Faster quantum subroutine for matrix chain multiplication via Chebyshev approximation

Matrix operations are crucial to various computational tasks in various fields, and quantum computing offers a promising avenue to accelerate these operations. We present a quantum matrix multiplication (QMM) algorithm that employs amplitude encoding and combines quantum walks with Chebyshev polynomial approximation to achieve quadratic acceleration for matrix chain multiplication where the same matrix is applied K times while maintaining logarithmic complexity in matrix dimension and precision. The algorithm can be applied to any complex matrix. Furthermore, we discuss integrating our QMM algorithm as a subroutine in other matrix operations and propose strategies to optimize QMM for matrices with large condition numbers with numerical simulation.