Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals

In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, noninteracting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor—a more general representation of a quantum stochastic process—describing the evolution. Then, we use this connection to motivate a tensor network algorithm for the simulation of multitime correlations in open systems, building on recent work where the influence functional is represented in terms of time evolving matrix product operators. By exploiting the symmetries of the influence functional, we are able to use our algorithm to achieve orders-of-magnitude improvement in the efficiency of the resulting numerical simulation. Our improved algorithm is then applied to compute exact phonon emission spectra for the spin-boson model with strong coupling, demonstrating a significant divergence from spectra derived under commonly used assumptions of memorylessness.

Figure 1

(a) An arbitrary process with interventions can be represented as a matrix product form tensor network, the process tensor (upper row) that contracts with a filter function consisting of a sequence of superoperators (lower row). This makes it possible to separate implemented control operations from the underlying uncontrolled process. (b) In the infinitesimal time step limit, the uncontrolled process can be further decomposed into free evolution of the system (middle row) and a generalized influence functional capturing the influence of the environment.

Figure 2

Tensor network representation of the influence functional on five time steps, with nodes representing influence tensors and labeled by time step separation. Before contraction, indices are constrained to be equal along rows and columns in the network; therefore, the open boundary can be shifted to any tensor in the same column [panels (a) and (b)] or row [panels (c) and (d)]. (a) With the nonlocal boundary choice of Ref. [35], free indices are attached to influence tensors encoding memory effects over all different timescales. (b) The network is contracted iteratively from below, row by row, down to a boundary MPO. (c) With the local boundary choice, the free indices are always attached to time-local influence tensors. (d) Contraction proceeds as indicated, with the causal influence of each open leg sequentially incorporated into the wider network. The influence functional on an open leg is fixed once the corresponding layer has been contracted over.

Figure 3

(a) Variation of computation time with the inverse of the cutoff frequency for the local algorithm at a coupling strength of $\alpha =0.7$, for an Ohmic spectral density with $T=0.01\mathrm{\Omega }$ and ${\lambda }_c={10}^{-6}$, the vertical line shows the (fixed) time step size used for the Trotter-Suzuki decomposition. (b) Comparison between the computation time of the nonlocal [Eq. (5)] and local [Eq. (6)] time-evolving MPO algorithms as a function of coupling strength with ${\omega }_c=10\mathrm{\Omega }$ (the remaining parameters are unchanged). (c) Steady state phonon emission spectrum at $\alpha =0.5$, the (numerically converged) non-Markovian spectrum is simulated using the local algorithm and is compared with the spectrum obtained using the regression theorem.