Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Kagwe A. Muchane

TL;DR

The paper develops a real, grade-preserving Clifford-algebra framework for $N$-qubit quantum information using $\mathcal{C}\ell_{2,0}(\mathbb{R})^{\otimes N}$, with the bivector $J$ supplying a complex structure via right multiplication and Pauli operations realized as left actions. It introduces a State--Operator Clifford Compatibility law, $U(A P_N) = (U A) P_N$, ensuring that left actions on right-encoded states align with the composition of Clifford elements, and demonstrates a canonical embedding of the $N$-qubit vacuum as a tensor product of real idempotents. The framework extends to $N$ qubits via a graded tensor product, preserving locality and enabling $O(N)$-time updates for stabilizer dynamics and density-operator evolution, thereby offering a geometric, algebraic basis for efficient stabilizer simulation and potential applications to error correction and quantum software tooling. Overall, the approach provides a structurally transparent, scalable alternative to global complex algebras for modeling and simulating Clifford circuits in quantum information.

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 structure $C\ell_{2,0}(\mathbb{R})^{\otimes N}$. In this setting the bivector $J = e_{12}$ satisfies $J^{2} = -1$ and supplies the complex structure on a minimal left ideal via right-multiplication, while Pauli operations arise as left actions of suitable Clifford elements. Adopting a canonical stabilizer mapping, the $N$-qubit computational basis state $|0\cdots 0\rangle$ is represented natively by a tensor product of real algebraic idempotents. This structural choice leads to a State-Operator Clifford Compatibility law that is stable under the geometric product for $N$ qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

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

TL;DR

The paper develops a real, grade-preserving Clifford-algebra framework for -qubit quantum information using , with the bivector supplying a complex structure via right multiplication and Pauli operations realized as left actions. It introduces a State--Operator Clifford Compatibility law, , ensuring that left actions on right-encoded states align with the composition of Clifford elements, and demonstrates a canonical embedding of the -qubit vacuum as a tensor product of real idempotents. The framework extends to qubits via a graded tensor product, preserving locality and enabling -time updates for stabilizer dynamics and density-operator evolution, thereby offering a geometric, algebraic basis for efficient stabilizer simulation and potential applications to error correction and quantum software tooling. Overall, the approach provides a structurally transparent, scalable alternative to global complex algebras for modeling and simulating Clifford circuits in quantum information.

Abstract

We revisit the Pauli-Clifford connection to introduce a real, grade-preserving algebraic framework for -qubit quantum computation based on the tensor product structure . In this setting the bivector satisfies and supplies the complex structure on a minimal left ideal via right-multiplication, while Pauli operations arise as left actions of suitable Clifford elements. Adopting a canonical stabilizer mapping, the -qubit computational basis state is represented natively by a tensor product of real algebraic idempotents. This structural choice leads to a State-Operator Clifford Compatibility law that is stable under the geometric product for qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

Paper Structure

This paper contains 6 sections, 1 theorem, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The State--Operator Clifford Compatibility
  3. Basis States and Operators
  4. From Single-Qubit to Multi-Qubit Systems
  5. Density Operators and Compatibility
  6. Conclusion

Key Result

Theorem 2.1

With the above definitions, $\rho$ is an algebra representation and, for all $g,h \in \mathcal{C}\ell_{2,0}$,

Figures (1)

  • Figure 1: Commutative diagram for the State--Operator Clifford Compatibility. Evolving a state via $\rho(g)$ (bottom path) is equivalent to first composing Clifford elements in $\mathcal{C}\ell_{2,0}$ and then mapping to the state space via $\vartheta$ (top path).

Theorems & Definitions (1)

  • Theorem 2.1: State--Operator Clifford Compatibility