Dissipative universal Lindbladian simulation

It is by now well understood that quantum dissipative processes can be harnessed and turned into a resource for quantum-information processing tasks. In this paper we demonstrate yet another way in which this is true by providing a dissipation-assisted protocol for the simulation of general Markovian dynamics. More precisely, we show how a suitable coherent coupling of a quantum system to a set of Markovian dissipating qubits allows one to enact an effective Liouvillian generator of any Lindbladian form. This effective dynamical generator arises from high-order virtual-dissipative processes and governs the system dynamics exactly in the limit of infinitely fast dissipation. Applications to the simulation of collective decoherence are discussed as an illustration.

Figure 1

A quantum system $S$ (blue ball) is coupled with coupling strengths $g_i$ to $M$ qubits (yellow balls). Each of these qubits is subject to amplitude damping with rates ${\tau }_i^{-1}$. Proposition 2 shows that in the limit of small ${\tau }_i$ the qubits can be adiabatically decoupled and the effective dynamics of $S$ is described by $M$ Lindblad operators of strength $g_i^2{\tau }_i$.

Figure 2

Distance from the exact evolution $\left({\mathcal{E}}_T:=e^{T{\mathcal{L}}_T}\right)$ and the effective evolution with Liouvillian (6), as a function of $1/\sqrt{T}$. $N=3,{\tau }_{+}=2,{\tau }_{-}=1$, and $g={\left({\tau }_{R}T\right)}^{-1/2}$ (where the relaxation time is ${\tau }_R=\frac{{\tau }_{+}{\tau }_{-}}{{\tau }_{+}+{\tau }_{-}})$. The linear fit is obtained using the least-squares fitting on all of the data points, and the norm is the maximum singular value of the maps realized as matrices. Time is measured in units of ${\tau }_R$.

Figure 3

Distance from the exact evolution $\left({\mathcal{E}}_t:=e^{t{\mathcal{L}}_T}\right)$ and the effective evolution with the Liouvillian Eq. (7) as a function of ${log}_{10}\left(t\right)$. $N=1,{\tau }_{+}=2,{\tau }_{-}=1$, and $g={\left({\tau }_{R}T\right)}^{-1/2}({\tau }_R=\frac{{\tau }_{+}{\tau }_{-}}{{\tau }_{+}+{\tau }_{-}})$. Note that for the dashed line we have extended $t$ past $T$, purely for convenience. The norm is the maximum singular value of the maps realized as matrices. Time is measured in units of ${\tau }_R$.