Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

Abstract

Abstract:One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time isÕ(nM(n)), where M(n) is the time required to multiply two n × n matrices.

Hand-waving and interpretive dance: an introductory course on tensor networks

“…If we consider the case of local and gapped Hamiltonians, it has been shown that the relevant ground states cannot be highly entangled [19,[37][38][39][40] (see [41] for a review). This restricted entanglement means that such states admit efficient MPS approximations [17], and moreover that they may be efficiently approximated [40,[42][43][44][45], showing that the presence of the gap causes the complexity to plummet from QMA-complete all the way down to P, removing the complexity barrier to simulation. We note that despite the challenges, both complexity theoretic and physical, in applying MPS to gapless models, they have been successfully utilised for this purpose [46][47][48].…”

“…2. It is known that the local hamiltonian problem is in P for 1D gapped Hamiltonians [40,[42][43][44][45]. DMRG and TEBD are the most common techniques for numerically finding the ground states of such systems.…”

Quantum Neural Network States: A Brief Review of Methods and Applications

“…For more examples, see refs. [3][4][5][6][7]12] A periodic-boundary-condition MPS state is just like the righthand side of Equation (26), which consists of many local rank-3 tensors. For the open boundary case, the boundary local tensor is replaced with rank-2 tensors, and the inner part remains the same.…”

“…Although the dimension of the Hilbert space of the system grows exponentially with the number of particles in general, fortunately, physical states frequently have some internal structures, for example, obeying the entanglement area law, making it easier to solve problems than in the general case. [3][4][5][6][7] Physical properties of the system usually restrict the form of the ground state, for example, area-law states, [8] ground states of local gapped systems. [9] Therefore, many-body localized systems can be efficiently represented by a tensor network, [3,[10][11][12] which is a new tool developed in recent years to attack difficulties in representing quantum many-body states efficiently.…”

Simple proof of the detectability lemma and spectral gap amplification

“…Together with the detectability lemma, the two results establish a form of duality between H and DL(H ), showing that their spectral gaps are always within a constant factor from each other. This converse to the DL has already been used for the purpose of proving lower bounds on the spectral gap of frustration-free Hamiltonians in forthcoming work on 1D area laws and efficient algorithms [25].…”