A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems

Abstract

Graphs and recently hypergraphs have been known as an important tool for considering different properties of quantum many-body systems. In this paper, we study a mapping between an important class of quantum systems namely quantum Calderbank-Shor-Steane (CSS) codes and Ising-like systems by using hypergraphs. We show that the Hamiltonian corresponding to a CSS code on a hypergraph $H$ which is perturbed by a uniform magnetic field is mapped to Hamiltonian of a Ising-like system on dual hypergraph $\tilde{H}$ in a transverse field. Interestingly, we show that a strong regime of couplings in one of the systems is mapped to a weak regime of couplings in another one. We also give some applications for such a mapping where we study robustness of different topological CSS codes against a uniform magnetic field including Kitaev's toric codes defined on graphs and color codes in different dimensions. We show that a perturbed Kitaev's toric code on an arbitrary graph is mapped to an Ising model in a transverse field on the same graph and a perturbed color code on a D-colex is mapped to a Ising-like model on a D-simplicial lattice in a transverse field. In particular, we use these results to explicitly compare the robustness of TC on different graphs in different dimensions. Interestingly, our results show that the robustness of such topological codes defined on graphs decreases with increasing dimension. Furthermore, we also use the duality mapping for some self-dual models where we exactly derive the point of phase transition.

9 pages, 10 figures, submitted to PRB

References in corpus (34)

• Fault-tolerant quantum computation by anyons
• Good Quantum Error-Correcting Codes Exist
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Topological quantum memory
• Multi-party entanglement in graph states
• Topological Quantum Distillation
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• Topological quantum order: stability under local perturbations
• Superconductors are topologically ordered
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Topological order in an exactly solvable 3D spin model
• Quantum memories at finite temperature
• Robustness of a perturbed topological phase
• Exactly solvable Kitaev model in three dimensions
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• Quantum Hypergraph States
• Fractal symmetries: Ungauging the cubic code
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Entanglement, fidelity and topological entropy in a quantum phase transition to topological order
• Encoding Hypergraphs into Quantum States
• Extreme violation of local realism in quantum hypergraph states
• Robustness of a topological phase: Topological color code in parallel magnetic field
• Quantum phase transitions out of a Z2 x Z2 topological phase
• On the robustness of topological quantum codes: Ising perturbation
• The Kitaev-Ising model, Transition between topological and ferromagnetic order
• Baxter-Wu model in a transverse magnetic field
• Locally inequivalent four qubit hypergraph states
• Multipartite Entanglement and Hypergraph states of three qubits
• Quantum phase transition in the $Z_3$ Kitaev-Potts model
• Local Pauli stabilizers of symmetric hypergraph states
• A quantum phase transition from $Z_2\times Z_2$ to $Z_2$ topological order
• General scheme for preparation of different topological states on the cluster states

Cited by in corpus (9)

• Quantum robustness and phase transitions of the 3D Toric Code in a field
• Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field
• Dual correspondence between classical spin models and quantum CSS states
• Duality family of scalar field
• Topological line in frustrated Toric code models
• Phase transition in a noisy Kitaev toric code model
• Classical criticality establishes quantum topological order
• Noisy Toric code and random bond Ising model: The error threshold in a dual picture
• Independent e- and m-anyon confinement in the parallel field toric code on non-square lattices