Statistical mechanics of quantum error correcting codes

By Yaodong Li, Matthew Fisher. Physical review B, 2020

162 citations recorded by the source corpus.

We study stabilizer quantum error-correcting codes (QECC) generated under hybrid dynamics of local Clifford unitaries and local Pauli measurements in one dimension. Building upon 1) a general formula relating the error-susceptibility of a subregion to its entanglement properties, and 2) a previously established mapping between entanglement entropies and domain wall free energies of an underlying spin model, we propose a statistical mechanical description of the QECC in terms of "entanglement domain walls". Free energies of such domain walls generically feature a leading volume law term coming from its "surface energy", and a sub-volume law correction coming from thermodynamic entropies of its transverse fluctuations. These are most easily accounted for by capillary-wave theory of liquid-gas interfaces, which we use as an illustrative tool. We show that the information-theoretic decoupling criterion corresponds to a geometric decoupling of domain walls, which further leads to the identification of the "contiguous code distance" of the QECC as the crossover length scale at which the energy and entropy of the domain wall are comparable. The contiguous code distance thus diverges with the system size as the subleading entropic term of the free energy, protecting a finite code rate against local undetectable errors. We support these correspondences with numerical evidence, where we find capillary-wave theory describes many qualitative features of the QECC; we also discuss when and why it fails to do so.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/419d68ef6b37f7e38739d79ec0ceb04c20ef2013

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. A. Nahum
2. Matthew Fisher
3. V. Khemani
4. B. Skinner
5. J. Ruhman
6. S. Vijay
7. Sthitadhi Roy
8. S. Gopalakrishnan

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. Matthew Fisher
3. V. Khemani
4. B. Skinner
5. J. Ruhman
6. S. Vijay
7. S. Gopalakrishnan
8. Sthitadhi Roy

Citations in the corpus

• Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits (2021, background): “For p > 0 one can still have many degenerate paths [17].”
• Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, methodology): “In the many-body case this random dynamics admits a “measurement phase transition” (MPT) between two qualitatively different, stable dynamical phases, with distinct entanglement properties [6–27].”
• Random Quantum Circuits (2022, background): “15In the min cut/domain wall picture, this is related to the cost of a horizontal domain wall that separates the initial from the final time (55, 56, 57), and which measures the “entanglement” between the initial and final time boundaries of the non-unitary circuit.”