| 内容 |
In quantum computation, one of the primary problems is to solve the local Hamiltonian problem, namely finding the ground state (i.e., the lowest energy state) for a given many-body Hamiltonian. The problem is known to be the QMA complete problem in general [1]. On the other hand, from many numerical studies, the most important class, where the ground state has a spectral gap, is expected to be efficiently solved. This class corresponds to non-critical ground states and determines quantum phases of matter. In the analysis of this class, the entanglement entropy (or Von-Neumann entropy in subsystem) plays a central role. The area-law conjecture states that it is proportional to the surface region of subsystem if the ground state is gapped. This conjecture is a backbone of most of the classical algorithms such as the density-matrix-renormalization group [2] as well as classification of the quantum phases. Despite much effort on this conjecture, the area law is mathematically proved in highly limited cases [3,4,5]. In the present talk, I will give an overview of the conjecture, and show our recent achievement if the time allows.
[1] J. Kempe, A. Kitaev, O. Regev, SIAM Journal on Computing, 35(5),1070-1097 (2006).
[2] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
[3] M. B. Hastings, J. Stat. Mech., P08024 (2007).
[4] Z. Landau, U. Vazirani and T.Vidick, Nature Physics, 11, 566–569 (2015)
[5] F. G. S. L. Brandao and M. Horodecki, Nature Physics, 9, 721–726 (2013) |
