Fig: Scheme of the hamiltonian construction.
the authors prove that deciding whetherthe Hamiltonian of a quantum many-body system
is gapped or gapless is an undecidable problem.
They construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Their result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.
MathQI Seminar: new schedule!!!
From March 30th on it's being hold on Wednesdays at 16:00.
Undecidability of the spectral gap in the media
28th Apr 2016 13:00 MathQI Seminar.
Angelo Lucia: "Estabilidad y ley de área para sistemas cuánticos disipativos con equilibración rápida".
20th Apr 2016 16:00 MathQI Seminar.Cécilia Lancien (UAB/Lyon): "Flexible constrained de Finetti reductions and parallel repetition of multi-player non local games".