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arXiv:1906.09077

Comparison of MPS based real time evolution algorithms for Anderson Impurity Models

Daniel Bauernfeind, Markus Aichhorn, Hans Gerd Evertz

Published Jun 21, 2019• 4 citations
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Abstract

We perform a detailed comparison of two Matrix Product States (MPS) based time evolution algorithms for Anderson Impurity Models. To describe the bath, we use both the star-geometry as well as the commonly employed Wilson chain geometry. For each bath geometry, we use either the Time Dependent Variational Principle (TDVP) or the Time Evolving Block Decimation (TEBD) to perform the time evolution. To apply TEBD for the star-geometry, we use a specially adapted algorithm that can deal with the long-range coupling terms. Analyzing the major sources of errors, one expects them to be proportional to the system size for all algorithms. Surprisingly, we find errors independent of system size except for TEBD in chain geometry. Additionally, we show that the right combination of bath representation and time evolution algorithm is important. While TDVP in chain geometry is a very precise approach, TEBD in star geometry is much faster, such that for a given accuracy it is superior to TDVP in chain geometry. This makes the adapted version of TEBD in star geometry the most efficient method to solve impurity problems.

DOI
10.48550/arxiv.1906.09077

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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.
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.
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Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO WI,II method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics.
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