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Optimizing over several sites in DMRG to find quantum ground states is NP-hard

Jens Eisert

Published Sep 7, 2006• 0 citations
OpenAlex

Abstract

The density matrix renormalization group (DMRG) approach is arguably the most successful method to numerically approximate ground states of quantum spin chains. It essentially amounts to iteratively locally optimizing matrix-product states, aiming at better and better approximating the true ground state. Despite considerable effort, to date both a proof of convergence of this procedure to the globally best approximation and an assessment of its complexity are lacking. Here, we rigorously establish a statement on the classical complexity of an approximation with matrix-product states: the surprising result is that when one globally optimizes matrix-product states with respect to several sites simultaneously - to find the optimal approximation over several sites at once, avoiding local optima - one encounters in the worst case a computationally difficult NP-hard problem. We prove this by exploiting a novel way of polynomially relating it to binary quadratic programming. This is true even for translationally invariant local hamiltonians. This result gives rise to intriguing ramifications on the difficulty of describing quantum many-body systems, which we discuss.

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The density matrix renormalization group (DMRG) approach is arguably the most successful method to numerically approximate ground states of quantum spin chains. It essentially amounts to iteratively locally optimizing matrix-product states, aiming at better and better approximating the true ground state. Despite considerable effort, to date both a proof of convergence of this procedure to the globally best approximation and an assessment of its complexity are lacking. Here, we rigorously establish a statement on the classical complexity of an approximation with matrix-product states: the surprising result is that when one globally optimizes matrix-product states with respect to several sites simultaneously - to find the optimal approximation over several sites at once, avoiding local optima - one encounters in the worst case a computationally difficult NP-hard problem. We prove this by exploiting a novel way of polynomially relating it to binary quadratic programming. This is true even for translationally invariant local hamiltonians. This result gives rise to intriguing ramifications on the difficulty of describing quantum many-body systems, which we discuss.
The density matrix renormalization group approach is arguably the most successful method to numerically find ground states of quantum spin chains. It amounts to iteratively locally optimizing matrix-product states, aiming at better and better approximating the true ground state. To date, both a proof of convergence to the globally best approximation and an assessment of its complexity are lacking. Here we establish a result on the computational complexity of an approximation with matrix-product states: The surprising result is that when one globally optimizes over several sites of local Hamiltonians, avoiding local optima, one encounters in the worst case a computationally difficult NP-hard problem (hard even in approximation). The proof exploits a novel way of relating it to binary quadratic programming. We discuss intriguing ramifications on the difficulty of describing quantum many-body systems.
In the approaches based on matrix-product states (MPSs), such as the density-matrix renormalization group (DMRG) method, the ordering of the sites crucially affects the computational accuracy. We investigate the performance of an algorithm that searches for the optimal site order by iterative local site rearrangement. We improve the algorithm by expanding the range of site rearrangement and apply it to a one-dimensional quantum Heisenberg model with random site permutation. The results indicate that increasing the range of the site rearrangement significantly improves the computational accuracy of the DMRG method. In particular, increasing the rearrangement range from two to three sites reduces the average relative error in the ground-state energy by 65% to 94% in the cases we tested. We also discuss the computational cost of the algorithm and its application as a preprocessing for MPS-based calculations.
Density matrix renormalization group (DMRG) algorithm has been extremely successful for computing the ground states of one-dimensional quantum many-body systems. For problems concerned with mixed quantum states, however, it is less successful in that either such an algorithm does not exist yet or that it may return unphysical solutions. Here we propose a positive matrix product ansatz for mixed quantum states which preserves positivity by construction. More importantly, it allows one to build a DMRG algorithm which, the same as the standard DMRG for ground states, iteratively reduces the global optimization problem to local ones of the same type, with the energy converging monotonically in principle. This algorithm is applied for computing both the equilibrium states and the nonequilibrium steady states and its advantages are numerically demonstrated.
The success of polynomial-time tensor network methods for computing ground states of certain quantum local Hamiltonians has recently been given a sound theoretical basis by Arad et al. [Math. Phys. 356, 65 (2017)]. The convergence proof, however, relies on ``rigorous renormalization group'' (RRG) techniques which differ fundamentally from existing algorithms. We introduce a practical adaptation of the RRG procedure which, while no longer theoretically guaranteed to converge, finds matrix product state ansatz approximations to the ground spaces and low-lying excited spectra of local Hamiltonians in realistic situations. In contrast to other schemes, RRG does not utilize variational methods on tensor networks. Rather, it operates on subsets of the system Hilbert space by constructing approximations to the global ground space in a treelike manner. We evaluate the algorithm numerically, finding similar performance to density matrix renormalization group (DMRG) in the case of a gapped nondegenerate Hamiltonian. Even in challenging situations of criticality, large ground-state degeneracy, or long-range entanglement, RRG remains able to identify candidate states having large overlap with ground and low-energy eigenstates, outperforming DMRG in some cases.
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 zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPSs). Instead of the site tensors themselves, we propose to optimize sequentially the ``message'' tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPSs closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep, while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins and on free electrons on the lattice.