Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits

Abstract

Measurement-only (quantum) circuit (MoC) gives possibility to realize the states with rich entanglements, topological orders and quantum memories. This work studies the MoC, in which the projective-measurement operators consist of stabilizers of the toric code and competitive local Pauli operators. The former correspond to terms of the toric code on a triangular lattice and the later to external magnetic and electric fields. We employ efficient numerical stabilizer algorithm to trace evolving states undergoing phase transitions. We elucidate the phase diagram of the MoC system with the observables such as, topological entanglement entropy (TEE), disorder parameters of 1-form symmetries and emergent logical operators. We clarify the locations of the phase transitions through the observation of the above quantities and obtain precise critical exponents to examine if the observables exhibit the critical behavior simultaneously under the MoC and transitions belong to the same universality class. In contrast to the TC Hamiltonian system and toric code MoC on a square lattice, the system on the triangular lattice is not self-dual nor bipartite, and then, coincidence by symmetries, such as critical behaviors across the TC and Higgs/confined phase, does not takes place. Then, the toric code MoC on the triangular lattice provides us a suitable playground to clarify the mutual relationship between the TEE, spontaneous symmetry breaking of the 1-form symmetries, and emergence of logical operators. Obtained results indicate that toric code MoC on the triangular lattice exhibits a few distinct phase transitions with different location and critical exponents, and some of them are closely related with the two-dimensional percolation transition.

15 pages, 7 main figures and 6 figures in Appendix. Version accepted for publication in Phys.Rev.B

References in corpus (34)

• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Topological quantum memory
• Generalized Global Symmetries
• Improved Simulation of Stabilizer Circuits
• Quantum Error Correction for Quantum Memories
• Quantum Entanglement Growth Under Random Unitary Dynamics
• Generalized Symmetries in Condensed Matter
• Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory
• A symmetry principle for Topological Quantum Order
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Entanglement phase transitions in measurement-only dynamics
• Low-energy effective theory of the toric code model in a parallel field
• Robustness of a perturbed topological phase
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models
• Symmetry and Topological Order
• Topological order and criticality in (2+1)D monitored random quantum circuits
• A quantum topological phase transition at the microscopic level
• Self-duality and bound states of the toric code model in a transverse field
• Phase diagram of the toric code model in a parallel magnetic field
• Anyon condensation and the color code
• Gapless Coulomb state emerging from a self-dual topological tensor-network state
• Topology, criticality, and dynamically generated qubits in a stochastic measurement-only Kitaev model
• Strong-to-weak symmetry breaking states in stochastic dephasing stabilizer circuits
• Critical behavior of Fredenhagen-Marcu string order parameters at topological phase transitions with emergent higher-form symmetries
• Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model
• Multipartite entanglement of the topologically ordered state in a perturbed toric code
• Bulk-Measurement-Induced Boundary Phase Transition in Toric Code and Gauge-Higgs Model
• Structured volume-law entanglement in an interacting, monitored Majorana spin liquid
• Phase transition and evidence of fast-scrambling phase in measurement-only quantum circuit
• Intrinsic mixed state topological order in a stabilizer system under stochastic decoherence: Strong-to-weak spontaneous symmetry breaking from percolation point of view
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Measurement-only dynamical phase transition of topological and boundary order in toric code and gauge-Higgs models
• Quantum robustness of the toric code in a parallel field on the honeycomb and triangular lattice