非线性狄拉克方程的数值求解：一种分裂变分量子算法
Numerical solution of the nonlinear Dirac equation by a splitting variational quantum algorithm
AI总结：
提出分裂变分量子算法Dirac-sVQA，通过线性狄拉克子步和非线性变分修正分解非线性狄拉克方程，利用自旋-傅里叶传播器和测量基变分更新实现数值模拟。
摘要：
本文提出一种算子分裂变分量子算法，称为Dirac-sVQA，用于模拟非线性狄拉克方程（NLDE）。主要困难来自状态相关的非线性相互作用，其时间离散更新显式依赖于中间旋量态，通常无法实现为固定的状态无关酉电路。为解决这一困难，我们将NLDE演化分解为结构化的线性狄拉克子步和非线性变分修正。线性子步通过在联合位置-自旋寄存器上的自旋-傅里叶狄拉克传播子实现，保留了狄拉克算子的自旋-动量耦合和质量诱导的自旋演化。非线性修正被重新表述为基于测量的变分更新，通过少量重叠、自通道和交叉通道可观测量实现。我们提供了相应的量子电路，并推导了考虑测量的资源和复杂度估计。在多个非线性区域中的数值实验表明，Dirac-sVQA准确捕捉了总密度和分量旋量动力学，与经典傅里叶伪谱分裂解吻合良好，并随时间表现出稳定的误差行为。这些结果为非线性相对论波动方程的算子分裂变分量子模拟的可行性提供了数值证据。
英文摘要：
In this work, we propose an operator-splitting variational quantum algorithm, termed Dirac-sVQA, for simulating the nonlinear Dirac equation (NLDE). The main difficulty arises from the state-dependent nonlinear interaction, its time-discrete update depends explicitly on the intermediate spinor state and, in general, cannot be implemented as a fixed state-independent unitary circuit. To address this difficulty, we decompose the NLDE evolution into a structured linear Dirac substep and a nonlinear variational correction. The linear substep is implemented by a spinor-Fourier Dirac propagator on a joint position-spin register, preserving the spin-momentum coupling and mass-induced spin evolution of the Dirac operator. The nonlinear correction is reformulated as a measurement-based variational update through a small set of overlap, self-channel, and cross-channel observables. We provide the corresponding quantum circuits and derive measurement-aware resource and complexity estimates. Numerical experiments in several nonlinear regimes show that Dirac-sVQA accurately captures both the total density and the componentwise spinor dynamics, agrees well with classical Fourier pseudospectral splitting solutions, and exhibits stable error behavior over time. These results provide numerical evidence for the feasibility of operator-splitting variational quantum simulation for nonlinear relativistic wave equations.
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