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International Conference on Simulation of Semiconductor Processes and Devices

Study of Using Variational Quantum Linear Solver for Solving Poisson Equation

Nithin Reddy Govindugari, Hiu Yung Wong

Published Sep 24, 2024• 5 citations
Publisher

Abstract

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.

DOI
10.1109/sispad62626.2024.10732984

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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.
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.
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.
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.
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.
We present a variational quantum linear solver (VQLS) for the Poisson equation built on an exact block encoding of the discrete Laplacian, and demonstrate its performance on physically motivated benchmarks. Unlike LCU-based VQLS where the number of distinct circuits required per cost-function evaluation is O(L2)\mathcal{O}(L^2)O(L2), where LLL is the number of terms in the LCU decomposition of the discrete Laplacian operator, this approach requires only a single circuit for cost evaluation. We further empirically demonstrate that the choice of classical optimizer materially affects where the variational optimization ceases to make progress. The solver is benchmarked on three problems: a Poisson equation with sinusoidal forcing and a steady-state heat conduction problem with a localized Gaussian source, both with Dirichlet boundaries, and the pressure-Poisson equation of a two-dimensional lid-driven cavity flow, in which the solver is invoked once per time step under Neumann boundary conditions.
Solving differential equations is one of the most promising applications of quantum computing. Recently we proposed an efficient quantum algorithm for solving one-dimensional Poisson equation avoiding the need to perform quantum arithmetic or Hamiltonian simulation. In this paper, we further develop this algorithm to make it closer to the real application on the noisy intermediate-scale quantum (NISQ) devices. To this end, we first optimize the quantum 1D-Poisson solver by developing a new way of performing the sine transformation. The circuit depth for implementing the sine transform is reduced from n2 to n without increasing the total qubit cost of the whole circuit, which is achieved by neatly reusing the additional ancillary quits. Then, we analyse the effect of common noise existing in the real quantum devices on our algorithm using the IBM Qiskit toolkit. We find that the phase damping noise has little effect on our algorithm, while the bit flip noise has the greatest impact. In addition, threshold errors of the quantum gates are obtained to make the fidelity of the circuit output being greater than 90%. The results of noise analysis will provide a good guidance for the subsequent work of error mitigation and error correction for our algorithm. The noise-analysis method developed in this work can be used for other algorithms to be executed on the NISQ devices.