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 attains near-optimal theoretical complexity, the fixed-point iteration may be better suited to near-term noisy hardware, as supported by our numerical experiments.

翻译：本文提出了一种新颖的高维非线性量子计算框架，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来求解非线性方程，并给出了误差容忍度相关的定量运行时间上界。结果表明，该方法保持了量子优势，其计算复杂度随问题维度呈对数尺度增长。虽然牛顿法达到了接近最优的理论复杂度，但我们的数值实验表明，定点迭代法可能更适合当前存在噪声的近期量子硬件。