Ground state of a class of noncritical one-dimensional quantum spin systems can be approximated efficiently

Abstract

We study families ${H}_{n}$ of one-dimensional (1D) quantum spin systems, where $n$ is the number of spins, which have a spectral gap $\ensuremath{\Delta}E$ between the ground-state and first-excited state energy that scales, asymptotically, as a constant in $n$. We show that if the ground state $\ensuremath{\mid}{\ensuremath{\Omega}}_{m}⟩$ of the Hamiltonian ${H}_{m}$ on $m$ spins, where $m$ is an $O(1)$ constant, is locally the same as the ground state $\ensuremath{\mid}{\ensuremath{\Omega}}_{n}⟩$, for arbitrarily large $n$, then an arbitrarily good approximation to the ground state of ${H}_{n}$ can be stored efficiently for all $n$. We formulate a conjecture that, if true, would imply our result applies to all noncritical 1D spin systems. We also include an appendix on quasiadiabatic evolutions.