Quantum Register Algebra: the mathematical language for quantum computing

Hrdina, J.; Hildenbrand, D.; Návrat, A.; Steinmetz, C.; Alves, R.; Lavor, C.; Vašík, P.; Eryganov, I.

doi:10.1007/s11128-023-04086-y

Cited by 4 publications

(3 citation statements)

References 12 publications

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“…For this reason, we added visual schematic depictions together with background GA preliminary technical details (including, for instance, what appears in Appendix A). Moreover, we were able to highlight most of the very interesting works we have partially inspired throughout these years [18][19][20][21][22][23][24][25][26][27][28]. Finally, we were able to suggest limitations and, at the same time, proposals for future research directions compatible with the current scientific knowledge at the boundaries between GA and quantum computing (with the concept of entanglement playing a prominent role).…”

Section: Introductionmentioning

confidence: 85%

“…[17] and reiterated in the above point [ii] of this paper, is that (complex) operators and operands (i.e., states) are elements of the same (real) space in the GA setting. This fact, in turn, is at the root of the increasing number of works advocating for the use of online calculators capable of performing quantum computing operations based on geometric algebra [20,21,26,27]. We are proud to see that our original work in Ref.…”

Section: Discussionmentioning

confidence: 99%

“…[17] as evident from Refs. [18][19][20][21][22][23][24][25][26][27][28], we present a critical revisitation of our work in Ref. [17].…”

Section: Introductionmentioning

confidence: 99%

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From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Cafaro,

Bahreyni,

Rossetti

2024

Symmetry

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Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.

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Section: Introductionmentioning

confidence: 85%

Section: Discussionmentioning

confidence: 99%

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From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Cafaro,

Bahreyni,

Rossetti

2024

Symmetry

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Complex Clifford algebra in repeated quantum prisoner's dilemma

Eryganov

Hrdina

2022

Math Methods in App Sciences

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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.

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Survey of new applications of geometric algebra

Hitzer,

Kamarianakis,

Papagiannakis

et al. 2023

Math Methods in App Sciences

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This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics.

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Quantum Register Algebra: the mathematical language for quantum computing

Cited by 4 publications

References 12 publications

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Complex Clifford algebra in repeated quantum prisoner's dilemma

Survey of new applications of geometric algebra

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