Quantum Gradient Flow Algorithm for Symmetric Positive Definite Systems via Quantum Eigenvalue Transformation: Towards Quantum CAE

TL;DR

The paper addresses SPD linear systems arising in computational engineering, where large condition numbers hinder conventional quantum inverse methods. It proposes the Quantum Gradient Flow Algorithm (QGFA), which solves $K\mathbf{u}=\mathbf{f}$ by simulating the gradient-flow dynamics of the quadratic energy $\Pi(\mathbf{u})$ and approximating the necessary matrix functions with Quantum Signal Processing implemented via Quantum Eigenvalue Transformation and Linear Combination of Unitaries. Key contributions include a variational formulation of SPD solvers in the quantum setting, a soft-absolute regularization to enable stable QSP polynomial approximations of exponential-type functions, and a circuit design that achieves the gradient-flow solution with cost scaling primarily with the polynomial degree $d$ rather than the condition number. Demonstrations on 2D FEM problems show QGFA converges to classical FEM solutions with modest phase factors and well-chosen initial states, often outperforming QMIA in relative error, signaling potential as a preconditioned quantum solver and a stepping stone toward Quantum CAE for nonlinear and multiphysics simulations.

Abstract

In this study, we propose the Quantum Gradient Flow Algorithm (QGFA), a novel quantum algorithm for solving symmetric positive definite (SPD) linear systems based on the variational formulation and time-evolution dynamics. Conventional quantum linear solvers, such as the quantum matrix inverse algorithm (QMIA), focus on approximating the matrix inverse through quantum signal processing (QSP). However, QMIA suffers from a crucial drawback: its computational efficiency deteriorates as the condition number increases. In contrast, classical SPD linear solvers, such as the steepest descent and conjugate gradient methods, are known for their fast convergence, which stems from the variational optimization principle of SPD systems. Inspired by this, we develop QGFA, which obtains the solution vector through the gradient-flow process of the corresponding quadratic energy functional. To validate the proposed method, we apply QGFA to the displacement-based finite element method (FEM) for two-dimensional linear elastic problems under plane stress conditions. The algorithm demonstrates accurate convergence toward classical FEM solutions even with a moderate number of QSP phase factors. Compared with QMIA, QGFA achieves lower relative errors and faster convergence when initialized with suitable initial states, demonstrating its potential as an efficient preconditioned quantum linear solver. The proposed framework provides a physically interpretable connection between classical iterative solvers and quantum computational paradigms. These findings suggest that QGFA can serve as a foundation for future developments in Quantum Computer-Aided Engineering (Quantum CAE), including nonlinear and multiphysics simulations.

Figures (6)

• Figure 1: Schematic overview of the QGFA. (a) Relationship between the number of phase factors $p$ and the time--evolution parameter $t$ in the QGFA solution. For a fixed tolerance $\theta$, QGFA exhibits a characteristic $p$--$t$ curve. Along this curve, one can choose an optimal pair $(p,t)$ that attains the same accuracy as QMIA while requiring a significantly smaller number of phase factors $p$. (b) Flowchart of QGFA. The input consists of the SPD matrix $\bm{K}$, the load vector $\bm{f}$, and the initial vector $\bm{u}(0)$. Using the gradient--flow formulation, two matrix functions $g_{1}\!\left(s(\bm{K},\varepsilon)\right)$ and $\tilde{g}_{2}\!\left(s(\bm{K},\varepsilon)\right)$ are implemented through Quantum Eigenvalue Transformation (QET). The Linear Combination of Unitaries (LCU) framework then coherently combines the two QET blocks, producing the quantum state proportional to $\alpha\,\text{P}\left[g_{1}(s(\bm{K},\varepsilon))\right]_{d-1}\,\bm{u}(0) + \beta\,\text{P}\left[\tilde{g}_{2}(s(\bm{K},\varepsilon))\right]_{d-1}\,\bm{f}$, which corresponds to the gradient--flow solution.
• Figure 2: Quantum circuit for the QGFA. From top to bottom, the four wires represent: (i) the QSP ancilla qubit, where the Pauli--$X$ and Hadamard gates generate the even--odd interference needed to extract the imaginary part of the QET polynomial, (ii) the block-encoding ancilla initialized in $\ket{0^{m}}$ and used to implement the block--encoding $\bm{U}_{\bm{K}}$ of $\tilde{\bm{K}}$, (iii) the system register initialized in $\ket{0^{n}}$, on which the state-preparation unitaries $\bm{W}^{\mathrm{I}}$ and $\bm{W}^{\mathrm{II}}$ prepare $\ket{\bm{u}(0)}$ and $\ket{\bm{f}}$ and where the QET uniraty operators act, and (iv) the LCU control qubit, which is rotated by $\bm{S}_{\alpha,\beta}$ to coherently select between the two QET sequences with phase sets $\bm{\Phi}^{\mathrm{I}}$ and $\bm{\Phi}^{\mathrm{II}}$.
• Figure 3: Tensile simulation with a $3\times3$ (9-element) quadrilateral mesh solved by the finite element method (FEM) using the QGFA. (a) Geometry and discretization of the unit square ($x,y\in[0,1]$) with the tensile boundary condition; the remaining edges are traction-free. (b) Distribution of the relative error $\mathcal{R}$ between the QGFA solution and the classical FEM solution, obtained by varying the time-evolution parameter $t$ (vertical axis) and the number of phase factors $p$ (horizontal axis). For comparison, the reference relative error $\mathcal{R}^{\text{inv}}$ from the QMIA is also indicated.
• Figure 4: Cantilever beam bending simulation with 3 quadrilateral elements solved by the FEM using QGFA. (a) Geometry and boundary conditions of the cantilever beam: the left edge is fully fixed, and a vertical load is applied at the right edge. (b) Distribution of the relative error $\mathcal{R}$ between the QGFA solution and the classical FEM solution, obtained by varying the time-evolution parameter $t$ (vertical axis) and the number of phase factors $p$ (horizontal axis). For comparison, the reference relative error $\mathcal{R}^{\text{inv}}$ obtained from the QMIA is also indicated.
• Figure 5: Results of the QSP response in cantilever beam bending simulation with 3 quadrilateral elements ($\kappa = 37.018$). Response of (a) $g_1\left(s(x,\varepsilon)\right)$, (b) $\tilde{g_2}\left(s(x,\varepsilon)\right)$, and (c) $g_{\kappa,\epsilon}^{\text{inv}}(x)$.