Phase diagram of the three-dimensional subsystem toric code

Abstract

Subsystem quantum error-correcting codes typically involve measuring a sequence of non-commuting parity check operators. They can sometimes exhibit greater fault-tolerance than conventional subspace codes, which use commuting checks. However, unlike subspace codes, it is unclear if subsystem codes -- in particular their advantages -- can be understood in terms of ground state properties of a physical Hamiltonian. In this paper, we address this question for the three-dimensional subsystem toric code (3D STC), as recently constructed by Kubica and Vasmer [Nat. Comm. 13, 6272(2022)], which exhibits single-shot error correction (SSEC). Motivated by a conjectured relation between SSEC and thermal stability, we study the zero and finite temperature phases of an associated non-commuting Hamiltonian. By mapping the Hamiltonian model to a pair of 3D Z_2 gauge theories coupled by a kinetic constraint, we find various phases at zero temperature, all separated by first-order transitions: there are 3D toric code-like phases with deconfined point-like excitations in the bulk, and there are phases with a confined bulk supporting a 2D toric code on the surface when appropriate boundary conditions are chosen. The latter is similar to the surface topological order present in 3D STC. However, the similarities between the SSEC in 3D STC and the confined phases are only partial: they share the same sets of degrees of freedom, but they are governed by different dynamical rules. Instead, we argue that the process of SSEC can more suitably be associated with a path (rather than a point) in the zero-temperature phase diagram, a perspective which inspires alternative measurement sequences enabling SSEC. Moreover, since none of the above-mentioned phases survives at nonzero temperature, SSEC of the code does not imply thermal stability of the associated Hamiltonian phase.

23 pages, 22 figures. abstract shortened to meet arxiv requirements, see pdf for full abstract. v2: 25 pages, 22 figures. new background section. accepted version

References in corpus (42)

• Fault-tolerant quantum computation by anyons
• Topological quantum memory
• Quantum Error Correction for Quantum Memories
• Local stabilizer codes in three dimensions without string logical operators
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• Topological quantum order: stability under local perturbations
• Stabilizer Formalism for Operator Quantum Error Correction
• Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Quantum memories at finite temperature
• Protected Qubits and Chern Simons theories in Josephson Junction Arrays
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• A short proof of stability of topological order under local perturbations
• Entanglement and topological entropy of the toric code at finite temperature
• Dynamically Generated Logical Qubits
• Single-shot fault-tolerant quantum error correction
• Topological Order at Non-zero Temperature
• Topological order in a 3D toric code at finite temperature
• On thermalization in Kitaev's 2D model
• Universal transversal gates with color codes - a simplified approach
• The Quantum Compass Model on the Square Lattice
• Fault-tolerant error correction with the gauge color code
• A theory of single-shot error correction for adversarial noise
• Statistical mechanical models for quantum codes with correlated noise
• Detecting Topological Order at Finite Temperature Using Entanglement Negativity
• Feasibility of self-correcting quantum memory and thermal stability of topological order
• Floquet codes without parent subsystem codes
• Symmetry protected topological order at nonzero temperature
• Qudit surface codes and gauge theory with finite cyclic groups
• Single-shot error correction of three-dimensional homological product codes
• Cellular-automaton decoders with provable thresholds for topological codes
• Anyon condensation and the color code
• Local topological order inhibits thermal stability in 2D
• Improved single-shot decoding of higher dimensional hypergraph product codes
• Symmetry-protected self-correcting quantum memories
• Cellular automaton decoders for topological quantum codes with noisy measurements and beyond
• 3-d topological quantum memory with a power-law energy barrier
• Universality Classes of Stabilizer Code Hamiltonians
• Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory
• Symmetry protected self correcting quantum memory in three space dimensions
• Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models
• Self-Duality and Phase Structure of the 4D Random-Plaquette Z_2 Gauge Model

Cited by in corpus (3)

• Efficient Detection of Strong-To-Weak Spontaneous Symmetry Breaking via the Rényi-1 Correlator
• Perturbative stability and error correction thresholds of quantum codes
• Simulating Topological Order on Quantum Processors