Quantum circuits for partial differential equations in Fourier space

TL;DR

This work presents non-variational quantum circuit constructions for solving certain high-dimensional PDEs by leveraging the quantum Fourier transform to diagonalize differential operators and quantum singular value transformation to implement required functions of operators. It provides explicit circuit templates for the incompressible advection, heat, isotropic acoustic wave, and Poisson equations, with detailed complexity analyses and strategies for both general (non-smooth) and smooth initial data. The approaches extend naturally to arbitrary spatial dimensions and include boundary-condition handling, plus alternative encoding schemes such as density-matrix diagonal encoding to mitigate postselection. Collectively, the paper demonstrates that, under suitable encodings and approximations, high-dimensional PDEs can be tackled with circuits whose depths scale polynomially in dimension, suggesting a feasible pathway for near-term quantum hardware to address PDEs beyond classical capabilities. The results offer concrete building blocks and performance guarantees that can guide future implementations and extensions to more complex or nonlinear PDEs.

Abstract

For the solution of partial differential equations (PDEs), we show that the quantum Fourier transform (QFT) can enable the design of quantum circuits that are particularly simple, both conceptually and with regard to hardware requirements. This is shown by explicit circuit constructions for the incompressible advection, heat, isotropic acoustic wave, and Poisson's equations as canonical examples. We utilize quantum singular value transformation to develop circuits that are expected to be of optimal computational complexity. Additionally, we consider approximations suited for smooth initial conditions and describe circuits that make lower demands on hardware. The simple QFT-based circuits are efficient with respect to dimensionality and pave the way for current quantum computers to solve high-dimensional PDEs.

Figures (9)

• Figure 1: Five-qubit circuit representation of the operator $\frac{1}{\sqrt{N}} \sum_{\ell} \text{e}^{\text{i} 2 \pi (k+a) (\ell+b) / N} \ket{\ell}\bra{k}$ realizing the QFT Co02NiCh10 with wavenumbers $k$ shifted by a constant $a$ and positions $\ell$ shifted by a constant $b$. More specifically, the circuit implements the shifted QFT without the global phase $\text{e}^{\text{i} 2 \pi a b / N}$. Here $n = 5$, $N = 2^{n}$, $R_{b}^{(\zeta)} = \text{diag}[1, \exp(\text{i} \pi 2^{-\zeta} b)]$, $H$ is the Hadamard gate, $R_{\zeta} = \text{diag}[1, \exp(\text{i} \pi 2^{-\zeta+1})]$ and $R_{a}^{(\zeta)} = \text{diag}[1, \exp(\text{i} \pi 2^{\zeta-n+1} a)]$, where $\text{diag}[\cdot]$ denotes the diagonal single-qubit operator with $[\cdot]$ on the diagonal.
• Figure 2: Four-qubit quantum circuit for the realization of Dirichlet and Neumann boundary conditions using the plane wave basis of our shifted QFT. (a) The input quantum state $\ket{\tilde{f}}$ stores the $8$ desired function values that fulfill the specific boundary conditions considered. We add one ancilla qubit initialized in $\ket{0}$, apply $R_{y}^{(\phi)} = \text{e}^{-\text{i} \phi Y / 2}$ to it, where $Y$ is the Pauli $Y$ matrix, and apply the CNOTs as shown. This circuit creates the state $\ket{f}$. (b) To impose Dirichlet boundary conditions, we set $\phi = -\pi/2$ such that $\ket{f}$ corresponds to an odd function. (c) To impose Neumann boundary conditions, we set $\phi = \pi/2$ such that $\ket{f}$ corresponds to an even function.
• Figure 3: Quantum circuits for the one-dimensional incompressible advection equation. (a) Circuit representation of the operator $U_{\hat{k}}$\ref{['eq:PlaneWaveBlockEncoding']} where $\mathcal{R}_{z}^{(\zeta)} = \text{e}^{-\text{i} \pi 2^{-\zeta - 1} Z_{\zeta}}$. (b) Circuit representation of \ref{['eq:SmoothAdvection']} where $\mathcal{U}_{z}^{(\zeta)} = \text{e}^{\text{i} \pi t r 2^{-\zeta+n-1} Z_{\zeta}}$ and we exclude the global phase $\text{e}^{\text{i} \pi t r}$.
• Figure 4: Quantum circuit realization of the operator product in Eq. \ref{['eq:SmoothHeat']} for system size $n = 3$ using three ancilla qubits that are assumed to be measured in $\ket{0}$ and then reused. The angles $\phi_{1}$, $\phi_{2}$, …, $\phi_{6}$ are calculated via Eq. \ref{['eq:ThetaToPhi']} using $\theta_{1} = -8 \pi^{2} t u$, $\theta_{2} = -4 \pi^{2} t u$, $\theta_{3} = -2 \pi^{2} t u$, $\theta_{4} = -16 \pi^{2} t u$, $\theta_{5} = -8 \pi^{2} t u$, $\theta_{6} = -4 \pi^{2} t u$, respectively. Note that one can readily reduce the required number of ancilla qubits from three to one by redefining the circuit such that only one ancilla qubit is sequentially measured and reused.
• Figure 5: Initial-state preparation circuits for the $d$-dimensional wave equation. Given efficient circuits to prepare $|f\rangle = U_{f} |\boldsymbol{0}\rangle$ and $|\partial_{t}f\rangle = U_{\partial_{t} f} |\boldsymbol{0}\rangle$, we prepare the state $|\psi\rangle$ in Eqs. \ref{['eq:WaveSchrodingerStateA']} and \ref{['eq:WaveSchrodingerStateB']} using the quantum circuit (a) and (b), respectively. In (a) the operator $\mathcal{U}$ transforms $|\zeta\rangle \otimes |f\rangle$ into $-\text{i} \mathcal{H}^{(d)} |\zeta\rangle \otimes |f\rangle$. In (b) the operator $\mathcal{V}$ transforms $|\zeta\rangle \otimes |\partial_{t}f\rangle$ into $\text{i} \left(\mathcal{H}^{(d)}\right)^{-1} |\zeta\rangle \otimes |\partial_{t}f\rangle$.