Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm

Zwolak, Michael; Vidal, Guifré

doi:10.1103/physrevlett.93.207205

Cited by 646 publications

(685 citation statements)

References 17 publications

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“…That is, the power of the t-DMRG approach can be rigorously grasped in terms of Lieb-Robinson bounds. For 1D local Liouvillian dynamics, variants of t-DMRG have also been proposed [66,72], usually as variational principles over matrix-product operators, the mixed state analogues of matrix-product states, or by means of suitable sampling employing classical stochastic processes in Hilbert space [52].…”

Section: Time-dependent Density-matrix Renormalization Group Methodsmentioning

confidence: 99%

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Kliesch

Gogolin

Eisert

2014

Many-Electron Approaches in Physics, Chemistry and Mathematics

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This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound that gives rise to a maximum speed of propagation of correlations in many body systems of locally interacting spins and fermions. Finally, we discuss a number of important consequences concerning the simulation of time evolution and properties of ground states and stationary states.

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Section: Time-dependent Density-matrix Renormalization Group Methodsmentioning

confidence: 99%

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Kliesch

Gogolin

Eisert

2014

Many-Electron Approaches in Physics, Chemistry and Mathematics

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“…The application of the MPS approach to non-zero temperatures requires its extension allowing for a description of operators, in this case density matrix operators [33][34][35]. One introduces matrix product operators (MPO) of the form:…”

Section: ψ|H|ψ (Upper Right)mentioning

confidence: 99%

Towards overcoming the Monte Carlo sign problem with tensor networks

et al. 2017

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Abstract. The study of lattice gauge theories with Monte Carlo simulations is hindered by the infamous sign problem that appears under certain circumstances, in particular at non-zero chemical potential. So far, there is no universal method to overcome this problem. However, recent years brought a new class of non-perturbative Hamiltonian techniques named tensor networks, where the sign problem is absent. In previous work, we have demonstrated that this approach, in particular matrix product states in 1+1 dimensions, can be used to perform precise calculations in a lattice gauge theory, the massless and massive Schwinger model. We have computed the mass spectrum of this theory, its thermal properties and real-time dynamics. In this work, we review these results and we extend our calculations to the case of two flavours and non-zero chemical potential. We are able to reliably reproduce known analytical results for this model, thus demonstrating that tensor networks can tackle the sign problem of a lattice gauge theory at finite density.

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“…(1-2), with no conditions imposed on the operators H(t) and V k (for example, they need not be local [34,35] and with no a priori knowledge of the attractor state. The are two important issues.…”

Section: Introductionmentioning

confidence: 99%

Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method

et al. 2017

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Quantum systems out of equilibrium are presently a subject of active research, both in theoretical and experimental domains. In this work we consider time-periodically modulated quantum systems which are in contact with a stationary environment. Within the framework of a quantum master equation, the asymptotic states of such systems are described by time-periodic density operators. Resolution of these operators constitutes a non-trivial computational task. Approaches based on spectral and iterative methods are restricted to systems with the dimension of the hosting Hilbert space dimH = N 300, while the direct long-time numerical integration of the master equation becomes increasingly problematic for N 400, especially when the coupling to the environment is weak. To go beyond this limit, we use the quantum trajectory method which unravels master equation for the density operator into a set of stochastic processes for wave functions. The asymptotic density matrix is calculated by performing a statistical sampling over the ensemble of quantum trajectories, preceded by a long transient propagation. We follow the ideology of eventdriven programming and construct a new algorithmic realization of the method. The algorithm is computationally efficient, allowing for long 'leaps' forward in time, and is numerically exact in the sense that, being given the list of uniformly distributed (on the unit interval) random numbers, {η1, η2, ..., ηn}, one could propagate a quantum trajectory (with ηi's as norm thresholds) in a numerically exact way. By using a scalable N -particle quantum model, we demonstrate that the algorithm allows us to resolve the asymptotic density operator of the model system with N = 2000 states on a regular-size computer cluster, thus reaching the scale on which numerical studies of modulated Hamiltonian systems are currently performed.

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Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm

Cited by 646 publications

References 17 publications

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Towards overcoming the Monte Carlo sign problem with tensor networks

Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method

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