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Telecommunication and information technologies

A VARIATIONAL QUANTUM ALGORITHM FOR SOLVING LINEAR SYSTEMS FOR DECISION SUPPORT SYSTEMS

Максим Майборода

Published Sep 30, 2026• 0 citations
Publisher

Abstract

A hybrid quantum–classical approach to solving the load balancing problem in real-time computer systems is proposed, with a focus on reliability assessment and control. The approach is based on the use of a Variational Quantum Linear Solver (VQLS) to process large sparse linear systems arising from the spatio-temporal discretization of load distribution models and task flows among computational nodes. An algorithmic scheme is presented for transforming an engineering model of a real-time system into a quantum-realizable problem formulation. Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. This scheme includes decomposing the system matrix into a linear combination of Pauli operators, preparing a quantum state corresponding to the right-hand side, and optimizing a variational ansatz. It is shown that the resulting quantum solution state is most effective when used not for full reconstruction of the load vector, but for direct estimation of aggregated and local metrics, such as peak load, slack to deadline violations, and integral stability indicators, which are directly related to system failures and performance degradation. The proposed approach is considered as a tool for quantum benchmarking of classical load balancing algorithms, as well as a potential foundation for building reliability management systems for computer systems operating in a near–real-time regime. Key words. Reliability of computer systems, Bayesian neural networks, ensemble methods, artificial intelligence, hybrid models, risk assessment, fault tolerance, uncertainty modeling, failure prediction, software security, system resilience, neural networks, fault trees, machine learning, quantum systems, quantum algorithms, quantum machine learning, hybrid models.

DOI
10.31673/2412-4338.2026.039705

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A hybrid quantum–classical approach to solving the load balancing problem in real-time computer systems is proposed, with a focus on reliability assessment and control. The approach is based on the use of a Variational Quantum Linear Solver (VQLS) to process large sparse linear systems arising from the spatio-temporal discretization of load distribution models and task flows among computational nodes. An algorithmic scheme is presented for transforming an engineering model of a real-time system into a quantum-realizable problem formulation. Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. This scheme includes decomposing the system matrix into a linear combination of Pauli operators, preparing a quantum state corresponding to the right-hand side, and optimizing a variational ansatz. It is shown that the resulting quantum solution state is most effective when used not for full reconstruction of the load vector, but for direct estimation of aggregated and local metrics, such as peak load, slack to deadline violations, and integral stability indicators, which are directly related to system failures and performance degradation. The proposed approach is considered as a tool for quantum benchmarking of classical load balancing algorithms, as well as a potential foundation for building reliability management systems for computer systems operating in a near–real-time regime. Key words. Reliability of computer systems, Bayesian neural networks, ensemble methods, artificial intelligence, hybrid models, risk assessment, fault tolerance, uncertainty modeling, failure prediction, software security, system resilience, neural networks, fault trees, machine learning, quantum systems, quantum algorithms, quantum machine learning, hybrid models.
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 .
Gaussian processes are widely known for their ability to provide probabilistic predictions in supervised machine learning models. Their non-parametric nature and flexibility make them particularly effective for regression tasks. However, training a Gaussian process model using standard methods requires matrix inversions with a cubic time complexity, which poses significant computational challenges for inference on larger datasets. Quantum algorithms, such as the HHL algorithm, have been proposed as solutions that overcome the need for classical matrix inversions by efficiently solving linear systems of equations using quantum computers. However, to gain a computational advantage over classical algorithms, these algorithms require fault-tolerant quantum computers with a large number of qubits, which are not yet available. The Variational Quantum Linear Solver is a hybrid quantum-classical algorithm that solves linear systems of equations by optimizing the parameters of a variational quantum circuit using a classical computer. This method is especially suitable for noisy intermediate-scale quantum computers, as it does not require many qubits. It can be used to compute the posterior distribution of a Gaussian process by reformulating the matrix inversion into a set of linear systems of equations. We empirically demonstrate that using the Variational Quantum Linear Solver to perform inference for Gaussian process regression delivers regression quality comparable to that of classical methods.
Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that they need to tackle. In fact, these algorithms have been run on real hardware only for problems in quantum physics, such as Hamiltonians of a few qubits systems. These problems are particularly favorable for quantum hardware, since their matrices are the sum of just a few unitary terms and since only shallow quantum circuits are required to estimate the cost function. However, for many interesting problems in linear algebra, it appears far less trivial to find an efficient decomposition and to trade it off with the depth of the cost quantum circuits. A first simple yet interesting instance to consider are tridiagonal systems of equations. These arise, for instance, in the discretization of one-dimensional finite element analyses. This work presents a method to solve a class of tridiagonal systems of equations with the variational quantum linear solver (VQLS), a recently proposed variational hybrid algorithm for solving linear systems. In particular, we present a new decomposition for this class of matrices based on both Pauli strings and multi--qubit gates, resulting in less terms than those obtained by just using Pauli gates. Based on this decomposition, we discuss the tradeoff between the number of terms and the near-term implementability of the quantum circuits. Furthermore, we present the first simulated and real-hardware results obtained by solving tridiagonal linear systems with VQLS, using the decomposition proposed.
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⟩|x\rangle∣x⟩ such that A∣x⟩∝∣b⟩A|x\rangle \propto |b\rangleA∣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⟩|b\rangle∣b⟩ and A∣x⟩A|x\rangleA∣x⟩, and we derive an operational meaning for this cost in terms of the solution precision ϵ\epsilonϵ. 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 32×3232 \times 3232×32. Furthermore, we numerically find that the complexity of our algorithm scales efficiently in both 1/ϵ1/\epsilon1/ϵ and κ\kappaκ, with κ\kappaκ the condition number of AAA. Our algorithm provides a heuristic for quantum linear systems that could make this application more near term.
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.
Abstract Finding solutions to systems of linear equations is a common problem in many areas of science and engineering, with much potential for a speed up on quantum devices. While the Harrow–Hassidim–Lloyd (HHL) quantum algorithm yields up to an exponential speed up over classical algorithms in some cases, it requires a fault-tolerant quantum computer, which is unlikely to be available in the near-term. Thus, attention has turned to the investigation of quantum algorithms for noisy intermediate-scale quantum (NISQ) devices where several near-term approaches to solving systems of linear equations have been proposed. This paper focuses on the Variational Quantum Linear Solvers (VQLS), and other closely related methods and adaptions. Several contributions are made in this paper, which include: the first application of the Evolutionary Ansatz to the VQLS (EAVQLS), the first implementation of the Logical Ansatz VQLS (LAVQLS), based on the Classical Combination of Quantum States (CQS) method, a proof of principle demonstration of the CQS method on real quantum hardware and a method for the implementation of the Adiabatic Ansatz on the VQLS (AAVQLS). These approaches are implemented and contrasted. The CQS method is run with moderate success on a real quantum device. The EAVQLS and AAVQLS show promise as possible improvements to the standard VQLS algorithm once refined.