A unitary matrix in quantum mechanics

Leo Georgy Sapogin

Yumi Nakajima A

A quantum computer is expected to solve some problems far faster than today's computers and may be a promising device with low power consumption.This computer is based on the physical theory of quantum mechanics.In the theory of quantum mechanics, quantum states and evolutions of quantum states are represented by vectors in a Hilbert space and unitary operators, respectively.A unitary matrix can then be regarded as an algorithm in the conventional computer.Since it is generally difficult to develop an arbitrary unitary matrix in a quantum system, quantum computation is carried out by combining elementary operations such as one-or two-qubit operations, where a qubit is the smallest unit of information in quantum computation.A unitary matrix is said to be performed effectively if there exists a quantum circuit composed of a polynomial number of elementary gates.Therefore, translating a unitary matrix into an efficient sequence of elementary gates is a fundamental problem in designing quantum circuits.In this dissertation, the problem involved in synthesizing minimal quantum circuits for carrying out any (large) quantum operation is described.The author considers two quantum systems, namely, the two-level quantum system and the d-level quantum system, where d is any integer greater than two.Here, the two-level quantum system can be regarded as a quantum mechanical analogue of conventional computation, and the d-level quantum system can be regarded as a quantum mechanical analogue of conventional multilevel logic.

Hsiang Shun Chou

Hollis Daniel Williams

Abstract We discuss the significance of superpositions of unitary operators in the formalism of quantum mechanics. We show that with this viewpoint, it can be demonstrated that one can observe a measurement with zero Ozawa uncertainty in a physically realisable feedback set-up which uses polarised photons coupled to spin. We derive a set of conditions under which a linear combination of unitary matrices is also a unitary matrix and confirm that the conditions are met by a realistic quantum gate.

S. G. Eckstein, B. B. Varga, S. Ø. Aks

Sergey Valentinovich Petoukhov, Elena S. Petukhova, Vitaly I. Svirin

Unitary operators are the basis of all calculations in quantum computers and play an important role in quantum mechanics. The presented results show that matrix forms of representation of structured DNA and RNA alphabets can be considered as a set of sparse unitary matrices. These results are useful for the development of models of quantum-algorithmic genetics, from the point of view of which living organisms are quantum-information entities.