Digital quantum simulation of many-body non-Markovian dynamics

We present an algorithmic method for the digital quantum simulation of many-body locally indivisible non-Markovian open quantum systems. It consists of two parts: first, a Suzuki-Lie-Trotter decomposition of the global system propagator into the product of subsystem propagators, which may not be quantum channels, and second, an algorithmic procedure for the implementation of the subsystem propagators through unitary operations and measurements on a dilated space. By providing rigorous error bounds for the relevant Suzuki-Lie-Trotter decomposition, we are able to analyze the efficiency of the method, and connect it with an appropriate measure of the local indivisibility of the system. In light of our analysis, the proposed method is expected to be experimentally achievable for a variety of interesting cases.

Figure 1

Given a 2-local global propagator $\tilde{T}=T^3{T}^2{T}^1$, with $\mathcal{C}\left(T^1\right)=\mathcal{C}\left(T^3\right)=1$ and $\mathcal{C}\left(T^2\right)=0$, this propagator can be implemented algorithmically through the implementation of the four circuits $\left\{C_r\right\}{|}_{r=0}^3$. Each circuit $C_r$ consists only of quantum channels and CPnTP maps.

Figure 2

Considering the same example as shown in Fig. 1, the circuit shown here illustrates the method, given by Eq. (51), for constructing ${\tilde{\rho }}^{\left(0\right)}\left(3\right)$, from which the state ${\rho }^{\left(0\right)}\left(t\right)=C_0(\rho \left(0\right))$ can be algorithmically reconstructed. Starting with the specified initial state $\rho \left(0\right)$, the first CPnTP map $T^{1,0}$ is implemented, as described in Sec. 4, through a unitary operation $U^{1,0}$ and a measurement $P^{1,0}$ on a dilated space. It is crucial to note that if measurement outcome 1 is obtained when performing the measurement, then the correct state ${\tilde{\rho }}^{\left(0\right)}\left(1\right)$ has been obtained and the procedure can continue, but if measurement outcome 2 is obtained then the procedure needs to be restarted. The quantum channel $T^2$ can then be implemented straightforwardly, via a conventional Stinespring dilation (not shown), before the second CPnTP map $T^{3,0}$ is implemented, analogously to $T^{1,0}$.