Constructing Tensor Network Influence Functionals for General Quantum Dynamics

Abstract

We describe an iterative formalism to compute influence functionals that describe the general quantum dynamics of a subsystem beyond the assumption of linear coupling to a quadratic bath. We use a space-time tensor network representation of the influence functional and investigate its approximability in terms of the bond dimensions and time-like entanglement in the tensor network description. We study two numerical models, the spin-boson model and a model of interacting hard-core bosons in a 1D harmonic trap. We find that the influence functional and the intermediates involved in its construction can be efficiently approximated by low bond dimension tensor networks in certain dynamical regimes, which allows the quantum dynamics to be accurately computed for longer times than with direct time evolution methods. However, as one iteratively integrates out the bath, the correlations in the influence functional can first increase before decreasing, indicating that the final compressibility of the influence functional is achieved via non-trivial cancellation.

Scaling of temporal entanglement in proximity to integrability

“…This surprising behavior is further confirmed by our computations in several limiting regimes [72]. It can be ascribed to including the subsystem-environment interaction terms fully into the influence functional [34], in contrast to earlier tensor-network approaches where the splitting of interaction gates as two-site MPOs changes the scaling of the IM with the Trotter step [22,71].…”

“…[33] and subsequently Refs. [71,72] exploited a similar numerical approach within the influence-functional formalism. Ref.…”

“…After at most t iterations the thermodynamic IM is reached due to the strict light cone effect in this model. To avoid intermediate states of high entanglement encountered during iterations [71,72] we choose a perfect dephaser [33] boundary IM, which makes the approach significantly more efficient [72].…”

Exact dynamics of non-additive environments in non-Markovian open quantum systems

“…The method we have introduced here can be taken even further by breaking down the MPOs for each bath into a series of MPOs representing the individual modes that make up the bath. This approach is similar in spirit to other recent works [34,50] where the process tensor can be constructed in a purely numerical manner, even when the environment is not Gaussian and the influence functional cannot be calculated analytically in closed form. Finally, we stress that process tensors calculated using different approaches for each environment can be combined like this, provided time is discretized in the same way.…”

“…where ρ O is the Gibbs state of the optical environment and A q (t) is the operator f q a † q + f * q a q in the interaction picture. Taking the continuum limit over the optical modes, we obtain an expression in terms of the absorption and emission effective spectral densities (50) where the vibrational influence is contained within [58][59][60] V…”