Celebratio Mathematica

article D. Das Sarma, M. H. Freedman, and C. Nayak: “Topologically-protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94 : 6 (2005), pp. 166802. ArXiv cond-mat/0412343

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at a Landau-level filling fraction \( \nu = 5/2 \). This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the \( \nu = 5/2 \) state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically protected qubit on which a logical Not operation is performed by quasiparticle braiding. Using the measured excitation gap at \( \nu = 5/2 \), we estimate the error rate to be \( 10^{-30} \) or lower.

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M. Freedman, S. Das Sarma, and C. Nayak: “Topological quantum computation,” Physics Today 59 : 7 (July 2006), pp. 32–38.

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article C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma: “Non-abelian anyons and topological quantum computation,” Rev. Modern Phys. 80 : 3 (2008), pp. 1083–1159. MR 2443722 Zbl 1205.81062 ArXiv 0707.1889

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the \( \nu =5/2 \) state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the \( \nu =5/2 \) fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

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Ady Stern	Related
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techreport P. Bonderson, S. Das Sarma, M. Freedman, and C. Nayak: A blueprint for a topologically fault-tolerant quantum computer. Preprint, March 2010. ArXiv 1003.2856

The advancement of information processing into the realm of quantum mechanics promises a transcendence in computational power that will enable problems to be solved which are completely beyond the known abilities of any “classical” computer, including any potential non-quantum technologies the future may bring. However, the fragility of quantum states poses a challenging obstacle for realization of a fault-tolerant quantum computer. The topological approach to quantum computation proposes to surmount this obstacle by using special physical systems — non-Abelian topologically ordered phases of matter — that would provide intrinsic fault-tolerance at the hardware level. The so-called “Ising-type” non-Abelian topological order is likely to be physically realized in a number of systems, but it can only provide a universal gate set (a requisite for quantum computation) if one has the ability to perform certain dynamical topology-changing operations on the system. Until now, practical methods of implementing these operations were unknown. Here we show how the necessary operations can be physically implemented for Ising-type systems realized in the recently proposed superconductor-semiconductor and superconductor-topological insulator heterostructures. Furthermore, we specify routines employing these methods to generate a computationally universal gate set. We are consequently able to provide a schematic blueprint for a fully topologically-protected Ising based quantum computer using currently available materials and techniques. This may serve as a starting point for attempts to construct a fault-tolerant quantum computer, which will have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and — most importantly — those applications which no one in our classical era has the prescience to foresee.

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