Quantum gates are traditionally described using matrix and tensor-product formalisms, representations that provide limited geometric intuition. In this work, we develop a formulation of the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework in order to obtain useful quantum gate decompositions. We show that the Pauli group is naturally identified with the group of blades up to a global phase, providing an intuitive geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of π/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical performance reveals remarkably compact decompositions of Clifford operators. Finally, we show that Clifford+T universality also acquires a natural geometric interpretation through π/8-rotors within this framework. This work highlights Geometric Algebra as a potential geometric tool for quantum computation applications.
This paper introduces the Geometric-Algebraic Hybrid Quantum Computing (GAHQ) framework, a novel algorithmic approach designed to efficiently perform high-precision geometric operations and leverage quantum entanglement for accelerated computation and storage. Traditional methods struggle with the complexities of accurate geometric calculations, prompting the development of this framework. GAHQ integrates geometric transformations with algebraic operations, employing quantum entanglement to enhance computational efficiency and storage capabilities. This paper details the design of the GAHQ algorithm, its theoretical advantages, and initial experimental results demonstrating its potential for significant advancements in geometric computation and quantum information processing.
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.
This paper explores the development of a novel probabilistic quantum computing framework, termed a Geometric Extension, designed to enhance the capabilities of quantum computation beyond traditional linear algebra. The core idea centers around leveraging quantum superposition and entanglement to model and simulate more complex geometric structures and quantum systems. We introduce a new geometric algorithm based on quantum superposition and entanglement, which allows for the efficient exploration of geometric spaces and the simulation of intricate patterns. The paper details the design of this algorithm, its potential advantages, and a preliminary examination of its efficacy in tackling specific geometric problems. We argue that this geometric extension represents a significant step towards unlocking new applications of quantum computation, particularly in areas requiring the representation and manipulation of geometric data.
Abstract: This paper introduces the Geometric-Algebraic Hybrid Quantum Acceleration (GAQA) framework, a novel computational model designed to efficiently perform high-precision geometric computations and leverage quantum entanglement for accelerated calculations and storage. The framework integrates geometric transformations with algebraic operations, utilizing quantum entanglement to achieve significant speedups compared to traditional methods. We present a detailed explanation of the GAQA architecture, including the core mechanism of entanglement-enhanced geometric manipulation and its impact on computational efficiency. The paper details the implementation of GAQA, showcases its performance on a range of geometric problems, and discusses potential future research directions.
Abstract The use of graphical language in quantum computing for the representation of algorithms, although intuitive, is not very useful for different tasks such as the description of quantum circuits in text environments, the calculation of quantum states or the optimization of quantum circuits. While classical circuits can be represented either by circuit graphs or by Boolean expressions, quantum circuits have until now predominantly been illustrated as circuit graphs because no formal language for quantum circuits that allows algebraic manipulations has so far been accepted. This work proposes a means to represent quantum circuits in a convenient and concise manner, similar to the way Boolean expressions are used in classical circuits. The proposed notation allows the consistent and parameterized description of quantum algorithms, as well as the easy handling of the elements that compose it to achieve powerful optimizations in the number of gates of the circuits. To visualize it, a software implementation of an algebraic quantum circuit framework has been made, which allows describing quantum circuits, as well as their respective state vectors, using the proposed algebraic language.