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.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.

quant-pharXiv:2512.07902Dec 9, 20250 citations

Peirce, Clifford, and Dirac

R. G. Beil, R. G. Beil

International Journal of Theoretical PhysicsMay 1, 20043 citations

Scientific Pride and Metaphysical Prejudice

Rosa Maria Mayorga

Abstract Metaphysicians and physicists both share the proud desire to understand the world and all the things in it, to fully comprehend the nature of reality. Almost one hundred years after it was proposed, quantum theory has radically transformed the science of physics, inviting a conception of reality that is drastically at odds with our most rooted metaphysical convictions. A scientist and metaphysician extraordinaire, Charles Peirce (1839–1914) was singularly poised to recognize some of the “metaphysical prejudices” about the world that quantum theory, developed years later, has revealed. This chapter proposes to trace how Peirce’s realism, inspired by the thirteenth-century Franciscan monk John Duns Scotus (c. 1266–1308) and adapted for a reconceived scientific metaphysics, parallels some of the familiar enigmas posed by quantum physics in the early twenty-first century.Abstract Metaphysicians and physicists both share the proud desire to understand the world and all the things in it, to fully comprehend the nature of reality. Almost one hundred years after it was proposed, quantum theory has radically transformed the science of physics, inviting a conception of reality that is drastically at odds with our most rooted metaphysical convictions. A scientist and metaphysician extraordinaire, Charles Peirce (1839–1914) was singularly poised to recognize some of the “metaphysical prejudices” about the world that quantum theory, developed years later, has revealed. This chapter proposes to trace how Peirce’s realism, inspired by the thirteenth-century Franciscan monk John Duns Scotus (c. 1266–1308) and adapted for a reconceived scientific metaphysics, parallels some of the familiar enigmas posed by quantum physics in the early twenty-first century.

Oxford University Press eBooksFeb 22, 20240 citations

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.