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arXiv · 2607.10473

Quantum algorithm for Clifford multiplication

Kagwe A. Muchane
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Abstract

Given two dense multivectors of the Clifford algebra $C\ell(V, Q)$ with $N=2^{p+q}$ coefficients, the fastest known classical algorithms compute their geometric product in $O(N^{\omega/2})$ arithmetic operations, where $\omega$ denotes the matrix multiplication exponent. I show that, under amplitude encoding, a quantum computer executes the geometric product in $O(\operatorname{polylog} N)$ 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.

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Kagwe A. Muchane. 2026-07-11. Quantum algorithm for Clifford multiplication. https://arxiv.org/abs/2607.10473

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