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

TL;DR

This paper develops a real algebraic framework using Clifford algebras for $n$-qubit quantum computation, providing a stable compatibility law that aligns Clifford algebra operations with quantum unitary evolution.

Contribution

It introduces a real, grade-preserving Clifford algebra approach to quantum computation, establishing a compatibility law stable under geometric product and linking algebraic and quantum structures.

Findings

The bivector J supplies the complex structure on the algebra.

Quantum states are represented as equivalence classes modulo the left annihilator.

The framework aligns Clifford multiplication with quantum unitary evolution.

Abstract

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.