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IEEE Transactions on Electron Devices

Variational Quantum Algorithm for Solving Quantum Transport Equation in Semiconductor Device

Qimao Yang, Jing Hua Guo

Published Apr 18, 2025• 1 citation
Publisher

Abstract

As semiconductor feature sizes continue to shrink, accurate and efficient simulations of quantum transport become increasingly critical in device design and manufacturing. The nonequilibrium Green’s function (NEGF) formalism is a widely used method for simulating quantum transport in semiconductor devices, but it is computationally demanding. Quantum computing offers a promising solution, in this work, we pioneer the application of the variational quantum linear solver (VQLS) to the NEGF problem, addressing the challenges associated with handling complex numbers inherent in quantum transport equations. We introduce a new cost function tailored to this framework, demonstrating improved performance over existing approaches. Furthermore, we show that VQLS can efficiently parallelize the computation across different energy levels, significantly reducing computational costs. Our results highlight the potential of variational quantum algorithms (VQAs) in enhancing the scalability and efficiency of quantum transport simulations.

DOI
10.1109/ted.2025.3558757

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As semiconductor feature sizes continue to shrink, accurate and efficient simulations of quantum transport become increasingly critical in device design and manufacturing. The nonequilibrium Green’s function (NEGF) formalism is a widely used method for simulating quantum transport in semiconductor devices, but it is computationally demanding. Quantum computing offers a promising solution, in this work, we pioneer the application of the variational quantum linear solver (VQLS) to the NEGF problem, addressing the challenges associated with handling complex numbers inherent in quantum transport equations. We introduce a new cost function tailored to this framework, demonstrating improved performance over existing approaches. Furthermore, we show that VQLS can efficiently parallelize the computation across different energy levels, significantly reducing computational costs. Our results highlight the potential of variational quantum algorithms (VQAs) in enhancing the scalability and efficiency of quantum transport simulations.
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.
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.
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.
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.
Based on a photonic computing platform, we experimentally validate a generalized variational quantum linear solver (VQLS) by systematically solving four-dimensional linear equation systems across different fields. In the complex field, in addition to solving non-singular systems that admit a unique solution, we investigate ill-conditioned problems arising from singularity--an issue frequently encountered in practical applications. To tackle these challenges, we introduce perturbation terms, a treatment inspired by Tikhonov regularization, and develop an algorithm capable of handling a wide range of systems. Furthermore, we extend the VQLS to the finite field F2 by redesigning the cost function to incorporate modulo 2 and imposing several constraints on the solution vector. This modulo 2 VQLS is inherently free from singularity. It is adapt to stabilizer coding theory and may find applications in areas such as decoders and the design of quantum gate sequences. Therefore, our work demonstrates the practical potential of VQLS in quantum computing, providing a solid experimental foundation and methodological guidance for its real-world applications.