Surface codes, quantum circuits, and entanglement phases

TL;DR

This paper links surface codes and entanglement phases by mapping 2D surface codes under errors to 1+1D fermionic circuits, revealing topological phases and entanglement properties relevant for quantum error correction.

Contribution

It establishes a novel connection between quantum error correction surface codes and entanglement phases through Ising model mappings and topological analysis.

Findings

Error-correcting phase shows a topologically nontrivial area law.

Above error threshold, incoherent errors lead to trivial area law.

Coherent errors result in logarithmic entanglement growth.

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 $|\Psi_\infty\rangle$. 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 $|\Psi_\infty\rangle$ 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.