User-Friendly Tail Bounds for Sums of Random Matrices
Foundations of Computational Mathematics
| Hi-index | 0.00 |
Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. The problem is known to be QMA-complete, even for one-dimensional Hamiltonians [1]. This means that we do not even expect that there is a sub-exponential size description of the ground state that allows efficient computation of local observables such as the energy. In sharp contrast, the heuristic density matrix renormalization group (DMRG) algorithm invented two decades ago [5] has been remarkably successful in practice on one-dimensional problems. The situation is reminiscent of the unexplained success of the simplex algorithm before the advent of ellipsoid and interior-point methods. Is there a principled explanation for this, in the form of a large class of one-dimensional Hamiltonians whose ground states can be provably efficiently approximated? Here we give such an algorithm for gapped one-dimensional Hamiltonians: our algorithm outputs an (inverse-polynomial) approximation to the ground state, expressed as a matrix product state (MPS) of polynomial bond dimension. The running time of the algorithm is polynomial in the number of qudits n and the approximation quality δ, for a fixed local dimension d and gap Δ 0. A key ingredient of our algorithm is a new construction of an operator called an approximate ground state projector (AGSP), a concept first introduced in [2] to derive an improved area law for gapped one-dimensional systems [3]. For this purpose the AGSP has to be efficiently constructed; the particular AGSP we construct relies on matrix-valued Chernoff bounds [4]. Other ingredients of the algorithm include the use of convex programming, recently discovered structural features of gapped 1D quantum systems [2], and new techniques for manipulating and bounding the complexity of matrix product states.