Quantum algorithm for Clifford multiplication

Given two dense multivectors of the Clifford algebra $C\mathrm{\ell }(V,Q)$ with $N=2^{p+q}$ coefficients, the fastest known classical algorithms compute their geometric product in $O(N^{\omega \mathrm{/}2})$ arithmetic operations, where $\omega$ denotes the matrix multiplication exponent. I show that, under amplitude encoding, a quantum computer executes the geometric product in $O(\mathrm{polylog}N)$ time, using logarithmic space with sublogarithmic circuit depth. This exponential speedup establishes Clifford multiplication as a quantum primitive, providing an efficient computational foundation for quantum geometric algorithms and relativistic simulations.