^T X + X A = -QATX+XA=−Q. Using the Kronecker product formulation, the matrix equation is transformed into the linear system Hz=bH z = bHz=b, where H=In⊗AT+AT⊗InH = I_n \otimes A^T + A^T \otimes I_nH=In​⊗AT+AT⊗In​ and z=vec(X)z = \mathrm{vec}(X)z=vec(X). VQLS approximates the solution via parameterized quantum circuits optimized in a hybrid quantum-classical loop. Numerical experiments on four control system benchmarks with Kronecker dimensions up to 25×2525 \times 2525×25 demonstrate residual norms as low as 10−910^{-9}10−9, with logarithmic qubit scaling (2-5 qubits) and shallow circuit depths (6-14 layers) suitable for near-term quantum devices. Compared to the HHL algorithm, VQLS offers reduced circuit complexity and qubit overhead, making it practical for noisy intermediate-scale quantum (NISQ) hardware. This paper demonstrates the potential of variational quantum algorithms for solving Lyapunov equations in control and dynamical systems applications.
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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.
Time-fractional diffusion equations have emerged as powerful models for describing anomalous transport phenomena in physics, biology and engineering. To address the computational challenges arising from their non-local operators, we employ the WEB-spline finite element method, which provides a flexible and accurate discretization framework. The resulting linear system of equations are then explored in the context of quantum computing. Specifically, we investigate three prominent quantum linear solvers: the variational quantum linear solver (VQLS), the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum annealing (QA). VQLS leverages hybrid variational techniques and shallow circuits, making it well-suited for noisy intermediate-scale quantum (NISQ) devices, while HHL offers a theoretically exponential speedup for sparse systems but requires deep fault-tolerant circuits. QA, in contrast, reformulates the problem into a quadratic unconstrained binary optimization (QUBO) instance, enabling approximate solutions through energy minimization on specialized hardware. We present a comparative analysis in terms of circuit depth, noise resilience, scalability and solution extraction, including the role of quantum state tomography in reconstructing classical information. Numerical experiments on a time-fractional diffusion problem highlight the complementary strengths and limitations of each method. This study bridges advanced numerical discretization with emerging quantum algorithms, providing insights into the feasibility and future potential of quantum-enhanced solvers for fractional partial differential equations.
Quantum power system state estimation (QPSSE) offers an inspiring direction for tackling the challenge of state estimation through quantum computing. Nevertheless, the current bottlenecks originate from the scarcity of practical and scalable QPSSE methodologies in the noisy intermediate-scale quantum (NISQ) era. This paper devises a NISQ-QPSSE algorithm that facilitates state estimation on real NISQ devices. Our new contributions include: (1) A variational quantum circuit (VQC)-based QPSSE formulation that empowers QPSSE analysis utilizing shallow-depth quantum circuits; (2) A variational quantum linear solver (VQLS)-based QPSSE solver integrating QPSSE iterations with VQC optimization; (3) An advanced NISQ-compatible QPSSE methodology for tackling the measurement and coefficient matrix issues on real quantum computers; (4) A noise-resilient method to alleviate the detrimental effects of noise disturbances. The encouraging test results on the simulator and real-scale systems affirm the precision, universality, and scalability of our QPSSE algorithm and demonstrate the vast potential of QPSSE in the thriving NISQ era.