On stability of k-local quantum phases of matter

• URL: http://arxiv.org/abs/2405.19412v2
• Date: Sat, 7 Sep 2024 18:14:14 GMT
• Title: On stability of k-local quantum phases of matter
• Authors: Ali Lavasani, Michael J. Gullans, Victor V. Albert, Maissam Barkeshli,
• Abstract summary: We analyze the stability of the energy gap to Euclids for Hamiltonians corresponding to general quantum low-density parity-check codes. We discuss implications for the third law of thermodynamics, as $k$-local Hamiltonians can have extensive zero-temperature entropy.
• Score: 0.4999814847776097
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The current theoretical framework for topological phases of matter is based on the thermodynamic limit of a system with geometrically local interactions. A natural question is to what extent the notion of a phase of matter remains well-defined if we relax the constraint of geometric locality, and replace it with a weaker graph-theoretic notion of $k$-locality. As a step towards answering this question, we analyze the stability of the energy gap to perturbations for Hamiltonians corresponding to general quantum low-density parity-check codes, extending work of Bravyi and Hastings [Commun. Math. Phys. 307, 609 (2011)]. A corollary of our main result is that if there exist constants $\varepsilon_1,\varepsilon_2>0$ such that the size $\Gamma(r)$ of balls of radius $r$ on the interaction graph satisfy $\Gamma(r) = O(\exp(r^{1-\varepsilon_1}))$ and the local ground states of balls of radius $r\le\rho^\ast = O(\log(n)^{1+\varepsilon_2})$ are locally indistinguishable, then the energy gap of the associated Hamiltonian is stable against local perturbations. This gives an almost exponential improvement over the $D$-dimensional Euclidean case, which requires $\Gamma(r) = O(r^D)$ and $\rho^\ast = O(n^\alpha)$ for some $\alpha > 0$. The approach we follow falls just short of proving stability of finite-rate qLDPC codes, which have $\varepsilon_1 = 0$; we discuss some strategies to extend the result to these cases. We discuss implications for the third law of thermodynamics, as $k$-local Hamiltonians can have extensive zero-temperature entropy.

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