Mathematical Models of Contemporary Elementary Quantum Computing Devices

TL;DR

This paper reviews mathematical models of elementary quantum gates in contemporary quantum devices like cavity QED, ion traps, and quantum dots, focusing on their physical properties and derivations of fundamental gates.

Contribution

It provides a self-contained survey of the mathematical derivations of 1-bit and 2-bit quantum gates for key quantum computing devices.

Findings

Derivation of 1-bit unitary rotation gates from laser-atom interactions.

Derivation of 2-bit quantum phase and CNOT gates.

Analysis of physical properties of cavity QED, ion traps, and quantum dots.

Abstract

Computations with a future quantum computer will be implemented through the operations by elementary quantum gates. It is now well known that the collection of 1-bit and 2-bit quantum gates are universal for quantum computation, i.e., any n-bit unitary operation can be carried out by concatenations of 1-bit and 2-bit elementary quantum gates. Three contemporary quantum devices--cavity QED, ion traps and quantum dots--have been widely regarded as perhaps the most promising candidates for the construction of elementary quantum gates. In this paper, we describe the physical properties of these devices, and show the mathematical derivations based on the interaction of the laser field as control with atoms, ions or electron spins, leading to the following: (i) the 1-bit unitary rotation gates; and (ii) the 2-bit quantum phase gates and the controlled-not gate. This paper is aimed at providing a sufficiently self-contained survey account of analytical nature for mathematicians, physicists and computer scientists to aid interdisciplinary understanding in the research of quantum computation.