The String-Net Surface Code: Quantum Circuits for Doubled Topological Phases

We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.

arXiv:2405.17379May 27, 20240 citations

Topological Quantum Computation with Surface Codes

Keisuke Fujii

SpringerBriefs in mathematical physicsJan 1, 20153 citations

Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases

Zhu-Xi Luo, Brendan Pankovich, Yuting Hu +1 more

We study the correspondence between boundary excitation distribution spectrum of nonchiral topological orders on an open surface $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$ . Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinder. When $\mathcal{M}$ has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, ${\mathbb{Z}}_{N}$ theories, and the simple non-Abelian case of doubled Fibonacci are demonstrated.We study the correspondence between boundary excitation distribution spectrum of nonchiral topological orders on an open surface $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$. Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinder. When $\mathcal{M}$ has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, ${\mathbb{Z}}_{N}$ theories, and the simple non-Abelian case of doubled Fibonacci are demonstrated.

arXiv:1806.07794Physical review. B./Physical review. BMay 22, 20196 citations

Topological Quantum Computing in Multiple Surface Codes

Paul Webster

Apr 6, 20203 citations

From String Nets to Nonabelions

Lukasz Fidkowski, Michael Hartley Freedman, Chetan Nayak +2 more

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.

arXiv:cond-mat/0610583Communications in Mathematical PhysicsFeb 13, 200937 citations

Quantum Codes for Topological Quantum Computation

Clarice Dias de Albuquerque, Eduardo Brandani da Silva, Waldir Silva Soares

SpringerBriefs in mathematicsJan 1, 20223 citations

We study the correspondence between boundary excitation distribution spectrum of nonchiral topological orders on an open surface

\mathcal{M}

with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as

\mathcal{M}

. Specifically, we focus on the case of generalized string-net models and discuss the cases where

\mathcal{M}

is a disk or a cylinder. When

\mathcal{M}

has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code,

{\mathbb{Z}}_{N}

theories, and the simple non-Abelian case of doubled Fibonacci are demonstrated.