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European Radar Conference (EuRAD)

Radar Cross Section Estimation using the Variational Quantum Linear Solver Algorithm

Thibaut Pellerin, Robin van Gaalen, Ronny I. A. Harmanny

Published Sep 25, 2024• 0 citations
Publisher

Abstract

Electromagnetic simulations are computationally intensive yet paramount in radar technologies. Finite Element Methods (FEM) require expensive linear algebra algorithms, with a polynomial complexity in the mesh size. Variational Quantum Linear Solver (VQLS) is a quantum algorithm that might be able to offer an exponential speed-up for such tasks. Despite this quantum algorithm not being relevant for every application, it might be a promising approach to tackle the problem of radar cross-section computation efficiently. In this paper, we explain how to apply such a quantum algorithm to this radar-related problem, with a focus on the statistical estimation part of VQLS.

DOI
10.23919/eurad61604.2024.10734862

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Electromagnetic simulations are computationally intensive yet paramount in radar technologies. Finite Element Methods (FEM) require expensive linear algebra algorithms, with a polynomial complexity in the mesh size. Variational Quantum Linear Solver (VQLS) is a quantum algorithm that might be able to offer an exponential speed-up for such tasks. Despite this quantum algorithm not being relevant for every application, it might be a promising approach to tackle the problem of radar cross-section computation efficiently. In this paper, we explain how to apply such a quantum algorithm to this radar-related problem, with a focus on the statistical estimation part of VQLS.
Many problems cannot be solved analytically, thus numerical methods are required. It turns out that many problems are too complex and therefore beyond the reach of current algorithms and computers. Quantum computation might offer fundamental improvements in solving these problems. However, it is a difficult task to estimate the computational power of quantum algorithms. The aim of this thesis is to find applications related to aerospace especially well suited for quantum computation and estimate their possible quantum advantage. In particular we focus on the calculation of radar cross sections, which has been suggested as a promising candidate for quantum speedup. It turns out that the main problem lies in the input of classical data. Hence, the reimplementation of classical finite element method is needed, requiring a substantial amount of quantum resources. Asymptotically a quantum speedup is achievable but it is far beyond the reach of today’s capabilities to successfully implement the quantum algorithm for this application. Additionally, a serious prediction regarding the timeframe is not foreseeable yet. Substantial breakthroughs are necessary to make quantum computation compete against classical computation in calculating radar cross sections using the finite element method.
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.
Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.
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 .
Quantum computing is promising in speeding up the system of linear equations (SLE) solving process. However, its performance is limited by noise. The variational quantum linear solver (VQLS) algorithm is expected to be more resilient to noise than gate-based quantum computing algorithms. This is because error correction is not available yet and VQLS is based on cost function minimization. In this paper, the gate insulator Poisson equation is solved using VQLS. The results are compared to technology computer-aided design (TCAD) results and gate-based quantum algorithm results. We show that, even without error-free qubits, the IBM-Q quantum computer hardware can solve a 2-variable SLE with high fidelity. We further demonstrate that, through VQLS simulation, an 8-variable SLE can be solved with fidelity as high as 0.96.
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.