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arXiv:2203.16350 · Physical review. B./Physical review. B

Density matrix renormalization group algorithm for mixed quantum states

Chu Guo

Published May 31, 2022• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.195152

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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 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.
We investigate the density matrix renormalization group (DMRG) discovered by White and show that in the case where the renormalization eventually converges to a fixed point the DMRG ground state can be simply written as a matrix-product form. This ground state can also be rederived through a simple variational ansatz making no reference to the DMRG construction. We also show how to construct the matrix-product states and how to calculate their properties, including the excitation spectrum. This paper provides details of many results announced earlier.
The density matrix renormalization group (DMRG) is a celebrated tensor network algorithm, which computes the ground states of one-dimensional quantum many-body systems very efficiently. Here we propose an improved formulation of continuous tensor network algorithms, which we name a matrix product renormalization group (MPRG). MPRG is a universal quantum many-body solver, which potentially works at both zero and finite temperatures, in two and higher dimensions, and is even applicable to open quantum systems. Furthermore, MPRG does not rely on any variational principles and thus supports any kind of non-Hermitian systems in any dimension. As a demonstration, we present critical properties of the Yang-Lee edge singularity in one dimension as a representative non-Hermitian system.
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 (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.
Despite the success of modern quantum chemistry in predicting properties of organic molecules, the treatment of inorganic systems, which have many close lying states, remains out of quantitative reach for current methods. To treat non-dynamic correlation, we take advantage of the density matrix renormalization group (DMRG) method that has become very successful in the field of solid state physics. We present a detailed study of the DMRG method, and we pay special attention to the evolution of the understanding behind the mathematical structure of the DMRG wave function. Our primary goal is to develop a density matrix renormalization group self--consistent--field (DMRG-SCF) approach, analogous to the complete active space self--consistent field (CASSCF) method, but dealing with large active spaces that are too demanding for the full configuration interaction (FCI) method. \nAs a first step towards such a DMRG-SCF procedure, we present a spin-adapted DMRG algorithm designed to target spin- and spatial-symmetry states that are hard to obtain while using an unrestricted algorithm. \nOur next step is a modification of the DMRG algorithm to obtain decreasing energy at every step during the sweep. This monotonically convergent DMRG scheme lets us obtain the two-body density matrix as a by--product of the existing procedure without any additional cost in storage. Additionally, the two-body density matrix produced at convergence using this scheme is free from the N-representability problem that is present when the two--body density matrix is produced with the two-site DMRG scheme without additional storage cost. Finally, taking advantage of the modifications developed herein, we present results obtained from our DMRG-SCF method. Lastly, we discuss possible ways of incorporating dynamical correlation into the DMRG scheme, in order to obtain a modern multireference approach.