Surface codes, quantum circuits, and entanglement phases

Abstract

Surface codes—leading candidates for quantum error correction (QEC)—and entanglement phases—a key notion for many-body quantum dynamics—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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>(</a:mo><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn><a:mo>)</a:mo><a:mi mathvariant="normal">D</a:mi></a:mrow></a:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mrow><c:mo>|</c:mo></c:mrow><c:msub><c:mi mathvariant="normal">Ψ</c:mi><c:mi>∞</c:mi></c:msub><c:mrow><c:mo>〉</c:mo></c:mrow></c:mrow></c:math>. 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mrow><e:mo>|</e:mo></e:mrow><e:msub><e:mi mathvariant="normal">Ψ</e:mi><e:mi>∞</e:mi></e:msub><e:mrow><e:mo>〉</e:mo></e:mrow></e:mrow></e:math> 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 (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>d</g:mi><g:mo>+</g:mo><g:mn>1</g:mn></g:mrow></g:math>)D topological codes and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>d</h:mi></h:math>-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. Published by the American Physical Society 2024