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

**Authors:** Kagwe A. Muchane  
**arXiv:** [2512.07902](https://arxiv.org/abs/2512.07902)  
**Date:** 2025-12-05  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2512.07902v1/

## Key points

- Kagwe A. Muchane explores the Pauli-Clifford connection in $N$-qubit quantum computation
- Utilizes the tensor product structure $C\ell_{2,0}(\mathbb{R})^{\otimes N}$ and bivector $J = e_{12}$ with $J^{2} = -1$
- Establishes a State-Operator Clifford Compatibility law for multi-qubit systems
- Law aligns symbolic Clifford multiplication with unitary evolution on Hilbert space
- Real algebraic idempotents represent the computational basis state $|0\cdots 0\rangle$
- Reveals connections between Pauli operations, Clifford elements, and quantum states in quantum information processing

## Summary

In "The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information," Kagwe A. Muchane delves into the Pauli-Clifford connection in $N$-qubit quantum computation through a real, grade-preserving algebraic framework. By utilizing the tensor product structure $C\ell_{2,0}(\mathbb{R})^{\otimes N}$ and the bivector $J = e_{12}$ satisfying $J^{2} = -1$, Muchane establishes a State-Operator Clifford Compatibility law that remains stable under the geometric product for multi-qubit systems. This law aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space and showcases how real algebraic idempotents can represent the computational basis state $|0\cdots 0\rangle$. Through this work, Muchane sheds light on the intricate connections between Pauli operations, Clifford elements, and quantum states in multi-qubit systems within quantum information processing.