基于时间推进表示的量子算法用于输运-扩散方程的格子玻尔兹曼模型
Equilibrium-energy-scaled quantum lattice Boltzmann method for variable-velocity linear advection--diffusion equations
AI总结：
本文提出基于时间推进表示的量子算法，用于格子玻尔兹曼模型中的输运-扩散方程，实现了无测量的量子模拟方法。
摘要：
本文介绍了一种新的框架，用于开发用于输运-扩散方程的格子玻尔兹曼方法（LBM）的量子算法。我们将LBM的碰撞-传播演化公式化为紧凑的时间推进方案，并严格建立了其在低马赫数条件下的稳定性。这种统一的公式消除了每个时间步需要经典测量的需要，从而实现了系统且完全的量子实现。在此表示基础上，我们研究了两种不同的量子算法方法。第一种是通过顺序演化算符实现的时间推进量子算法，我们提供了详细的实现，包括块编码和扩张单位化，并附有完整的复杂度分析。第二种方法采用量子线性系统算法，将整个时间演化编码到一个全局线性系统中。我们证明了两种方法在渐近时间复杂度上具有可比性。所提出的算法通过一维和二维基准问题的数值模拟进行了验证。这项工作为通过格子玻尔兹曼范式实现输运-扩散过程的量子模拟提供了一条系统且无测量的路径。
英文摘要：
We develop a quantum lattice Boltzmann method for linear advection--diffusion equations with prescribed space- and time-dependent velocities. Direct block-encoding can accumulate normalization factors that suppress success probability exponentially, even when the classical propagator remains bounded. We control this loss by scaling each distribution component by the inverse square root of its local equilibrium weight. For uniformly positive weights and dimensionless relaxation times $τ^*\geq 1/2$, this equilibrium energy scaling gives the Bhatnagar--Gross--Krook collision operator unit spectral norm while preserving the discrete dynamics. Streaming normalization depends on weight ratios at the departure and arrival positions and times of each step; these ratios equal one for uniform steady advection. For periodic streaming, cumulative normalization stays bounded under mesh refinement at fixed physical time if logarithmic derivatives of the weights along streaming trajectories are bounded uniformly in space, time, and mesh resolution. A single-step block-encoding with two flag qubits supports coherent propagation of the full kinetic distribution without intermediate measurements. We compare oracle query bounds for time marching with those for a history-state linear system built from the same scaled operator, under specified assumptions on initial-state preparation and coefficient access. Tests in one and two dimensions and noiseless circuit simulations verify the construction. In a controlled comparison after 50 steps, scaling increases the probability of preparing the quantum state encoding the terminal distribution by approximately $450\times$. Classical refinement diagnostics up to $N_t=16{,}384$ steps confirm bounded cumulative normalization. The scaling controls additional normalization losses while retaining physical diffusive attenuation.
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