狄拉克方程的格子玻尔兹曼实现的精确量子电路
Exact quantum circuits for lattice Boltzmann realization of the Dirac equation
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中文总结 AI 辅助
本文将Succi-Dellar量子格子玻尔兹曼方案映射为门模型量子电路，在模拟器上以极高精度复现经典求解器，证实该理论可在量子计算机上实现并报告门数量，未声称计算优势。
中文摘要 AI 辅助
Succi和Dellar提出的量子格子玻尔兹曼（QLB）方案通过固定序列的局部、严格保范操作在格子上推进四分量狄拉克旋量：基旋转、碰撞、流移位和逆旋转。这种幺正性是该方案的结构属性，而非近似，这表明QLB时间步长应映射为量子门序列。本文明确构建了这种映射，给出了三维狄拉克QLB方案所有操作的门级构造：固定旋转门、碰撞门、作为位置寄存器受控增量的流移位、作为相位 oracle 的位置相关势，以及作为幺正电路的周期性和反射性（反弹）边界条件，随后将其组合成单轴、二维和三维时间步长。在状态矢量模拟器上，所得电路以机器精度复现经典QLB求解器（一维、二维和三维测试中的最大密度偏差在3.7×10⁻¹²至1.0×10⁻¹⁷之间），因此这些电路是该方案本身，而非其近似。研究范围较窄：本文证实Succi-Dellar理论可在（门模型）量子计算机上实现，并报告了相关门数量；未声称具有计算优势，状态制备、测量和渐近成本作为开放问题讨论。所有算子、电路、测试和图均可从开源quantumKineticMethods库复现。
英文摘要
The quantum lattice Boltzmann (QLB) scheme of Succi and Dellar advances a four-component Dirac spinor on a lattice by a fixed sequence of local, exactly norm-preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation. This unitarity is a structural property of the scheme, not an approximation, which suggests that a QLB time step should map onto a sequence of quantum gates. Here we make that mapping explicit. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting (bounce-back) boundary conditions as unitary circuits. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision (maximum density deviation between $3.7\times10^{-12}$ and $1.0\times10^{-17}$ across the one-, two-, and three-dimensional tests), so the circuits are the scheme rather than an approximation of it. The scope is narrow: we establish that the Succi-Dellar theory can be implemented on a (gate-model) quantum computer, and report the associated gate counts. We make no claim of computational advantage; state preparation, measurement, and asymptotic cost are discussed as open questions. All operators, circuits, tests, and figures are reproducible from the open-source quantumKineticMethods library.
