Abstract
A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the lengthNof the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-ksegments of the chain, the reduced density matrices of our approximations areϵ-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like(k/ϵ)1+o(1), and at the expense of worse but still poly(k,1/ϵ)scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimensionexp(O(k/ϵ)), which is exponentially worse, but still independent ofN. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to findO(1)-accurate local approximations to the ground state inT(N)time implies the ability to estimate the ground state energy toO(1)precision inO(T(N)log(N))time.
Highlights
In nature, interactions between particles act locally, motivating the study of many-body Hamiltonians consisting only of terms involving particles spatially near each other
An important method that has emerged from this course of study is the Density Matrix Renormalization Group (DMRG) algorithm [36, 37], which aims to find a description of the ground state of local Hamiltonians on a one-dimensional chain of sites
If we attempt to perform variational optimization over the set of constant bond dimension matrix product operator (MPO), we can be certain that our search space contains a good local approximation to the ground state, but we have no way of restricting our search only to the set of valid quantum states; the minimal energy MPO we find may not correspond to any quantum state at all
Summary
Interactions between particles act locally, motivating the study of many-body Hamiltonians consisting only of terms involving particles spatially near each other. Entire infinite chain — have been implemented with successful results, but these methods lack the second ingredient that justified DMRG: it is not clear how large we must make the bond dimension to guarantee that the set over which we are optimizing contains a good approximation to the ground state Progress toward this ingredient can be found work by Huang [17] (and later by Schuch and Verstraete [27]), who showed that the ground state of a gapped local 1D Hamiltonian can be approximated locally by a matrix product operator (MPO) — a 1D tensor network object that corresponds to a (possibly mixed) density operator as opposed to a quantum state vector — with bond dimension independent of N and polynomial in the inverse local approximation error. Since strategies for estimating the ground state energy typically involve constructing a globally accurate approximation to the ground state, this observation gives us an intuition that it may not be possible to find the local approximation much more quickly than the global approximation, despite the fact that the bond dimensions required for the two approximations are drastically different
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