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arXiv:2303.09426 · Physical Review A

Comparing bipartite entropy growth in open-system matrix-product simulation methods

Guillermo Preisser, David Wellnitz, Thomas Botzung, Johannes Schachenmayer

Published Jul 18, 2023• 18 citations
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Abstract

The dynamics of one-dimensional quantum many-body systems is often numerically simulated with matrix-product states (MPSs). The computational complexity of MPS methods is known to be related to the growth of entropies of reduced density matrices for bipartitions of the chain. While for closed systems the entropy relevant for the complexity is uniquely defined by the entanglement entropy, for open systems it depends on the choice of the representation. Here, we systematically compare the growth of different entropies relevant to the complexity of matrix-product representations in open-system simulations. We simulate an $XXZ$ spin-1/2 chain in the presence of spontaneous emission and absorption and dephasing. We compare simulations using a representation of the full density matrix as a matrix-product density operator (MPDO) with a quantum trajectory unraveling, where each trajectory is itself represented by an MPS (QT + MPS). We show that the bipartite entropy in the MPDO description (operator entanglement, OE) generally scales more favorably with time than the entropy in QT + MPS (trajectory entanglement, TE): (i) For spontaneous emission and absorption the OE vanishes while the TE grows and reaches a constant value for large dissipative rates and sufficiently long times; (ii) for dephasing the OE exhibits only logarithmic growth while the TE grows polynomially. Although QT + MPS requires a smaller local state space, the more favorable entropy growth can thus make MPDO simulations fundamentally more efficient than QT + MPS. Furthermore, MPDO simulations allow for easier exploitation of higher-order Trotter decompositions and translational invariance, allowing for larger time steps and system sizes.

