Quantum Computation with Abelian Anyons

A universal quantum computer can be constructed using abelian anyons. Two qubit quantum logic gates such as controlled-NOT operations are performed using topolog-ical effects. Single-anyon operations such as hopping from site to site on a lattice suffice to perform all quantum logic operations. Quantum computation using abelian anyons shares some but not all of the robustness of quantum computation using non-abelian anyons. A wide variety of methods can be used to construct quantum computers in principle (1-18). Essentially any interaction between two quantum degrees of freedom suffices to construct universal quantum logic gates (10-11). In particular, Kitaev (12) has shown that quantum computation can be effected using non-abelian anyons. The resulting quantum computation is intrinsically fault-tolerant. This paper shows that universal quantum computation can be effected using abelian anyons: two-qubit quantum logic gates, such as the controlled-NOT gate, are enacted topologically via the phase factor of e iφ that occurs when one anyon is moved around another. Quantum computation using abelian anyons shares some but not all of the robustness of quantum computation using non-abelian anyons. Possible realizations in terms of solid-state systems and interferometers are discussed, and issues of noise and decoherence are investigated. A quantum logic gate is an operation that transforms quantum-information bearing degrees of freedom. A set of quantum logic gates is universal if arbitrary quantum computations can be built up by repeatedly applying gates from the set to different qubits. To prove that abelian anyons are capable of universal quantum computation, we will show 1 that single anyon quantum logic gates such as those that move an anyon from one site to another form a universal set: universal quantum computation can be effected simply by moving fermions around a lattice or network. Consider the case of anyons moving on a two-dimensional lattice. Each site j of the lattice corresponds to a local mode that can be either occupied by an anyon, |+ j , or unoccupied |− j. For example, the anyons could be quantum-Hall effect excitations on a two-dimensional spatial lattice. Now consider operations that can be performed on anyons. Let b j , b † j be the annihilation and creation operators for the j th mode. First, applying the Hamiltonian A j = b † j b j leaves |− j unchanged and multiplies the state |+ j by a phase. Second, if j and k are two adjacent sites or modes, …