This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game.This paper introduces an application of complex Clifford algebra in a representation of the quantum prisoner's dilemma. The authors propose a novel modification of the Eisert–Lewenstein–Wilkens protocol to represent a repeated version of the quantum game. This repeated modification allows to embed entanglement into players' strategy sets and to see how players will operate with it. The apparatus of complex Clifford algebra enables an intuitive representation of the suggested protocol and efficient computation of the resulting payoff functions. The presented findings provide a new point of view on the interpretation of entanglement as a measure of information transition between rounds of the game.

Mathematical Methods in the Applied SciencesSep 6, 20223 citations

A Unified Clifford Algebra Framework for Quantum Computation: From Specific Spinor Realizations to General Operator Representations

Luciano Silva

This paper presents a unified algebraic framework for $n$ -qubit quantum computation rooted in the complex Clifford algebra $Cl(2n, \mathbb{C})$ . We leverage the fundamental isomorphism $Cl(2n, \mathbb{C}) \cong M(2^n, \mathbb{C})$ to directly identify the algebra of all $n$ -qubit quantum operators (including states, gates, and observables) with elements of $Cl(2n, \mathbb{C})$ . This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors $\rho \in Cl(2n, \mathbb{C})$ satisfying standard physical constraints ( $\rho^\dagger = \rho$ , $2^n \langle \rho \rangle_0 = 1$ , positivity), and unitary gates $U$ as multivectors obeying $U^\dagger U = I$ . Quantum evolution $\rho' = U\rho U^\dagger$ and measurements $p(m) = 2^n \langle P_m \rho \rangle_0$ are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals ( $S = Cl(2n, \mathbb{C})\prod_j f_j f_j^\dagger$ ), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.This paper presents a unified algebraic framework for $n$-qubit quantum computation rooted in the complex Clifford algebra $Cl(2n, \mathbb{C})$. We leverage the fundamental isomorphism $Cl(2n, \mathbb{C}) \cong M(2^n, \mathbb{C})$ to directly identify the algebra of all $n$-qubit quantum operators (including states, gates, and observables) with elements of $Cl(2n, \mathbb{C})$. This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors $\rho \in Cl(2n, \mathbb{C})$ satisfying standard physical constraints ($\rho^\dagger = \rho$, $2^n \langle \rho \rangle_0 = 1$, positivity), and unitary gates $U$ as multivectors obeying $U^\dagger U = I$. Quantum evolution $\rho' = U\rho U^\dagger$ and measurements $p(m) = 2^n \langle P_m \rho \rangle_0$ are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals ($S = Cl(2n, \mathbb{C})\prod_j f_j f_j^\dagger$), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.

May 23, 20250 citations

Clifford algebra and quantum logic gates

Wuming Li

Advances in Applied Clifford AlgebrasOct 1, 20043 citations

Quantum Gates and Clifford Algebras

Alexander Yu. Vlasov

Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical properties of the algebras those also justifies such applications. First, any complex Clifford algebra with 2n generators, Cl(2n,C), has representation as algebra of all 2^n x 2^n complex matrices and so includes unitary matrix of any quantum n-gate. An arbitrary element of whole algebra corresponds to general form of linear complex transformation. The last property is also useful because linear operators are not necessary should be unitary if they used for description of restriction of some unitary operator to subspace. The second advantage is simple algebraic structure of Cl(2n) that can be expressed via tenzor product of standard "building units" and similar with behavior of composite quantum systems. The compact notation with 2n generators also can be used in software for modeling of simple quantum circuits by modern conventional computers.Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical properties of the algebras those also justifies such applications. First, any complex Clifford algebra with 2n generators, Cl(2n,C), has representation as algebra of all 2^n x 2^n complex matrices and so includes unitary matrix of any quantum n-gate. An arbitrary element of whole algebra corresponds to general form of linear complex transformation. The last property is also useful because linear operators are not necessary should be unitary if they used for description of restriction of some unitary operator to subspace. The second advantage is simple algebraic structure of Cl(2n) that can be expressed via tenzor product of standard "building units" and similar with behavior of composite quantum systems. The compact notation with 2n generators also can be used in software for modeling of simple quantum circuits by modern conventional computers.

arXiv:quant-ph/9907079Jul 26, 19996 citations

Quantum computing models for algebraic applications

Nikolay Raychev

International Journal of Scientific and Engineering ResearchAug 25, 20158 citations

This paper presents a unified algebraic framework for

n

-qubit quantum computation rooted in the complex Clifford algebra

Cl(2n, \mathbb{C})

. We leverage the fundamental isomorphism

Cl(2n, \mathbb{C}) \cong M(2^n, \mathbb{C})

to directly identify the algebra of all

n

-qubit quantum operators (including states, gates, and observables) with elements of

Cl(2n, \mathbb{C})

. This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors

\rho \in Cl(2n, \mathbb{C})

satisfying standard physical constraints (

\rho^\dagger = \rho

2^n \langle \rho \rangle_0 = 1

, positivity), and unitary gates

U

as multivectors obeying

U^\dagger U = I

. Quantum evolution

\rho' = U\rho U^\dagger

and measurements

p(m) = 2^n \langle P_m \rho \rangle_0

are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals (

S = Cl(2n, \mathbb{C})\prod_j f_j f_j^\dagger

), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.