Kagwe A. Muchane | Searcharxiv

arXiv2026

Quantum algorithm for Clifford multiplication

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 $O(N^{ω/2})$ arithmetic operations, where $ω$ 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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arXiv2026

The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information

We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for $n$-qubit quantum computation based on the tensor product $C\ell_{2,0}(\mathbb{R})^{\otimes n}$. In this setting, the bivector $J = e_{12}$ satisfies $J^{2} = -1$ and supplies the complex structure on the $J$-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the $n$-qubit computational basis state $|0\cdots 0\rangle$ is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for $n$ qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

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