Quantum Register Algebra: the mathematical language for quantum computing

We introduce Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present GAALOP (Geometric Algebra Algorithms Optimizer) implementation of our approach. Using the QRA basis vectors definitions given in Section 4 and the framework based on the de Witt basis presented in Section 5, we are able to fully describe and compute with QRA in GAALOP using the geometric product. We illustrate the intuitiveness of this computation by presenting the QRA form for the well known SWAP operation on a two qubit register.Comment: 11 page

Basis-free solution to Sylvester equation in Clifford algebra of arbitrary dimension

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case $n=4$, the proofs for the case $n=5$ and the case of arbitrary dimension $n$. The results can be used in symbolic computation.Comment: 19 page

Clifford Multivector Toolbox (for MATLAB)

matlab ® is a numerical computing environment oriented towards manipulation of matrices and vectors (in the linear algebra sense, that is arrays of numbers). Until now, there was no comprehensive toolbox (software library) for matlab to compute with Clifford algebras and matrices of multivectors. We present in the paper an account of such a toolbox, which has been developed since 2013, and released publically for the first time in 2015. The paper describes the major design decisions made in implementing the toolbox, gives implementation details, and demonstrates some of its capabilities, up to and including the LU decomposition of a matrix of Clifford multivectors

Quantization of Two- and Three-player Cooperative Games Based on QRA

In this paper, a novel quantization scheme for cooperative games is proposed. The considered circuit is inspired by the Eisert-Wilkens-Lewenstein protocol modified to represent cooperation between players and extended to $3$-qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. To distribute players' payoffs after the measurement, the expected value of the Shapley value with respect to quantum probabilities is employed. We study how entanglement, representing the level of pre-agreement between players, affects the final distribution of utility. The paper also demonstrates how all necessary calculations can be automatized using the Quantum Register Algebra and GAALOP software