Solving Helmholtz problems with finite elements on a quantum annealer

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.

💡 Research Summary

The paper investigates whether quantum annealers can be used to solve the linear systems that arise when the Helmholtz equation is discretized with the finite‑element method (FEM). Classical solvers struggle with these systems because they are often large, indefinite, and ill‑conditioned, especially when high‑order elements are required to capture high‑frequency oscillations. The authors first reformulate the FEM‑derived linear system as a generalized eigenvalue problem (gEVP) of the form H ϕ = λ M ϕ, where H is Hermitian and M is positive‑definite (or made positive‑definite by using normal equations). They then express the variational formulation of the gEVP as a continuous optimization problem with a Lagrange multiplier λ, and finally map the continuous variables onto binary variables using a linear binary encoding. This yields a quadratic unconstrained binary optimization (QUBO) problem:

 q* = arg min q (qᵀ Q_H q − λ qᵀ Q_M q),

where Q_H and Q_M are binary matrices derived from H and M, and λ is adjusted during a bisection search.

Two algorithms are considered. The Quantum Annealer Eigensolver (QAE) performs a fixed number of λ‑bisection steps, solving a QUBO on a quantum annealer at each step. Because the binary representation limits precision and hardware noise (integrated control errors, ICE) perturbs the programmed couplings, convergence can be slow. To mitigate this, the authors introduce the Adaptive Quantum Annealer Eigensolver (AQAE). AQAE repeatedly narrows the search interval for the continuous variables using a “box algorithm” with a geometric reduction factor r. After each box iteration the λ‑bisection is repeated, allowing the user to specify a target precision ε; the number of box iterations scales as N_δ ≈ log_r(1/ε). A classical counterpart, the Adaptive Classical Annealer Eigensolver (ACAE), replaces the quantum annealer with simulated annealing and serves as a reference benchmark.

The experimental section focuses on two types of Helmholtz problems in one dimension. The first is the homogeneous eigenvalue problem (ϕ″ + k² ϕ = 0) with Dirichlet/Neumann boundaries, where the eigenvalue λ = −ω². The second is a non‑homogeneous problem with a sinusoidal source term, leading to a linear system A ϕ = b. For the latter, the authors consider both the direct gEVP (−f f† w = λ(K+M) w) and the normal‑equation formulation (−b b† w = λ A w, with A = (K+M)†(K+M) positive‑definite). They vary the mesh size N and polynomial order p, which affect the condition number κ of the matrices. Results show that a larger κ forces AQAE to use finer meshes and more box iterations to achieve convergence, leading to a variable computational overhead. Moreover, the sensitivity to ICE depends on the gEVP formulation: the direct indefinite formulation is more vulnerable, while the normal‑equation version is more tolerant.

Hardware experiments are performed on D‑Wave 2000Q and Advantage processors. By calibrating ICE to be below 1 % of the programmed coupling values, AQAE achieves eigenvalue errors comparable to ACAE (within a few percent) for moderate κ. However, when κ grows large, the quantum solution deteriorates unless the number of reads (samples per QUBO) is increased, which raises the total runtime.

A theoretical analysis of the minimum annealing time t_a is also provided. The authors relate t_a to the minimum energy gap Δ between the ground state and first excited state of the problem Hamiltonian, citing bounds of the form t_a ≫ Δ⁻², Δ⁻³, or Δ⁻²|log Δ|^β found in the adiabatic theorem literature. Empirical measurements of Δ on the hardware suggest that t_a must lie in the 10–100 µs range for the studied instances. Longer annealing times improve the probability of reaching the ground state but also increase exposure to decoherence and thermal noise, creating a trade‑off.

In conclusion, the study demonstrates a complete pipeline—from FEM discretization to gEVP formulation, binary encoding, and adaptive quantum annealing—that can solve Helmholtz problems on current quantum annealers. Success hinges on two key parameters: the condition number of the discretized system and the magnitude of ICE in the hardware. While AQAE can tolerate moderate ICE for well‑conditioned gEVPs, it struggles with highly ill‑conditioned or indefinite formulations. The authors suggest future work on preconditioning techniques to reduce κ, higher‑precision qubit control to lower ICE, and next‑generation annealers with larger qubit counts and improved coherence. Such advances could narrow the gap between quantum annealing and classical linear solvers, potentially delivering a quantum advantage for large‑scale wave‑propagation simulations.