This paper presents a unified algebraic framework for
n-qubit quantum computation rooted in the complex Clifford algebra
Cl(2n,C). We leverage the fundamental isomorphism
Cl(2n,C)≅M(2n,C) to directly identify the algebra of all
n-qubit quantum operators (including states, gates, and observables) with elements of
Cl(2n,C). This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors
ρ∈Cl(2n,C) satisfying standard physical constraints (
ρ†=ρ,
2n⟨ρ⟩0=1, positivity), and unitary gates
U as multivectors obeying
U†U=I. Quantum evolution
ρ′=UρU† and measurements
p(m)=2n⟨Pmρ⟩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,C)∏jfjfj†), 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.
In this work, we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially, we present a general formulation through elements of minimal left ideals in tensor products of Clifford algebras. Posteriorly, we perform some applications in quantum computation: qubits, entangled states, quantum gates, representations of the braid group, quantum teleportation, Majorana operators and supersymmetry. Finally, we discuss advantages compared to standard Hilbert space formulation.
This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
This paper aims at explaining that the key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R 4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R 3. In the three-dimensional rotation group the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding acquired to derive the free-space Dirac equation from scratch proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine's statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
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.