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arXiv:2110.13841 · Physical Review A

SU(N) toric code and non-Abelian anyons

Manu Mathur, Atul Rathor

Published May 18, 2022• 4 citations
arXivHTMLPDF

Abstract

We construct an $\mathrm{SU}(N)$ toric code model describing the dynamics of $\mathrm{SU}(N)$ electric and magnetic fluxes on a two-dimensional torus. We show that the model has ${N}^{2}$ 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 $\mathrm{SU}(N)$ electric charges and magnetic fluxes are constructed. We show that the braiding statistics of these $\mathrm{SU}(N)$ electric or magnetic quasiparticles or non-Abelian anyons are encoded in the Wigner rotation matrices.

DOI
10.1103/physreva.105.052423

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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 ∣\ensuremathψ0\ensuremath⟩(p,q)|{\ensuremath{\psi}}_{0}{\ensuremath{\rangle}}_{(\mathsf{p},\mathsf{q})}∣\ensuremathψ0​\ensuremath⟩(p,q)​, which are the loop states characterized by ZN\ensuremath⨂ZN{Z}_{N}\ensuremath{\bigotimes}{Z}_{N}ZN​\ensuremath⨂ZN​ center charges (p,q=0,1,2,...,N\ensuremath−1)(\mathsf{p},\mathsf{q}=0,1,2,...,N\ensuremath{-}1)(p,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.
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.
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.
FINDING: Quantinuum's universal topological gate via non-Abelian anyons; Kitaev toric code; anyon statistics as continuous parameter κ; torus fundamental group abelian (π₁(T²)=ℤ²). MATH: - Toric code ground state degeneracy (GSD) on genus-g surface: GSD = 4^g (for ℤ₂ anyons). For torus g=1, GSD=4 — matches π₁(T²)=ℤ² (two independent loops). - Verlinde formula: N_{ab}^c = Σ_x (S_{ax} S_{bx} S_{cx}^*)/S_{0x}, where S is modular S-matrix. For Fibonacci anyons (non-Abelian), quantum dimension φ = (1+√5)/2 = 1.618… satisfies φ² = φ+1. - Anyon statistical parameter κ (from arXiv 2210.10776): continuous shift κ → κ+δκ; braiding phase e^{iπκ}. - Kitaev toric code Hamiltonian: H = -Σ_v A_v - Σ_p B_p, with A_v = Π_{i∈v} σ_i^x, B_p = Π_{i∈p} σ_i^z. Excitations: e (charge), m (flux), ε = e×m (fermion). CONNECTION: - Fibonacci anyon quantum dimension φ = 1.618 — the golden ratio. Verlinde coefficients for Fibonacci: N_{ττ}^τ = 1, N_{ττ}^1 = 1, giving fusion τ×τ = 1+τ. This is the gold Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
We exploit the spin network properties of the magnetic eigenstates of SU(2) Hamiltonian lattice gauge theory and use the Wilson loop operators to obtain a wide class of new identities amongst 3nj Wigner coefficients. We also show that the topological ground states of the SU(2) toric code Hamiltonian lead to Wigner 3nj identities with non-trivial phases. The method is very general and involves only the eigenvalue equations of any gauge invariant operator and their solutions. Therefore, it can be extended to any higher dimensional spin networks as well as larger SU(N) groups.