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arXiv:0711.0014

Quantum loop models and the non-abelian toric code

Paul Fendley

Published Nov 1, 2007• 5 citations
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Abstract

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 $d$ in the ground state. For d=\sqrt{2}, a gapped non-abelian topological phase can occur with only four-spin interactions.

DOI
10.48550/arxiv.0711.0014

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•Jan 1, 2016•0 citations
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.
Condensation of quantum loops naturally leads to topological phases with Abelian excitations. Here, I propose that non-Abelian topological phases can arise from merging two (or several) identical Abelian quantum loop condensates. I define merging through a symmetrization operation, which makes the two loop condensates indistinguishable, inducing the possibility of a topological degeneracy in the space of quasiparticles. To illustrate the construction, the case of two identical toric-code quantum loop gases is considered. A spin-1 model for the two-dimensional square lattice is presented for which the resulting merged state is the exact unique ground state. This Hamiltonian involves four-body interactions between spins located at the same plaquette or vertex, which are quadratic in the spin-1 operators. Vertex and plaquette interaction terms are not mutually commuting. The model displays gapped excitations that result from merging anyons in both toric-code copies. These excitations exhibit non-Abelian braiding properties.
Two-dimensional quantum loop gases are elementary examples of topological ground states with Abelian or non-Abelian anyonic excitations. While Abelian loop gases appear as ground states of local, gapped Hamiltonians such as the toric code, we show that gapped non-Abelian loop gases require nonlocal interactions (or nontrivial inner products). Perturbing a local, gapless Hamiltonian with an anticipated "non-Abelian" ground-state wave function immediately drives the system into the Abelian phase, as can be seen by measuring the Hausdorff dimension of loops. Local quantum critical behavior is found in a loop gas in which all equal-time correlations of local operators decay exponentially.
Decohering topological order (TO) is central to the many-body physics of open quantum matter and decoding transitions. We identify statistical mechanical models for decohering non-Abelian TOs, which have been crucial for understanding the error threshold of Abelian stabilizer codes. The decohered density matrix can be described by loop models, whose topological loop weight N is the quantum dimension of the decohering anyon-reducing to the Ising model if N=1. In particular, the Rényi-n moments correspond to n coupled O(N) loop models. Moreover, by diagonalizing the density matrix at maximal error rate, we connect the fidelity between two logically distinct ground states to random O(N) loop and spin models. We find a remarkable stability to quantum channels which proliferate non-Abelian anyons with large quantum dimension, with the possibility of critical phases for smaller dimensions. Intuitively, this stability is due to non-Abelian anyons not admitting finite-depth string operators. We confirm our framework with exact results for Kitaev quantum double models, and with numerical simulations for the non-Abelian phase of the Kitaev honeycomb model. Our work opens up the possibility of non-Abelian TO being robust against maximally proliferating certain anyons, which can inform error-correction studies of these topological memories.
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
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.
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.