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The Toric Code as a Phase of Matter and a TQFT

Steven H. Simon

Published Sep 29, 2023• 0 citations
Publisher

Abstract

Abstract The Toric Code can also be interpreted as a phase of matter. The vertex and plaquette operators become terms of the Hamiltonian, and the ground state space becomes the code space. The various types of errors become the various types of quasiparticle excitations. We explore the braiding properties of these excitations and identify properties of the TQFT described by this phase of matter such as the S- and T-matrices. We also describe this anyon theory with a simple charge-flux model. We repeat the exercise for the ZN toric code.

DOI
10.1093/oso/9780198886723.003.0028

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Topological Quantum Error Correction via Toric Codes and Anyon Braiding — E8 Intelligence Research

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FINDING: Topological quantum error correction (TQEC) uses lattice geometry—specifically the toric code and surface codes—to encode logical qubits non-locally, with error thresholds tied to lattice percolation and anyon braiding. MATH: - Toric code: Hamiltonian \( H = -\sum_v A_v - \sum_p B_p \), with star operators \( A_v = \prod_{i \in v} X_i \) and plaquette operators \( B_p = \prod_{i \in p} Z_i \). Ground state degeneracy on genus-\(g\) surface: \( 4^g \). - Logical operators: non-contractible loops on torus — \( \bar{X}, \bar{Z} \) each have weight \( L \) (lattice length), minimum weight scales as \( O(L) \) for \( L \times L \) lattice. - Error threshold: surface code ~1.1% (per-qubit depolarizing), toric code ~10.1% (bit-flip only) — related to percolation threshold on square lattice \( p_c \approx 0.593 \) (site) and \( p_c \approx 0.5 \) (bond). - Anyon statistics: \( e \) (charge) and \( m \) (flux) excitations obey \( \mathbb{Z}_2 \) fusion: \( e \times m = \psi \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 24, 2026•0 citations
Abstract The Toric Code can also be interpreted as a phase of matter. The vertex and plaquette operators become terms of the Hamiltonian, and the ground state space becomes the code space. The various types of errors become the various types of quasiparticle excitations. We explore the braiding properties of these excitations and identify properties of the TQFT described by this phase of matter such as the S- and T-matrices. We also describe this anyon theory with a simple charge-flux model. We repeat the exercise for the ZN toric code.
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.
Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.
Kitaev's toric code is an exactly solvable model with Z2{\mathbb{Z}}_{2}Z2​-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental realization remains an open challenge. Here, we propose a building block for Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically nontrivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we outline experimental signatures of the ground-state wave function and introduce a minimal braiding protocol. Detecting a $\ensuremath{\pi}$-phase shift between Ramsey fringes in this protocol reveals the anyonic excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators.
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.
Abstract Topologically ordered matter is famously robust to small perturbations. If a Hamiltonian has a topologically ordered (TQFT) ground state with a gap, then adding a small pertrubation (on the scale of the gap) to the Hamiltonian cannot change its TQFT properties. We explore this robustness with the example of the toric code. We show that the robustness is guaranteed by the fact that the toric code has a protected code space. We further discuss how the properties of the quasiparticles are also unchanged under perturbations of the Hamiltonian. These properties might be unsurprising, given the rigidity of the properties of TQFTs. We define the notion of topological order and give a definition of a topological phase of matter.
FINDING: Topological quantum error correction (TQEC) uses lattice geometry—specifically the toric code and surface codes—to encode logical qubits non-locally, with error thresholds tied to lattice percolation and anyon braiding. MATH: - Toric code: Hamiltonian H=−∑vAv−∑pBp H = -\sum_v A_v - \sum_p B_p H=−∑v​Av​−∑p​Bp​, with star operators Av=∏i∈vXi A_v = \prod_{i \in v} X_i Av​=∏i∈v​Xi​ and plaquette operators Bp=∏i∈pZi B_p = \prod_{i \in p} Z_i Bp​=∏i∈p​Zi​. Ground state degeneracy on genus-ggg surface: 4g 4^g 4g. - Logical operators: non-contractible loops on torus — Xˉ,Zˉ \bar{X}, \bar{Z} Xˉ,Zˉ each have weight L L L (lattice length), minimum weight scales as O(L) O(L) O(L) for L×L L \times L L×L lattice. - Error threshold: surface code ~1.1% (per-qubit depolarizing), toric code ~10.1% (bit-flip only) — related to percolation threshold on square lattice pc≈0.593 p_c \approx 0.593 pc​≈0.593 (site) and pc≈0.5 p_c \approx 0.5 pc​≈0.5 (bond). - Anyon statistics: e e e (charge) and m m m (flux) excitations obey Z2 \mathbb{Z}_2 Z2​ fusion: e×m=ψ e \times m = \psi e×m=ψ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com