DOI
10.1103/physreva.108.012616

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•Apr 25, 2018•0 citations
The dynamics of one-dimensional quantum many-body systems is often numerically simulated with matrix-product states (MPSs). The computational complexity of MPS methods is known to be related to the growth of entropies of reduced density matrices for bipartitions of the chain. While for closed systems the entropy relevant for the complexity is uniquely defined by the entanglement entropy, for open systems it depends on the choice of the representation. Here, we systematically compare the growth of different entropies relevant to the complexity of matrix-product representations in open-system simulations. We simulate an XXZXXZXXZ spin-1/2 chain in the presence of spontaneous emission and absorption and dephasing. We compare simulations using a representation of the full density matrix as a matrix-product density operator (MPDO) with a quantum trajectory unraveling, where each trajectory is itself represented by an MPS (QT + MPS). We show that the bipartite entropy in the MPDO description (operator entanglement, OE) generally scales more favorably with time than the entropy in QT + MPS (trajectory entanglement, TE): (i) For spontaneous emission and absorption the OE vanishes while the TE grows and reaches a constant value for large dissipative rates and sufficiently long times; (ii) for dephasing the OE exhibits only logarithmic growth while the TE grows polynomially. Although QT + MPS requires a smaller local state space, the more favorable entropy growth can thus make MPDO simulations fundamentally more efficient than QT + MPS. Furthermore, MPDO simulations allow for easier exploitation of higher-order Trotter decompositions and translational invariance, allowing for larger time steps and system sizes.
Quantum trajectories and superoperator algorithms implemented within the matrix product state (MPS) framework are powerful tools to simulate the real-time dynamics of open dissipative quantum systems. As for the unitary case, the reachable time-scales as well as system sizes are limited by the (possible) build-up of entanglement entropy. The aforementioned methods constitute complementary approaches how Lindblad master equations can be integrated relying either on a quasi-exact representation of the full density matrix or a stochastic unraveling of the density matrix in terms of pure states. In this work, we systematically benchmark both methods by studying the dynamics of a Bose-Hubbard chain in the presence of local as well as global dephasing. The build-up as well as system-size scaling of entanglement entropy strongly depends on the method and the parameter regime and we discuss the applicability of the methods for these cases as well as study the distribution of observables and time discretization errors that can become a limiting factor for global dissipation.
Abstract We present a collection of methods to simulate entangled dynamics of open quantum systems governed by the Lindblad master equation with tensor network methods. Tensor network methods using matrix product states (MPSs) have been proven very useful to simulate many-body quantum systems and have driven many innovations in research. Since the MPS design is tailored for closed one-dimensional systems governed by the Schrödinger equation, the next step for many-body quantum dynamics is the simulation of one-dimensional open quantum systems. We review the three dominant approaches to the simulation of open quantum systems via the Lindblad master equation: quantum trajectories, matrix product density operators (MPDOs), and locally purified tensor networks. Selected examples guide possible applications of the methods and serve moreover as a benchmark between the techniques. These examples include the finite temperature states of the transverse quantum Ising model, the dynamics of an exciton traveling under the influence of spontaneous emission and dephasing, and a double-well potential simulated with the Bose–Hubbard model including dephasing. We analyze which approach is favorable leading to the conclusion that a complete set of all three methods is most beneficial, pushing the limits of different scenarios. The convergence studies using analytical results for macroscopic variables and exact diagonalization methods as comparison, show, for example, that MPDOs are favorable for the exciton problem in our study. All three methods access the same library, i.e., the software package Open Source Matrix Product States , allowing us to have a meaningful comparison between the different approaches based on the selected examples. For example, tensor operations are accessed from the same subroutines and with the same optimization eliminating one possible bias in a comparison of such numerical methods.
We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states(MPSs) and argue that it is intimately related to that of encoding temporal transition matrices and their partial traces. We show that we can upper-bound the rank of these reduced transition matrices by one of the Heisenberg evolution of local operators, thus making aconnection between two apparently different quantities, the temporal entanglement and the local operator entanglement(OE). As a result, whenever the local OE grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal MPSs is likely advantageous with respect to computing the same quantities using standard MPS techniques. Published by the American Physical Society 2024
We analyze the modification of entanglement dynamics in the Grover algorithm when the qubits are subjected to single-qubit amplitude-damping or phase-flip noise. We compare quantum trajectories with full density-matrix simulations, analyzing the dynamics of averaged trajectory entanglement (TE) and operator entanglement (OE), in the respective state representation. Although not a genuine entanglement measure, both TE and OE are connected to the efficiency of matrix product state simulations and thus of fundamental interest. As in many quantum algorithms, at the end of the Grover circuit entanglement decreases as the system converges towards the target product state. While we find that this is well captured in the OE dynamics, quantum trajectories rarely follow paths of reducing entanglement. Optimized unraveling schemes can lower TE slightly, however we show that deep in the circuit OE is generally smaller than TE. This implies that matrix product density operator (MPDO) simulations of quantum circuits can in general be more efficient than quantum trajectories. In addition, we investigate the noise-rate scaling of success probabilities for both amplitude-damping and phase-flip noise in Grover's algorithm.
The operator space entanglement entropy, or simply "operator entanglement" (OE), is an indicator of the complexity of quantum operators and of their approximability by matrix product operators (MPOs). We study the OE of the density matrix of 1D many-body models undergoing dissipative evolution. It is expected that, after an initial linear growth reminiscent of unitary quench dynamics, the OE should be suppressed by dissipative processes as the system evolves to a simple stationary state. Surprisingly, we find that this scenario breaks down for one of the most fundamental dissipative mechanisms: dephasing. Under dephasing, after the initial "rise and fall," the OE can rise again, increasing logarithmically at long times. Using a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, we demonstrate that this growth is inherent to a U(1) conservation law. We argue that in an XXZ spin model and a Bose-Hubbard model the OE grows universally as 1/4log_{2}t at long times and as 1/2log_{2}t for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.
We present an open source code to simulate entangled dynamics of open quantum systems governed by the Lindblad master equation with tensor network methods. Tensor network methods using matrix product states have been proven very useful to simulate many-body quantum systems and have driven many innovations in research. Since the matrix product state design is tailored for closed one-dimensional systems governed by the Schr\odinger equation, the next step for many-body quantum dynamics is the simulation of one-dimensional open quantum systems. We review the three dominant approaches to the simulation of open quantum systems via the Lindblad master equation: quantum trajectories, matrix product density operators, and locally purified tensor networks. The package Open Source Matrix Product States combines these three major methods to simulate one-dimensional open quantum systems with the Lindblad master equation. We access the same underlying tensor network algorithms for all techniques, e.g., the tensor contractions optimized in the same way, allowing us to have a meaningful comparison between the different approaches based on selected examples. These examples include the finite temperature states of the transverse quantum Ising model, the dynamics of an exciton traveling under the influence of spontaneous emission and dephasing, and a double-well potential simulated with the Bose-Hubbard model including dephasing. We analyze which approach is favorable leading to the conclusion that a complete set of all three methods is most beneficial, pushing the limits of different scenarios. The convergence studies using analytical results for macroscopic variables and exact diagonalization methods as comparison, show, for example, that matrix product density operators are favorable for the exciton problem in our study.