2015/02/16 by Toby Cubitt, Toby S. Cubitt, David Pérez-Garcı́a +2 · 2 voices · 1 citation
Mathematics · Physics and Astronomy · #Bounded function #Computer science #Condensed matter physics #Constant (computer programming) #Decidability #Discrete mathematics #Gapless playback #Ground state #Hamiltonian (control theory) #Invariant (physics) #Ising model #Lattice (music) #Mathematical analysis #Mathematics #Physics #Quantum #Quantum and electron transport phenomena #Quantum chaos and dynamical systems #Quantum many-body systems #Quantum mechanics #Spectral gap #Square lattice #Thermodynamic limit #Undecidable problem #cond-mat.other #hep-th #math-ph #quant-ph
paper · pdf · doi:10.1017/fmp.2021.15
arxiv published 2015/02/16 · openalex publication_date 2022/01/01 · arxiv updated 2022/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.