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arXiv:1103.1885 · Annals of Physics

Feasibility of self-correcting quantum memory and thermal stability of topological order

Beni Yoshida

Published Jun 29, 2011• 86 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2011.06.001

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Recent studies have shown that topological models with interacting anyonic quasiparticles can be used as self-correcting quantum memories. Here we study the behavior of these models at thermal equilibrium. It is found that the interactions allow topological order to exist at finite temperature, not only in an extension of the ground-state phase, but also in a novel form of topologically ordered phase. Both phases are found to support self-correction in all models considered, and the transition between them corresponds to a change in the scaling of memory lifetime with system size.
We construct a family of infinitely many candidate non-Abelian self-correcting topological quantum memories in D ≥ 5 + 1 spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of Z 2 3 higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group D 8 gauge theory in ( 2 + 1 ) D . The simplest example includes a non-Abelian self-correcting memory in ( 5 + 1 ) D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability and devise a probabilistic local cellular-automaton decoder. We also construct fault-tolerant non-Clifford CCZ logical gate using constant-depth circuits from higher-cup products in the 5D non-Abelian code. The use of higher-cup products and non-Pauli stabilizers allows us to get an O ( n 2 / 5 ) distance overcoming the O ( n 1 / 3 ) distance barrier in conventional topological stabilizer codes, including the 3D color code and the 6D self-correcting color code.
We examine two proposals for marginally-self-correcting quantum memory: the cubic code by Haah and the welded code by Michnicki. In particular, we prove explicitly that they are absent of topological order above zero temperature, as their Gibbs ensembles can be prepared via a short-depth quantum circuit from classical ensembles. Our proof technique naturally gives rise to the notion of free energy associated with excitations. Further, we develop a framework for an ergodic decomposition of Davies generators in CSS codes which enables formal reduction to simpler classical memory problems. We then show that memory time in the welded code is doubly exponential in inverse temperature via the Peierls argument. These results introduce further connections between thermal topological order and self-correction from the viewpoint of free energy and quantum circuit depth.