Quantum simulations of dissipative dynamics: Time dependence instead of size

The simulation of quantum systems has been a key aim of quantum technologies for decades, and generalization to open systems is necessary to include physically realistic systems. We introduce an approach for quantum simulations of open system dynamics in terms of an environment of minimal size and a time-dependent Hamiltonian. This enables the implementation of a continuous-time simulation with a finite environment, whereas state-of-the-art methods require an infinite environment or only match the simulation at discrete times. We find the necessary and sufficient conditions for this Hamiltonian to be well behaved, and when these are not met, we show that there exists an approximate Hamiltonian that is well behaved and look into its applications.

Figure 1

Schematic of how to construct the dilation from a quantum channel. All the steps preserve the analyticity in time apart from separating the diagonalized Choi state into Kraus operators, which introduces a square root. It is differentiating this square root that introduces the possibility of discontinuities and divergences.

Figure 2

Decay of the $|+\rangle \langle +|$ state when subjected to a dilation of the dephasing channel, where the divergent Hamiltonian is replaced with a finite cutoff at short times. Inset: The behavior for short times.

Figure 3

Dilation for a qubit subjected to an amplitude-damping Lindbladian and a resonant sinusoidal driving field is detailed in Eq. (19). We plot here the real part of the time-dependent functions in that Hamiltonian with ${\omega }_0/\gamma =2$. Only one of the terms diverges at $t=0$, while all the terms decay to 0 quickly at long times.