Reconstructing non-Markovian open quantum evolution from multiple time measurements

For a quantum system undergoing non-Markovian open quantum dynamics, we demonstrate a tomography algorithm based on multitime measurements of the system, which reconstructs a minimal environment coupled to the system, such that the system plus environment undergoes unitary evolution and that the reduced dynamics of the system is identical to the observed dynamics of it. The reconstructed open quantum evolution model can be used to predict any future dynamics of the system when it is further assumed to be time-independent. We define the memory size and memory complexity for the non-Markovian open quantum dynamics which characterize the complexity of the reconstruction algorithm.

Figure 1

(a) Demonstration of a three-step process tensor under a hidden OQE model, for which $\mathcal{U}$ is the unitary evolutionary operator for the system plus the environment, and $|{\psi }^{SE}\rangle$ is the system-environment initial state. $i_j$ and $o_j$ are the $j\mathrm{th}$ input and output indices, respectively. (b) The matrix product state representation of the purified process tensor, for which the environment is not traced out in the end. (c) The quantum circuit implementation of the purified process tensor as a many-body pure state, where $|\mathrm{\Psi }\rangle$ is the maximally entangled state. The process tensor is obtained by tracing out the environment index. ${\rho }_j^E$ in (b) and (c) denotes the $j\mathrm{th}$ effective environment state in Eq. (15).

Figure 2

Demonstration of the efficient purified process tensor tomography algorithm for a four-step purified process tensor, where we have assumed that the environment size $D\le d^4$. The unitary gate $O_j$ is the $j\mathrm{th}$ disentangling operator in Eq. (7). $|a_s\rangle$ is the $s\mathrm{th}$ eigenstate of the reduced density matrix on the last two sites, and ${\lambda }_s$ is the square root of the corresponding eigenvalue.

Figure 3

Demonstration the tomography algorithm for reconstructing the hidden OQE model. The $x$ axis is the time step $j$. The $y$ axis is the infidelity $\mathcal{I}=1-\mathcal{F}$ ($\mathcal{F}(\rho ,\sigma )={\mathrm{tr}}^2\left(\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)$ is the quantum fidelity between two mixed states $\rho$ and $\sigma$) between the reduced four-step process tensor ${\mathrm{{\rm Y}}}_{j+3:j}$ computed from the exact OQE model, and ${\widetilde{\mathrm{{\rm Y}}}}_{j+3:j}$ predicted from the reconstructed OQE model. The black solid and dashed lines correspond to the results obtained by minimizing the loss function in Eq. (10) with $k=2,3$ respectively. The non-Markovian open quantum dynamics of the system is generated by a hidden OQE model with the unitary evolutionary operator $U$ in Eq. (11) (with $d=2, D=5, \eta =0.1$) and the SE initial state in Eq. (12).

Figure 4

(a) The memory complexity ${\mathcal{C}}_j^1$ and (b) the memory size ${\mathcal{D}}_j$ as a function of the time step $j$. The yellow (upper) and green (lower) dashed lines are results for pure and mixed system-environment initial states, respectively, while the yellow (upper) and green (lower) solid lines are the corresponding theoretical limits. The unitary evolutionary operator $U$ is generated by Eq. (11) in the same way as Fig. 3, while the pure and mixed system-environment initial states are randomly generated and are entangled.