Undecidability of the Spectral Gap. (10th June 2022)

• Record Type: Journal Article
• Authors: Cubitt, Toby; Perez-Garcia, David; Wolf, Michael M.
• Is Part Of: Forum of mathematics. Volume 10(2022)
• Publication Date: 2022-06-10
• Subjects: quantum many-body physics -- quantum computation -- undecidability -- spectral theory; 81Q10 -- 03D35 -- 81P68 -- 03D10 -- 03D80 -- 05b45; Mathematics -- Periodicals; 510
• DOI: 10.1017/fmp.2021.15
• Languages: English
• ISSNs: 2050-5086

Abstract

Abstract: We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d -level quantum systems ( d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant. Moreover, this constant can be taken equal to the operator norm of the local operator that generates the Hamiltonian (the local interaction strength). The result still holds true if one restricts to arbitrarily small quantum perturbations of classical Hamiltonians. The proof combines a robustness analysis of Robinson's aperiodic tiling, together with tools from quantum information theory: the quantum phase estimation algorithm and the history state technique mapping Quantum Turing Machines to Hamiltonians.