Algebraic representation of Three Qubit Quantum Circuit Problems (original) (raw)

Representation of mathfraksu(8)\mathfrak{su}(8)mathfraksu(8) in Pauli Basis

arXiv: Quantum Physics, 2020

Quantum computation started to become significant field of studies as it hold great promising towards the upgrade of our current computational power. Studying the evolution of quantum states serves as a good fundamental in understanding quantum information which lead to quantum computation. This was assisted with the respective mathematical tools such as Lie group and Lie algebra. In this study, the Lie algebra of mathfraksu(8)\mathfrak{su}(8)mathfraksu(8) is represented in tensor product between three Pauli matrices. This is done by constructing the generalized Gell-Mann matrices and compared to the Pauli basis. This study will explicitly shows the one-to-one correlation of Gell-Mann matrices with the Pauli basis resembled change of coordinates. This is particularly useful when dealing with quantum circuit problems.

Three-dimensional quantum algebras: a Cartan-like point of view

Journal of Physics A: Mathematical and General, 2004

A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related results recovered. Our results give a quantization of all existing three dimensional Lie algebras and reproduce, in the classical limit, the most relevant sector of the complete classification for real three dimensional Lie bialgebra structures given in [1].

Algebras and universal quantum computations with higher dimensional systems

2002

Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It is shown next, how for quantum computation with qubits can be used two-dimensional analog of this Cayley-Weyl matrix algebras, i.e. Clifford algebras, and discussed well known applications to product operator formalism in NMR, Jordan-Wigner construction in fermionic quantum computations. It is introduced universal set of quantum gates for higher dimensional system (``qudit''), as some generalization of these models. Finally it is briefly mentioned possible application of such algebraic methods to design of quantum processors (programmable gates arrays) and discussed generalization to quantum computation with continuous variables.

An Extended Approach for Generating Unitary Matrices for Quantum Circuits

Computers, Materials & Continua, 2019

In this paper, we do research on generating unitary matrices for quantum circuits automatically. We consider that quantum circuits are divided into six types, and the unitary operator expressions for each type are offered. Based on this, we propose an algorithm for computing the circuit unitary matrices in detail. Then, for quantum logic circuits composed of quantum logic gates, a faster method to compute unitary matrices of quantum circuits with truth table is introduced as a supplement. Finally, we apply the proposed algorithm to different reversible benchmark circuits based on NCT library (including NOT gate, Controlled-NOT gate, Toffoli gate) and generalized Toffoli (GT) library and provide our experimental results.

An Introduction to Quantum Algebras and Their Applications

Arxiv preprint math-ph/0003018, 2000

I shall give a very elementary introduction to the topic of quantum algebras and mention a few physical applications. Quantum algebras, or quantum groups, ex-tend the domain of classical group theory and constitute a new and growing field of mathematics with vast potential for ...

A Geometric Algebra Perspective on Quantum Computational Gates and Universality in Quantum Computing

Advances in Applied Clifford Algebras, 2011

We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic configuration space), we present an explicit algebraic description of one and two-qubit quantum states together with a MSTA characterization of one and two-qubit quantum computational gates. Second, using the above mentioned characterization and the GA description of the Lie algebras SO (3) and SU (2) based on the rotor group Spin + (3, 0) formalism, we reexamine Boykin's proof of universality of quantum gates. We conclude that the MSTA approach does lead to a useful conceptual unification where the complex qubit space and the complex space of unitary operators acting on them become united, with both being made just by multivectors in real space. Finally, the GA approach to rotations based on the rotor group does bring conceptual and computational advantages compared to standard vectorial and matricial approaches.

Matrix Representation of Quantum Gates

International Journal of Computer Applications, 2017

The field of quantum computing is growing rapidly and there is a surprisingly large literature. Research in this area includes the design of quantum reversible circuits and developing quantum algorithms for the models of quantum computing. This paper is focused on representing quantum reversible gates in matrix form. In turn these matrices can be used to develop quantum circuits with help of K-Map. Also this paper gives the historical development of quantum algorithms and basics concepts in quantum compuation.

A mathematica program for constructing quantum circuits and computing their unitary matrices

Physics of Particles and Nuclei Letters, 2009

One reason for this is the potential ability of a quantum computer to do certain computational tasks much more efficiently than they can be done by any classical computer Two the most famous examples of such calculations are Shor's algorithm for efficient factorization of large integers and Grover's algorithm of element search in an unsorted list.

Quantum-Logic Synthesis of Hermitian Gates

ACM Journal on Emerging Technologies in Computing Systems, 2015

In this article, the problem of synthesizing a general Hermitian quantum gate into a set of primary quantum gates is addressed. To this end, an extended version of the Jacobi approach for calculating the eigenvalues of Hermitian matrices in linear algebra is considered as the basis of the proposed synthesis method. The quantum circuit synthesis method derived from the Jacobi approach and its optimization challenges are described. It is shown that the proposed method results in multiple-control rotation gates around the y axis, multiple-control phase shift gates, multiple-control NOT gates, and a middle diagonal Hermitian matrix, which can be synthesized to multiple-control Pauli Z gates. Using the proposed approach, it is shown how multiple-control U gates, where U is a single-qubit Hermitian quantum gate, can be implemented using a linear number of elementary gates in terms of circuit lines with the aid of one auxiliary qubit in an arbitrary state.