Variational quantum algorithm for solving Helmholtz problems with high order finite elements

TL;DR

This work addresses solving Helmholtz problems discretized with high-order finite elements, which yield challenging linear systems for classical solvers. It develops a variational quantum algorithm that encodes the FE solution in a parameterized quantum state and employs block-encoding and a linear-combination-of-units (LCU) framework to efficiently evaluate the necessary expectation values, including $A$ and $A^ abla A$. By exploiting a regular mesh, the authors obtain a block-structured decomposition with $ ext{O}(p^2)$ terms for both $A$ and $A^ abla A$ that is independent of the number of elements $N$, enabling circuit depths of at most $ ext{O}(p^3 ext{polylog}(Np))$ for key operations. They validate the approach on a one-dimensional Helmholtz problem with Dirichlet and Neumann boundaries across several wavenumbers, analyze the expressiveness of the quantum ansatz via KL-divergence, and discuss the scaling of gradient computations in the VQA. The results indicate a viable path toward quantum-accelerated Helmholtz solvers for high-order FE discretizations and highlight the potential for extensions to more complex boundary conditions and piecewise-constant wavenumbers.

Abstract

Discretizing Helmholtz problems via finite elements yields linear systems whose efficient solution remains a major challenge for classical computation. In this paper, we investigate how variational quantum algorithms could address this challenge. We first show that, for regular meshes, a block encoding of the operators $A$ and $A^\dagger A$ arising from the high-order finite element discretisation of Helmholtz problems can be designed, resulting in a quantum circuit of depth $\mathcal{O}(p^3\mathrm{poly}\log(Np))$ with $N$ the number of elements and $p$ the order of the finite elements. Then we apply our algorithm to a one-dimensional Helmholtz problem with Dirichlet and Neumann boundary conditions for various wavenumbers.

Figures (4)

• Figure 1: Quantum circuits for the linear combination of unitaries of $A$. The LCU writes $A_k = \sum_{m}\alpha_{km}U_{km}$ with $k = 1, ..., 5$. The quantum circuits implement the unitary operators required for the LCU of (a) $A_1$, (b) $A_2$, (c) $A_3$, (d) $A_4$, and (e) $A_5$, respectively.
• Figure 2: Quantum circuit for the measurement of $\mathrm{Re}(\bra f A\ket \phi)$. (a) Hadamard test. The measuerment of the ancilla qubit (register a) in the computational basis leads to $(1+\mathrm{Re}(\bra{f}A\ket\phi/\eta))/2$. (b) Block encoding $U$ of the non unitary matrix $A$. The operator $U_\alpha$ acts on the register s as follows : $U_\alpha \ket{0}_\mathrm{s} = \sum_j (\alpha_j/\eta)^{1/2}\ket{j}_\mathrm{s}$, while $U_s$ acts jointly on registers s and w as follows : $U_s\ket j_\mathrm{s} \ket \phi_\mathrm{w} = \ket j_\mathrm{s} U_j \ket \phi_\mathrm{w}$.
• Figure 3: Variational quantum algorithm solution for the Helmholtz problem \ref{['eq:linear_system']}. The PQC architecture is the hardware efficient ansatz (HEA) kandala2017hardware, with $7$ layers of $R_Y(\theta_j)$ rotation gates with linear sequences of CNOT entanglement gates. The order of the finite elements is (a) $p=1$, (b) $p=2$, (c) $p=4$. The number of degrees of freedom is kept constant. (d) Square norm of the residual.
• Figure 4: Kullback-Leibler divergence between states sampled from the HEA and Haar random states. (a) HEA with $R_Y$ and $R_Z$ rotation gates compared to Haar random states over $SU(2^n)$. (b) HEA with $R_Y$ rotation gates compared to Haar random states over $SO(2^n)$.