带连续源注入的变系数输运方程的高效量子模拟
Efficient Quantum Simulation of Variable-Coefficient Transport with Continuous Source Injection
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中文总结 AI 辅助
本文提出一种带连续源注入的变系数平流-扩散方程的高效量子模拟算法，采用Gray码Trotter序列与Strang分裂，实现二阶时间收敛，可稳定演化$5\times10^4$时间步，为近期量子硬件研究提供量子比特高效的构建模块。
中文摘要 AI 辅助
针对输运偏微分方程的量子时间推进算法，常通过寄存器扩展伸缩、块编码预言机或重复后选择来表示变系数和强迫项。本文提出一种针对受迫变系数平流-扩散方程的替代算法，该方程采用流动启发的斜对称形式，包含空间变化的速度、粘性耗散和持续源注入，峰值逻辑量子比特需求为$n_q+1$。中心斜对称离散化使平流算子对任意速度剖面均为严格斜厄米，可使用Gray码Trotter序列的受控-$R_y$旋转实现无辅助量子比特的幺正实现。扩散项在傅里叶基下应用，通过对一个后选择辅助量子比特的均匀受控旋转实现，该辅助量子比特在两次扩散半步之间被测量、重置并复用，而源项通过二阶Strang分裂经典引入。对$N=16$和$32$的态矢量模拟，相对于高精度经典解恢复了二阶时间收敛性，而Richardson外推法给出四阶精度，并将核调用减少了4至14倍。通过$N=256$的独立测试确认了二阶空间一致性。进一步表明，每步辅助量子比特失败概率与瞬时粘性耗散率成正比，使后选择成本在50倍粘度范围内自调节。在$5\times10^4$时间步的稳定演化被证实，未观察到明显的长期误差增长，而Gray码平流占编译后受控-NOT门的71%至95%。该固定宽度核为近期硬件研究提供了量子比特高效的构建模块，尽管经典读出和态重新制备仍是相干多步演化的主要障碍。
英文摘要
Quantum time-marching algorithms for transport PDEs often represent variable coefficients and forcing through register-expanding dilations, block-encoding oracles, or repeated postselection. We present an alternative algorithm for a forced variable-coefficient advection-diffusion equation in flow-inspired skew-symmetric form that incorporates spatially varying velocity, viscous dissipation, and persistent source injection with a peak logical requirement of $n_q+1$ qubits. A centered skew-symmetric discretization makes the advection operator strictly skew-Hermitian for arbitrary velocity profiles, enabling an ancilla-free unitary realization using a Gray-code Trotter sequence of controlled-$R_y$ rotations. Diffusion is applied in the Fourier basis through a uniformly controlled rotation on one postselected ancilla, which is measured, reset, and reused between the two diffusion half-steps, while the source is incorporated classically through second-order Strang splitting. Statevector simulations for $N=16$ and $32$ recover second-order temporal convergence against high-accuracy classical solutions, while Richardson extrapolation gives fourth-order accuracy and reduces kernel calls by factors of four to fourteen. Independent tests through $N=256$ confirm second-order spatial consistency. We further show that the per-step ancilla failure probability is proportional to the instantaneous viscous dissipation rate, making postselection cost self-regulating over a fifty-fold viscosity range. Stable evolution is demonstrated for $5\times10^4$ time steps without observable secular error growth, while Gray-code advection accounts for $71$--$95\%$ of transpiled controlled-NOT gates. The fixed-width kernel provides a qubit-efficient building block for near-term hardware studies, although classical readout and state re-preparation remain the main obstacles to coherent multistep evolution.
