Quantum algorithms for solving a drift-diffusion equation: A complexity analysis

TL;DR

This work demonstrates quantum computational advantages for solving the multidimensional drift-diffusion equation by analyzing four quantum approaches (quantum linear systems, time evolution, quantum random walks, and quantum Fourier transform diagonalization) against four classical baselines (linear equations, time stepping, random walk, FFT diagonalization). A central contribution is showing that diagonalization by the quantum Fourier transform can outperform classical FFT-based diagonalization for computing the solution at a fixed time, provided the quantum distribution extraction error $\epsilon_q$ is appropriately bounded and combined with a discretization error $\epsilon_c$ (with total error $\epsilon=\epsilon_c+\epsilon_q$). The authors introduce a multidimensional amplitude-estimation framework to recover the full probability distribution from the quantum state and derive detailed, dimension-aware complexity bounds for each method, including space costs on qubits. The results offer practical guidance on when quantum methods can surpass classical counterparts for solving linear PDEs like the DDE and suggest potential extensions to other linear PDEs and higher-dimensional problems. Overall, the study highlights a path to quantum-accelerated PDE solvers with explicit trade-offs between $\epsilon_c$ and $\epsilon_q$ and emphasizes the role of QFT-based diagonalization in achieving quantum advantage at fixed final times.

Abstract

We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare the complexities of these methods to their classical counterparts, finding that diagonalization via the quantum Fourier transform offers a quantum computational advantage for solving linear partial differential equations at a fixed final time. We employ a multidimensional amplitude estimation process to extract the full probability distribution from the quantum computer.

Figures (8)

• Figure 1: A flowchart demonstrating the contributing factors to the overall time complexity of the classical linear equations method described by Theorem\ref{['thm: Lineqn']}. The complexity for the conjugate gradient method is provided in Ref. Shewchuk1994AnPain.
• Figure 2: A flowchart demonstrating the contributing factors to the overall time complexity of the classical time-evolution method described by Theorem\ref{['thm: timestepping']}.
• Figure 3: A flowchart demonstrating the contributing factors to the overall time complexity of the classical random-walk method described by Theorem\ref{['thm: timestepping']}. Here CFTP represents the coupled from the past method Propp1998HowGraph.
• Figure 4: A flowchart demonstrating the contributing components to the overall time complexity of the classical time-stepping method in Theorem\ref{['thm: Classical_FFT']}. The complexity for the FFT (and its inverse) is as described in Ref. Frigo2005TheFFTW3. $\Lambda$ denotes the diagonal matrix made up of the eigenvalues of $\mathcal{L}$ and the complexity is derived in Lemma \ref{['lemma: eigen_L']}.
• Figure 5: A flowchart demonstrating the contributing factors to the overall time complexity of the quantum linear equations solution method for the DDE in Theorem\ref{['thm: QLEM']}. The "solving linear equations" complexity is based on Theorem\ref{['thm: Solv LE']} and the complexity of the "measurement protocol" is from Theorem\ref{['thm: q_meas']}.

Theorems & Definitions (45)

• Theorem 1: Approximation up to $\infty$-norm error
• Corollary 2
• proof
• Lemma 3
• Lemma 4
• Theorem 5
• Theorem 6: Classical linear equations method
• proof
• Theorem 7: Classical time-evolution method
• proof