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arXiv:1912.06127 · Physical review. B./Physical review. B

Parallel time-dependent variational principle algorithm for matrix product states

Paul Secular, Nikita Gourianov, Michael Lubasch, Sergey Dolgov, Stephen R. L. Clark, Dieter Jaksch

Published Jun 5, 2020• 40 citations
arXivHTMLPDF

Abstract

Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as $86%$. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with $1/{r}^{2}$ interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems.

DOI
10.1103/physrevb.101.235123

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Time evolution of tensor-network states is a central tool for studying nonequilibrium dynamics in strongly correlated quantum systems. In the past decade, using the time-dependent variational principle (TDVP) discretized by projector-splitting has become the gold standard for evolving matrix product states (MPS) with long-range Hamiltonians. More recently, a conceptually different class of discretizations of the TDVP, the class of Basis Update and Galerkin (BUG) integrators, has been developed within the framework of dynamical low-rank approximation, offering an alternative, more flexible approach. In this work, we formulate two BUG integrators explicitly for MPS. The BUG integrators result in "single-site" MPS integrators that are robust with respect to small singular values, avoid backward-in-time substeps, and naturally allow for the adaptive choice of bond dimensions. We consider the augmented BUG integrator of first order and the midpoint BUG integrator of second order and clarify their relation to the projector-splitting discretization of the TDVP. Using representative models of strongly correlated quantum systems, like the XY spin model with long-range interaction and two-dimensional lattice models of fermions and spins of various lengths, we perform a systematic numerical comparison between BUG and projector-splitting time discretizations of the continuous-time TDVP, focusing on scalability, numerical complexity, and accuracy. We find that BUG integrators, due to their reduced numerical cost and the flexibility of the concept, provide a competitive and, in certain regimes, advantageous alternative to the well-established time-stepping by projector-splitting in TDVP, especially in models with large local dimensions, and in cases where observables are extracted with high temporal resolution.
In recent years, the time-dependent variational principle (TDVP) method based on the matrix product state (MPS) wave function formulation has shown its great power in performing large-scale quantum dynamics simulations for realistic chemical systems with strong electron-vibration interactions. In this work, we propose a stochastic adaptive single-site TDVP (SA-1TDVP) scheme to evolve the bond-dimension adaptively, which can integrate the traditional advantages of both the high efficiency of the single-site TDVP (1TDVP) variant and the high accuracy of the two-site TDVP (2TDVP) variant. Based on the assumption that the level statistics of entanglement Hamiltonians, which originate from the reduced density matrices of the MPS method, follows a Poisson or Wigner distribution, as generically predicted by random-matrix theory, additional random singular values are generated to expand the bond-dimension automatically. Tests on simulating the vibrationally resolved quantum dynamics and absorption spectra in the pyrazine molecule and perylene bisimide (PBI) J-aggregate trimer as well as a spin-1/2 Heisenberg chain show that it can be automatic and as accurate as 2TDVP but reduce the computational time remarkably.
Abstract Classical simulations of quantum circuits are vital for assessing potential quantum advantage and benchmarking devices, yet they require sophisticated methods to avoid the exponential growth of resources. Tensor network approaches, in particular matrix product states (MPS) combined with the time-evolving block decimation (TEBD) algorithm, currently dominate large-scale circuit simulations. These methods scale efficiently when entanglement is limited but suffer rapid bond dimension growth with increasing entanglement and handle long-range gates via costly SWAP insertions. Motivated by the success of the time-dependent variational principle (TDVP) in many-body physics, we reinterpret quantum circuits as series of discrete time evolutions, using gate generators to construct an MPS-based circuit simulation via a local TDVP formulation. This addresses TEBD's key limitations by (1) naturally accommodating long-range gates and (2) optimally representing states on the MPS manifold. By diffusing entanglement more globally, the method suppresses local bond growth and reduces memory and runtime costs. We benchmark the approach on five 49-qubit circuits: three Hamiltonian circuits (1D open and periodic Heisenberg, 2D 7x7 Ising) and two algorithmic ones (quantum approximate optimization, hardware-efficient ansatz). Across all cases, our method yields substantial resource reductions over standard tools, establishing a new state-of-the-art for circuit simulation and enabling advances across quantum computing, condensed matter, and beyond.
Theoretical understanding of strongly correlated systems in one spatial dimension has been greatly advanced by the density-matrix renormalization group (DMRG) algorithm, which is a variational approach using a class of entanglement-restricted states called matrix product states (MPSs). However, DMRG suffers from inherent accuracy restrictions when multiple states are involved due to multi-state targeting and also the approximate representation of the Hamiltonian and other operators. By formulating the variational approach of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state targeting approach. Furthermore, by using the matrix product operator (MPO) formalism, one can exactly represent the Hamiltonian and other operators relevant for the calculation. The MPO approach allows one-dimensional (1D) Hamiltonians to be templated using a small set of finite state automaton rules without reference to the particular microscopic degrees of freedom. We present two algorithms which take advantage of these properties: eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution of a generic time-dependent 1D Hamiltonian. We properly account for time-ordering of the propagator such that the error does not depend on the rate of change of the Hamiltonian. Our algorithms are applicable to microscopic degrees of freedom of any variety, and do not require Hamiltonian-specialized implementation. We benchmark our algorithms with a case study of the Ising model, where the critical point is located using entanglement measures. We then study the dynamics of this model under a time-dependent quench of the transverse field through the critical point. Finally, we present studies of a dipolar, or long-range Ising model, again using entanglement measures to find the critical point and study the dynamics of a time-dependent quench through the critical point.
We propose an improved scheme to do the time-dependent variational principle (TDVP) in finite matrix product states (MPSs) for two-dimensional systems or one-dimensional systems with long-range interactions. We present a method to represent the time-evolving state in a MPS with its basis enriched by state averaging with global Krylov vectors. We show that the projection error is significantly reduced so precise time evolution can still be obtained even if a larger time step is used. Combined with the one-site TDVP, our approach provides a way to dynamically increase the bond dimension while still preserving unitarity for real-time evolution. Our method can be more accurate and exhibit slower bond dimension growth than the conventional two-site TDVP.
We compare the accuracy of two prime time evolution algorithms involving matrix product states---tDMRG (time-dependent density matrix renormalization group) and TDVP (time-dependent variational principle). The latter is supposed to be superior within a limited and fixed auxiliary space dimension. Surprisingly, we find that the performance of algorithms depends on the model considered. In particular, many-body localized systems as well as the crossover regions between localized and delocalized phases are better described by tDMRG, contrary to the delocalized regime where TDVP indeed outperforms tDMRG in terms of accuracy and reliability. As an example, we study many-body localization transition in a large size Heisenberg chain. We discuss drawbacks of previous estimates [Phys. Rev. B 98, 174202 (2018)] of the critical disorder strength for large systems.