Surface codes, quantum circuits, and entanglement phases

Abstract

Surface codes$\unicode{x2014}$leading candidates for quantum error correction (QEC)$\unicode{x2014}$and entanglement phases$\unicode{x2014}$a key notion for many-body quantum dynamics$\unicode{x2014}$have heretofore been unrelated. Here, we establish a link between the two. We map two-dimensional (2D) surface codes under a class of incoherent or coherent errors (bit flips or uniaxial rotations) to $(1+1)$D free-fermion quantum circuits via Ising models. We show that the error-correcting phase implies a topologically nontrivial area law for the circuit's 1D long-time state $|{\Psi }_{\infty }⟩$. Above the error threshold, we find a topologically trivial area law for incoherent errors and logarithmic entanglement in the coherent case. In establishing our results, we formulate 1D parent Hamiltonians for $|{\Psi }_{\infty }⟩$ via linking Ising models and 2D scattering networks, the latter displaying respective insulating and metallic phases and setting the 1D fermion gap and topology via their localization length and topological invariant. We expect our results to generalize to a duality between the error-correcting phase of ($d+1$)D topological codes and $d$-dimensional area laws; this can facilitate assessing code performance under various errors. The approach of combining Ising models, scattering networks, and parent Hamiltonians can be generalized to other fermionic circuits and may be of independent interest.

16 pages, 9 figures; v2: accepted manuscript

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• Error-resilience Phase Transitions in Encoding-Decoding Quantum Circuits
• Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions
• Robust teleportation of a surface code and cascade of topological quantum phase transitions
• Measurement-induced phase transitions in the toric code
• Coherent information as a mixed-state topological order parameter of fermions
• Near-Term Spin-Qubit Architecture Design via Multipartite Maximally-Entangled States
• Anomalous distribution of magnetization in an Ising spin glass with correlated disorder
• Topological Modes in Monitored Quantum Dynamics
• Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors
• Emergent unitary designs for encoded qubits from coherent errors and syndrome measurements
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Free-Fermion Dynamics with Measurements: Topological Classification and Adaptive Preparation of Topological States
• Symmetry and Topology of Monitored Quantum Dynamics
• Competing automorphisms and disordered Floquet codes
• Phases of Floquet code under local decoherence
• Free-Fermion Measurement-Induced Volume- to Area-Law Entanglement Transition in the Presence of Fermion Interactions