A Quantum Spectral Method for Non-Periodic Boundary Value Problems

TL;DR

The paper develops a quantum spectral solver for non-periodic Dirichlet boundary value problems on axis-aligned hyperrectangles, achieving polylogarithmic complexity in the problem size by encoding forcing and solution vectors into quantum states and performing the diagonal inverse in Fourier space via a polynomial-encoded operator. It rigorously handles homogeneous Dirichlet BCs through antisymmetric domain doubling and enforces inhomogeneous BCs via superposition with a boundary-conforming function, while replacing the classical DST with a quantum sine transform derived from a unitary reflection. The authors demonstrate the method on Dirichlet-Poisson problems and fractional stochastic PDEs in 1D and 2D, with numerical evidence of polylogarithmic gate counts and controllable accuracy through polynomial degree and domain partitioning. They also discuss circuit-level implementations, scalability considerations, and potential extensions such as approximate QFT and QSVT, highlighting the method’s suitability for structured, well-conditioned problems and stochastic simulations where repeated solves are required.

Abstract

Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.

Figures (18)

• Figure 1: The extended domain $\Omega_{\text{E}}=(0, \, 2L)$ and its discretisation with $N = 8$ cells and $N = 8$ grid points. The original problem domain is $\Omega=(0, \, L)$. A sample reflected source term $f_\text{E}(x)$ is depicted in blue, and the components of the respective force vector $\vec{f}$ are depicted as circles. The extended domain problem is $2L$ periodic.
• Figure 2: Quantum circuit for QFT. An input vector $\sum_{k=0}^{N-1}f_k \ket k$ is mapped to the output vector $\sum_{k=0}^{N-1} \hat{f}_k \ket k$ (pursuant to some relabelling of the components).
• Figure 3: Quantum circuit for the reflection unitary $U_R= U_{R_3} U_{R_2} U_{R_1} U_{R_0}$. An input vector $\sum_{k=0}^{7} f_k \ket 0 \ket k$ is mapped to the antisymmetrically extended output vector $\sum_{k=0}^{15} f_k \ket k$.
• Figure 4: Two alternative quantum circuits for the forward shift unitary $U_F$, (a) using a cascade of controlled $X$ gates and (b) based on ripple-carry addition. In (b), the top two qubits are ancilla qubits. Both circuits map an input vector $\sum_{k=0}^{7} f_k \ket k$ to an output vector $\sum_{k=0}^{7} \ket {(k+1) \mod 2^3}$. The ancilla qubits have the value $\ket 0$ before and after execution of the circuit.
• Figure 5: Total number of universal gates for (a) the shift unitary $U_S$ and (b) the reflection unitary $U_R$. The label $MCX$ refers to the implementation using multi-controlled $X$ gates and the label ripple-carry to the implementation based on ripple-carry adder in Figures \ref{['fig:matp_mcx']} and \ref{['fig:matp_cascade']}, respectively.