The ground state of a class of noncritical 1D quantum spin systems can be approximated efficiently

We study families H_n of 1D quantum spin systems, where n is the number of spins, which have a spectral gap \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 |\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 |\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 quasi-adiabatic evolutions.