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Non-abelian anyons and topological quantum computation. (English) Zbl 1205.81062
Summary: Topological quantum computation is thought to be one of the most promising approaches to constructing a fault-tolerant quantum computer, due to the existence of non-abelian anyons abiding by non-abelian braiding statistics. This survey article, in a rather sweeping way, introduces non-abelian anyons and topological quantum computation together with their interconnections. It is shown how they can be realized in physical systems, particularly in several fractional quantum Hall states. The article gives the reader a glimpse of knot theory, topological quantum field theory, conformal field theory, quantum Hall effects and quantum computation along with the physics of gallium arsenide devices.
The main body of the article consists of three parts II-IV. The authors have made Part II accessible to all possible readers, introducing concepts only at a qualitative level. Beginning with the concept of braiding statistics in \(2+1\) dimension and the very definition of a non-abelian anyon followed by the review of basic ideas of quantum computation and the problems of errors and decoherence, they discuss how non-abelian statistics naturally leads to the idea of topological quantum computation, which is supposed to be error-free. It is argued that the non-abelian quantum Hall systems are the most likely arena for observing non-abelian anyons and so for producing a topological quantum computer.
“Topological quantum computation can become a reality only if some physical system condenses into a non-abelian topological phase,” where a system is said to be in a topological phase if its low energy effective field theory is a topological quantum field theory. The universal low-energy long-distance physics of such phases is described in Part III, where the focus is on a sequence of universality classes of non-abelian topological phases associated with \(\mathrm{SU}(2)_{k}\) Chern-Simon theory. It is also discussed how such phases can experimentally be detected in the quantum Hall regime and some others.
It is discussed in Part IV how quasiparticles in topological phases are to be used in quantum computation. Beginning with the case of \(\mathrm{SU}(2)_{2}\), the leading candidate for the \(\upsilon=5/2\) fractional quantum Hall state, the authors show how qubits and gates can be manipulated in a gated GaA device supporting this quantum Hall state. It is discussed why quasiparticle braiding alone does not suffice for universal quantum computation and how this limitation of the \(\upsilon=5/2\) state can be overcome. It is also debated how topological quantum computations are to be performed within the simplest non-abelian theory capable of universal topological quantum computation, which is usually called the Fibonacci anyon theory. It is then argued that the \(\mathrm{SU}(2)_{k}\) theories underlie universal topological quantum computation for all integers \(k\) except for \(k=1,2,4\). Finally the authors investigate the physical processes that may cause errors in a topological quantum computer.
Part V is concerned with concluding remarks. The 1980s has witnessed some collaborative efforts to revolutionize computation by use of quantum mechanics. The old paradigm of digital computers is based upon the physics of a single electron in semiconductors. The information revolution that is in progress demands the understanding and manipulation of systems of strongly interacting electrons. Modern condensed matter physics is powerful enough to provide us with effective tools to analyze them, namely, a mashup of renormalization group, conformal field theory, Bethe ansatz, dualities and numerics. The outstanding problem is to create, manipulate and classify topological states of matter. The quantum revolution will allow us to compute in superposition, which admits efficient solutions to such problems as factoring, finding units in number fields, etc. The authors discuss fundamental issues of physics to be tackled in order to realize this new paradise.

MSC:
81P68 Quantum computation
81V70 Many-body theory; quantum Hall effect
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