An Online Calculator for Quantum Computing Operations Based on Geometric Algebra

Abstract

In this paper, we present Geometric Algebra as a powerful language to describe quantum operations using its geometric intuitiveness. Using the web-based GAALOPWeb, an online geometric algebra algorithm optimizer for computing with qubits, we describe new formulations for the NOT operation, as well as a strategy to describe the Z gate and especially the Hadamard operation both for one and multiple qubits.

Current survey of Clifford geometric algebra applications

“…GA is a powerful language to describe quantum operations using its geometric intuitiveness 238 . Using the web‐based GAALOPWeb, an online GA algorithm optimizer for computing with qubits, and new formulations for the NOT operation are described, as well as a strategy to formulate Z gates and Hadamard operations both for one and multiple qubits.…”

Quantum Register Algebra: the mathematical language for quantum computing

“…Quantum computing, currently, is based on working with complex numbers, matrices of complex numbers and tensor computing in order to handle operations with arbitrary numbers of qubits. Some papers on geometric algebras and quantum computing demonstrate application of geometric algebras G 4,1 (relativistic case) and G 3,0 (nonrealtivistic case), [3], geometric algebra G n,n , [1] and finally a complex Clifford algebra C n , [2].…”

“…In order to compute with QRA, we use GAALOP, a stand alone geometric algebra algorithm optimizer [Ref for GAALOP]. We extended the GAALOP tool presented in [1] for the QRA support based on the definitions presented in Sections 4 and 5. The full implementation is shown in Sect.…”

Complex Clifford algebra in repeated quantum prisoner's dilemma

“…Clifford algebra is considered to be an appropriate language for various applications in physics and geometry [5][6][7] and can be efficiently implemented for the above-mentioned purposes. 8 Complex Clifford algebra (CCA) allows us to work with the quantum states intuitively, as well as to conveniently describe qubit systems of higher dimensions. This can be achieved through geometric properties of the considered algebras.…”

“…Since the quantum game theory is closely related to quantum computing, 4 it is crucial to choose the appropriate apparatus for the representation of the quantum operations, that is, application of strategies, and qubits present in the quantum system, that is, players' decisions. Clifford algebra is considered to be an appropriate language for various applications in physics and geometry 5–7 and can be efficiently implemented for the above‐mentioned purposes 8 . Complex Clifford algebra (CCA) allows us to work with the quantum states intuitively, as well as to conveniently describe qubit systems of higher dimensions.…”