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arXiv:0804.0931 · Physical Review B

Non-Abelian statistics from an Abelian model

James R. Wootton, Ville Lahtinen, Zhenghan Wang, Jiannis K. Pachos

Published Oct 30, 2008• 26 citations
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Abstract

It is well known that the Abelian ${Z}_{2}$ anyonic model (toric code) can be realized on a highly entangled two-dimensional spin lattice, where the anyons are quasiparticles located at the end points of stringlike concatenations of Pauli operators. Here we show that the same entangled states of the same lattice are capable of supporting the non-Abelian Ising model, where the concatenated operators are elements of the Clifford group. The Ising anyons are shown to be essentially superpositions of the Abelian toric code anyons, reproducing the required fusion, braiding, and statistical properties. We propose a string framing and ancillary qubits to implement the nontrivial chirality of this model.

DOI
10.1103/physrevb.78.161102

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It is well known that the Abelian Z2{Z}_{2}Z2​ anyonic model (toric code) can be realized on a highly entangled two-dimensional spin lattice, where the anyons are quasiparticles located at the end points of stringlike concatenations of Pauli operators. Here we show that the same entangled states of the same lattice are capable of supporting the non-Abelian Ising model, where the concatenated operators are elements of the Clifford group. The Ising anyons are shown to be essentially superpositions of the Abelian toric code anyons, reproducing the required fusion, braiding, and statistical properties. We propose a string framing and ancillary qubits to implement the nontrivial chirality of this model.
Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons, i.e., the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.
I define quantum loop models whose degrees of freedom are Ising spins on the square lattice as in the toric code, but where the excitations should have non-abelian statistics. The inner product is topological, allowing a direct implementation of the anyonic fusion matrix on the lattice. It also makes deconfined anyons possible for a variety of values of the weight per loop ddd in the ground state. For d=\sqrt{2}, a gapped non-abelian topological phase can occur with only four-spin interactions.
We construct an SU(N)\mathrm{SU}(N)SU(N) toric code model describing the dynamics of SU(N)\mathrm{SU}(N)SU(N) electric and magnetic fluxes on a two-dimensional torus. We show that the model has N2{N}^{2}N2 topologically distinct ground states $|{\ensuremath{\psi}}_{0}{\ensuremath{\rangle}}_{(\mathsf{p},\mathsf{q})}$, which are the loop states characterized by ${Z}_{N}\ensuremath{\bigotimes}{Z}_{N}$ center charges $(\mathsf{p},\mathsf{q}=0,1,2,...,N\ensuremath{-}1)$. We explicitly construct them in terms of coherent superpositions of all possible spin network states on a torus with Wigner coefficients as their amplitudes. All excited quasiparticle states with SU(N)\mathrm{SU}(N)SU(N) electric charges and magnetic fluxes are constructed. We show that the braiding statistics of these SU(N)\mathrm{SU}(N)SU(N) electric or magnetic quasiparticles or non-Abelian anyons are encoded in the Wigner rotation matrices.
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Non-Abelian anyons can exist as pointlike particles in two-dimensional systems and have particle exchange statistics which are neither bosonic nor fermionic. Like in spin systems, the role of fusion (Heisenberg-like) interactions between anyons has been well studied. However, unlike our understanding of the role of bosonic and fermionic statistics in the formation of different quantum phases of matter, little is known concerning the effect of non-Abelian braid statistics. We explore this physics using an anyonic Hubbard model on a two-legged ladder which includes braiding and nearest-neighbor Heisenberg interactions among anyons. We study two of the most prominent non-Abelian anyon models: the Fibonacci and Ising types. We discover rich phase diagrams for both anyon models and show the different roles of their fusion and braid statistics.
Abstract Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.