This paper presents a novel framework for high-dimensional nonlinear quantum computation that exploits tensor products of amplified vector and matrix encodings to efficiently evaluate multivariate polynomials. The approach enables the solution of nonlinear equations by quantum implementations of the fixed-point iteration and Newton's method, with quantitative runtime bounds derived in terms of the error tolerance. These results show that a quantum advantage, characterized by a logarithmic scaling of complexity with the dimension of the problem, is preserved. While Newton's method achieves near-optimal theoretical complexity, the fixed-point iteration already shows practical feasibility, as demonstrated by numerical experiments solving simple nonlinear problems on existing quantum devices. By bridging theoretical advances with practical implementation, the framework of amplified encodings offers a new path to nonlinear quantum algorithms.

翻译：本文提出了一种用于高维非线性量子计算的新型框架，该框架利用放大向量与矩阵编码的张量积来高效计算多元多项式。该方法通过量子实现定点迭代法和牛顿法求解非线性方程，并依据误差容限推导出定量运行时界限。结果表明，量子优势——即计算复杂度随问题维度呈对数缩放的特征——得以保持。虽然牛顿法达到了接近最优的理论复杂度，但定点迭代法已展现出实际可行性，这在现有量子设备上求解简单非线性问题的数值实验中得到了验证。通过连接理论进展与实际实现，放大编码框架为非线性量子算法开辟了新路径。