Solving Helmholtz problems with finite elements on a quantum annealer

TL;DR

Solving 1D Helmholtz problems via finite elements is reframed as a generalized eigenvalue problem (gEVP) and then as a QUBO solvable by adaptive quantum annealing methods. The authors compare adaptive quantum annealer eigensolver (AQAE) and adaptive classical annealer eigensolver (ACAE) across homogeneous and non-homogeneous cases, showing convergence is sensitive to the system condition number and to integrated control errors (ICE) in hardware. They derive lower bounds on annealing time and discuss scaling, concluding that a quantum advantage is not guaranteed in current regimes and may depend on problem frequency and dimensionality, with potential gains from Hamiltonian engineering and hybrid preconditioning. The results illuminate hardware limitations, guide parameter choices (e.g., discretization density D), and suggest directions for achieving robust quantum-accelerated PDE solvers.

Abstract

Solving Helmholtz problems using finite elements leads to the resolution of a linear system which is challenging to solve for classical computers. In this paper, we investigate how quantum annealers could address this challenge. We first express the linear system arising from the Helmholtz problem as a generalized eigenvalue problem (gEVP). The obtained gEVP is mapped into quadratic unconstrained binary optimization problems (QUBOs) which we solve using an adaptive quantum annealing eigensolver (AQAE) and its classical equivalent. We identify two key parameters in the success of AQAE for solving Helmholtz problems: the system condition number and the integrated control errors (ICE) in the quantum hardware. Our results show that a large system condition number implies a finer discretization grid for AQAE to converge, leading to a variable overhead, and that AQAE is either tolerant or not with respect to ICE depending on the gEVP. Finally, we establish lower bounds on the annealing time, narrowing the possibility of a quantum advantage for solving Helmholtz problems.

Figures (10)

• Figure 1: Solution of the one-dimensional homogeneous Helmholtz problem\ref{['eq:homogeneous_gevp']}. Results obtained using ACAE (disks and stars) and AQAE (crosses and pluses) for the three first eigenmodes (labelled with $n = 0,1,2$), for first order ($p=1$) and fifth order ($p=5$) finite elements, with $D=2$. The dashed lines represent the numerical reference solution obtained via a classical solver on a refined mesh. Other parameters are: $N_\delta=25$, $N_\lambda=10$, $r=0.5$, numreads$=100$ (AQAE) and 1000 (ACAE), $t_\mathrm{a} = 100\,\mu$s, and beta_schedule=geometric dwavedocumentation.
• Figure 2: Solution of the one-dimensional non-homogeneous Helmholtz problem\ref{['eq:helmholtz_normal']}. Results obtained using AQAE (crosses and pluses) and ACAE (disks and stars), for first order ($p=1$) and fifth order ($p=5$) finite elements with (a) $k_0=0$, (b) $k_0=\pi$, and (c) $k_0=2\pi$. AQAE results are generated with $D=D^\star$ listed in Table \ref{['tab:K_helmholtz']}. Thin dashed lines correspond to the finite element solution with the same ($N,p$) combinations as for AQAE and ACAE. Error bars correspond to the standard deviation of the measurements. Other parameters are: $N_\delta=25$, $N_\lambda=10$, $r=0.5$, numreads$=1000$ (AQAE) and 10000 (ACAE), $t_\mathrm{a} = 100\,\mu$s, and beta_schedule=geometric dwavedocumentation.
• Figure 3: Box algorithm examples. Results obtained by solving \ref{['eq:convexoptimization']} with $D=2$ and $r = 0.5$ for (a) $\sigma(A) \approx 5$, (b) $\sigma(A) \approx 50$. The discrete variables are represented as black points, the approximate optimum is represented as a black star, and the global optimum is represented as a blue star.
• Figure 4: Relative residual as a function of the perturbation $\sigma_\eta$ (a) in a homogeneous material for first order finite elements ($N=10$, $p=1$) (b) for fifth order finite elements ($N=2$, $p=5$). Error bars correspond to the standard deviation of the measurements. ACAE parameters: $D=2$, $N_\delta=25$, $N_\lambda=10$, $r=0.5$numreads$=1000$, beta_schedule=geometric dwavedocumentation.
• Figure 5: Relative residual as a function of the perturbation $\sigma_\eta$ for $D=D^\star$ and (a) $k_0=0$, (b) $k_0=\pi$ , (c) $k_0=2\pi$. Error bars correspond to the standard deviation of the measurements. ACAE parameters: $N_\delta=25$, $N_\lambda=10$, $r=0.5$numreads$=10000$, beta_schedule=geometric dwavedocumentation.

Theorems & Definitions (4)

• Lemma 1
• proof
• Theorem 1: Min-Max Theorem
• proof