AI 中文总结
本文针对非线性波直接量子模拟的测量成本问题，提出混合分步法求解器，验证其在超导硬件的适用性，指出量子成本随网格规模增大超经典成本，明确无测量非线性更新为端到端优势的定量目标。
AI 中文摘要
量子处理器可将N点场编码到log₂(N)个量子比特中，这使得非线性波动方程成为量子模拟的重要应用方向。然而，非线性演化需要场值本身，而若不进行量子测量就无法直接获取这些场值。现有算法通过线性嵌入和状态拷贝规避了该测量过程，却将其成本隐藏在截断阶数、辅助维度和状态制备中。为了揭示这一成本，本文提出了一种混合分步法求解器，该求解器每一步都会对场进行测量、经典更新并重新加载，且所有采样次数（shots）和量子门都被纳入统一的成本与误差模型中。由于每一步都能获取完整场，这是强非线性区域线性近似无法实现的特性，该求解器的设计可简化为对时间步长、多项式阶数和采样次数的预算分配问题。该求解器的相干核在超导硬件上得到验证，且一维和二维粘性伯格斯方程遵循相同的结构与瓶颈。由于每一步都读取完整场，以电路深度乘以测量采样次数衡量的单步量子成本会随网格规模增大而超过经典成本。因此，该框架确定了相干无测量非线性更新是任何端到端优势必须达到的定量目标。
英文摘要
Quantum processors encode an N-point field in log_2(N) qubits, which renders nonlinear wave equations an important application for quantum simulation. Nonlinear evolution, however, requires the field values themselves, and these are not directly accessible without quantum measurement. Existing algorithms circumvent this measurement through linear embeddings and state copies, thereby obscuring its cost within the truncation order, the auxiliary dimensions, and the state preparation. In order to expose this cost, a hybrid split-step solver is proposed in which the field is measured, updated classically, and reloaded at every step, with all shots and gates accounted for in a single cost-and-error model. Since the entire field is available at every step, a property unavailable to linear approximations in strongly nonlinear regimes, the design of the solver reduces to a budgeting problem over the timestep, the polynomial degree, and the shot count. The coherent kernels of the solver are validated on superconducting hardware. An identical structure and bottleneck govern the viscous Burgers' equation in one and two dimensions. Because every step reads the full field, the quantum cost per step, measured as circuit depth multiplied by measurement shots, exceeds the classical cost with increasing grid size. The framework consequently identifies a coherent, measurement-free nonlinear update as the quantitative target that any end-to-end advantage must meet.
