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arXiv:1112.5385 · Physical Review A

Universal quantum computation with continuous-variable Abelian anyons

Darran F. Milne, Natalia Korolkova, Peter van Loock

Published May 29, 2012• 7 citations
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Abstract

We describe how continuous-variable Abelian anyons, created on the surface of a continuous-variable analog of Kitaev's lattice model can be utilized for quantum computation. In particular, we derive protocols for the implementation of quantum gates using topological operations. We find that the topological operations alone are insufficient for universal quantum computation, which leads us to study additional nontopological operations such as offline squeezing and single-mode measurements. It is shown that these in conjunction with a non-Gaussian element allow for universal quantum computation using continuous-variable Abelian anyons.

DOI
10.1103/physreva.85.052325

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We describe how continuous-variable Abelian anyons, created on the surface of a continuous-variable analog of Kitaev's lattice model can be utilized for quantum computation. In particular, we derive protocols for the implementation of quantum gates using topological operations. We find that the topological operations alone are insufficient for universal quantum computation, which leads us to study additional nontopological operations such as offline squeezing and single-mode measurements. It is shown that these in conjunction with a non-Gaussian element allow for universal quantum computation using continuous-variable Abelian anyons.
We consider topological quantum memories for a general class of abelian anyon models defined on spin lattices. These are non-universal for quantum computation when restricting to topological operations alone, such as braiding and fusion. The effects of additional non-topological operations, such as spin measurements, are studied. These are shown to allow universal quantum computation, while still utilizing topological protection. Our work gives an insight into the relation between abelian models and their non-abelian counterparts.
In this thesis we explore the possibility of creating continuousvariable \nquantum systems that are capable of supporting universal \nquantum computation. We begin by examining the measurement-based \nmodel, which employs sequences of measurements on highly entangled \nresource states, known as a cluster states. We suggest a method for \nthe construction of Gaussian cluster states based on ensembles of atoms \nand quantum non-demolition interactions. We then go on to expand \nour model to allow for the inclusion of light modes as part of the cluster. \nThis yields a new class of states, the composite cluster states. \nThis leads us to propose a new architecture for the measurement-based \nmodel that uses these composite clusters to increase resource e ciency \nand reduce computational errors. \nThe second part of this thesis concerns topological quantum computation. \nIn states exhibiting topological degrees of freedom, quantum \ninformation can be stored as a non-local property of the physical system \nand manipulated by braiding quasiparticles known as anyons. Here \nwe show how these ideas can be extended to continuous variables. We \nestablish a continuous variable analogue of the Kitaev toric code, show \nthat excitations correspond to continuous versions of Abelian anyons \nand investigate their behaviour under the condition of nite squeezing \nof the resource state. \nFinally, we expand our continuous variable topological model to \ninclude non-abelian excitations by constructing superpositions of CV \ntoric code anyons. We derive the fusion and braiding behaviour of \nthese non-abelian excitations and nd that they correspond to a CV \nanalog of Ising anyons. Using these resources, we go on to suggest \na computational scheme that encodes qubits within the fusion spaces \nof the CV Ising anyons and derive one- and two-qubit quantum gates \noperations that are implemented in a topological manner.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
We consider topological quantum memories for a general class of abelian anyon models defined on spin lattices. These are non-universal for quantum computation when restricting to topological operations alone, such as braiding and fusion. The effects of additional non-topological operations, such as spin measurements, are studied. These are shown to allow universal quantum computation, while still utilizing topological protection. Our work gives an insight into the relation between abelian models and their non-abelian counterparts.
Anyons are quasiparticles that may be realized in two dimensional systems. They come in two types, the simpler Abelian anyons and the more complex non-Abelian anyons. Both of these have been considered as a means for quantum computation, but non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. First the means to perform quantum computation with Abelian anyon models is considered. These models, like many non-Abelian models, cannot realize universal quantum computation by braiding alone. Non-topological operations must be used in addition, whose complexity depends on the physical means by which the anyons are realized. Here we consider anyons based on spin lattice models, with single spin measurements playing the role of non-topological operations. The computational power achieved by various kinds of measurement is explored and the requirements for universality are determined. The possibility to simulate non-Abelian anyons using Abelian ones is then considered. Finally, a non-Abelian quantum memory is dissected in order to determine the means by which it provides fault-tolerant storage of information. This understanding is then employed to build equivalent quantum memories with Abelian anyon models. The methodology provides with the means to demonstrate that Abelian models have the capability to simulate non-Abelian anyons, and to realize the same computational power and fault-tolerance as non Abelian models. Apart from the intellectual interest in relating topological models with each other, and of understanding the properties of non-Abelian anyons in terms of the simpler Abelian ones, these results can also be applied in the lab. The simpler structure of Abelian anyons means that their physical realization is more straightforward. The demonstration of non-Abelian properties with Abelian models therefore allows features of non-Abelian anyons to be realized with present and near future technology. Based on this possibility, proposals are made here for proof of principle experiments.
Quantum Computing with Z2\mathbb{Z}_2Z2​ Abelian anyon system
Topological quantum computers provide a fault-tolerant method for performing quantum computation. Topological quantum computers manipulate topological defects with exotic exchange statistics called anyons. The simplest anyon model for universal topological quantum computation is the Fibonacci anyon model, which is a non-abelian anyon system. In non-abelian anyon systems, exchanging anyons always results a unitary operations instead of a simple phase changing in abelian anyon systems. So, non-abelian anyon systems are of the interest to topological quantum computation. Up till now, most people still hold the belief that topological quantum computions can be implemented only on the non-abelian anyon systems. But actually this is not true. Inspired by extrinsic semiconductor technology, we suggest that abelian anyon systems with defects also support topological quantum computing. In this letter, we report a topological quantum computer prototype based on Z2\mathbb{Z}_2Z2​ abelian anyon system.