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Register a new userSystematic distillation of composite Fibonacci anyons using one mobile quasiparticle
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Authors
Ben W. Reichardt
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A topological quantum computer should allow intrinsically fault-tolerant quantum computation, but there remains uncertainty about how such a computer can be implemented. It is known that topological quantum computation can be implemented with limited quasiparticle braiding capabilities, in fact using only a single mobile quasiparticle, if the system can be properly initialized by measurements. It is also known that measurements alone suffice without any braiding, provided that the measurement devices can be dynamically created and modified. We study a model in which both measurement and braiding capabilities are limited. Given the ability to pull nontrivial Fibonacci anyon pairs from the vacuum with a certain success probability, we show how to simulate universal quantum computation by braiding one quasiparticle and with only one measurement, to read out the result. The difficulty lies in initializing the system. We give a systematic construction of a family of braid sequences that initialize to arbitrary accuracy nontrivial composite anyons. Instead of using the Solovay-Kitaev theorem, the sequences are based on a quantum algorithm for convergent search.
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In a topological quantum computer, universal quantum computation is performed by dragging quasiparticle excitations of certain two dimensional systems around each other to form braids of their world lines in 2+1 dimensional space-time. In this paper we show that any such quantum computation that can be done by braiding $n$ identical quasiparticles can also be done by moving a single quasiparticle around n-1 other identical quasiparticles whose positions remain fixed.
Recent experiments have focused attention on the properties of chains of atoms in which the atoms are either in their ground states or in highly excited Rydberg states which block similar excitations in their immediate neighbors. As the low energy Hilbert space of such chains is isomorphic to that of a chain of Fibonacci anyons, they have been proposed as a platform for topological quantum computation and for simulating anyon dynamics. We show that generic local operators in the Rydberg chain correspond to non-local anyonic operators that do not preserve a topological symmetry of the physical anyons. Consequently, we argue that Rydberg chains do not yield Fibonacci anyons and quantum computation with Rydberg atoms is not topologically protected.
Topological systems, such as fractional quantum Hall liquids, promise to successfully combat environmental decoherence while performing quantum computation. These highly correlated systems can support non-Abelian anyonic quasiparticles that can encode exotic entangled states. To reveal the non-local character of these encoded states we demonstrate the violation of suitable Bell inequalities. We provide an explicit recipe for the preparation, manipulation and measurement of the desired correlations for a large class of topological models. This proposal gives an operational measure of non-locality for anyonic states and it opens up the possibility to violate the Bell inequalities in quantum Hall liquids or spin lattices.
The Fibonacci topological order is the simplest platform for a universal topological quantum computer, consisting of a single type of non-Abelian anyon, $tau$, with fusion rule $tautimestau=1+tau$. While it has been proposed that the anyon spectrum of the $ u=12/5$ fractional quantum Hall state includes a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a quantum Hall system has been lacking. Here we use recently proposed non-Abelian dualities to construct a Fibonacci state of bosons at filling $ u=2$ starting from a trilayer of integer quantum Hall states. Our parent theory consists of bosonic composite vortices coupled to fluctuating $U(2)$ gauge fields, which is related to the standard theory of Laughlin quasiparticles by duality. The Fibonacci state is obtained by clustering the composite vortices between the layers, along with flux attachment, a procedure reminiscent of the clustering picture of the Read-Rezayi states. We further use this framework to motivate a wave function for the Fibonacci fractional quantum Hall state.
An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same effective integer filling factor. We consider such anyons on a torus, and construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and ground state and find that it remains open as the statistical phase is varied. Finally, we obtain from these wave functions, both analytically and numerically, various topological quantities, such the ground state degeneracy, the Chern number and the Hall viscosity.
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