Undecidability of the spectral gap

Emiliano Ippoliti, S. Caprara

Laura Castilla-Castellano, Angelo Lucia

The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] or more dimensions [T. S. Cubitt et al., "Undecidability of the spectral gap", Nature 528 (2015) and Forum of Mathematics Pi, 10 (2022)]. In this work, we focus on the 1-dimensional result, showing that the constructed family with undecidable behavior is extremely sensitive to perturbations. In particular, for any $\varepsilon > 0$, there exists a 1-local, rank 1, perturbation with norm $O(\varepsilon)$, such that the spectral gap problem for the family in [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] now becomes decidable.

Doug Doucette

J.B. Houben

We treat the results of the 2015 paper by Cubitt et al. entitled “Undecidability of the Spectral Gap”, and the constructions required to arrive at these results. This paper showed that a specific quantum mechanical problem (the “spectral gap problem”) is algorithmically undecidable, using techniques from computer science. We also treat the theory from quantum mechanics and computer science that is required to understand these results and constructions.

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Matthew B. Hastings, Tohru Koma