Undecidability of the Spectral Gap in 1D

QuICS Seminar

Johannes Bausch (University of Cambridge)
May 9, 2018
PSC 3150
In this work, we show that for local Hamiltonians in 1D with translationally invariant nearest neighbour interactions, deciding whether or not the system is gapped or gapless, isĀ undecidable. In order to prove this, we need to go beyond the already-known 2D case, and replace a fractal Robinson tiling---only possible in more than one dimension---with a specific Hamiltonian that has a type of attractive force in its ground state. We further need to augment the quantum phase estimation algorithm to self-detect whether it succeded in its expansion.

Joint work with T. Cubitt, A. Lucia and D. Perez-Garcia