Spin qubits in solid-state structures

Abstract

t is remarkable that today's computers, after the tremendous development during the last 50 years, are still essentially described by the mathematical model formulated by AIan Turing in the 1930's. Turing's model describes computers which operate according to the laws ofclassical physics. What would happen ifa computer was operating according to the quantum laws? Physi­ cists and computer scientists have been interested in this question since the early 1980's, but research in quantum computation real­ ly started to flourish after 1994 when Peter Shor discovered a quantum algorithm to find prime factors oflarge integers effi­ ciently, a problem which is intrinsically hard for any classical computer (see (1) for an introduction into quantum computa­ tion). The lack ofan algorithm for efficient factoring on a classical machine is actually the basis ofthe widely used RSA encryption scheme. Phase coherence needs to be maintained for a sufficient­ ly long time in the memory of a quantum computer. This may sound like a harmless requirement, but in fact it is the main rea­ son why the physical implementation of quantum computation is so difficult. Usually, a quantum memory is thought of as a set oftwo-level systems, named quantum bits, or qubits for short. In analogy to the classical bit, two orthogonal computational basis states 10) and 11) are defined. The textbook example of a quan­ tum two-level system is the 1/2 of, say, an electron, where one can identify the spin up state with 10) and the spin down state with \1). While several other two-level systems have been proposed for quantum computing, we will devote the majority of our discussion to the potential use ofelectron spins in nanostruc­ tures (such as quantum dots) as qubits. Shor's fadoring algorithm We return to Shor's algorithm, since it allows us to explain many important concepts. At the heart of it lies the quantum Fourier transform (QFT). Given n qubits with an orthonormal basis 10), ...,12-1), the QFT is a unitary2x2 matrix UQFTsuch that