Stationary states of a dissipative two-qubit quantum channel and their applications for quantum machine learning

Abstract

Entangled-state preparation and preservation are the cornerstones of any quantum information platform. However, the strongest adversaries in quantum information science are unwanted environmental effects such as decoherence and dissipation. Here, we address how to control and harness these unwanted effects that arise from the coupling of a system with its environment, to provide stationary entangled states for quantum machine learning (QML). To do so, we design a dissipative quantum channel, i.e., a two-qubit system interacting with a squeezed vacuum field reservoir, and study the output state of the channel by solving the corresponding master equation, especially, in the small squeezing regime. We show that the time-dependent output state of the channel is the so-called two-qubit X-states that generalize many families of entangled two-qubit states. Also, by considering a general Bell diagonal state as the initial state of the system, we reveal that this dissipative channel generates two well-known classes of entangled mixed state and Werner-like states in the steady-state regime. Moreover, this channel provides an efficient way to determine whether a given initial state results in a stationary entangled state or not. Finally, we examine the potential application of the designed two-qubit channel for QML. In this line, we propose a general theoretical scheme for quantum neural networks (QNNs) implemented with the variational quantum circuits, which encode data in continuous variables (CVs) of the two-qubit states. The linear (also referred to as affine) and nonlinear (activation function) transformations are enacted in the QNN using the stationary states of the two-qubit channel and measurement process, respectively. Finally, we proceed to test our proposed model by solving some supervised binary classification tasks. Integrating the non-unitary transformation and parallel-processed neural computing on such a two-qubit channel establishes the requirements for a meaningful QNN. Such a CV-QNN model with sufficient layers may execute any algorithm implementable on a universal CV quantum computer.

Appendix.: Dissipative dynamics of a two-qubit system

In order to solve the master equation (5), we assume the following initial state for the two-qubit system

$$ \rho (0)=\begin{bmatrix} \rho_{11} & \rho_{12} & \rho_{13} & \rho_{14}\\ \rho_{21} & \rho_{22} & \rho_{23} & \rho_{24}\\ \rho_{31} & \rho_{32} & \rho_{33} & \rho_{34}\\ \rho_{41} & \rho_{42} & \rho_{43} & \rho_{44}\\ \end{bmatrix}. $$

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The general solution of the master equation (5) is too cumbersome. However, for small squeezing r ≪ 1, the analytical expression of the density matrix of the two-qubit system at time t can be obtained as

$$ \begin{array}{@{}rcl@{}} \rho_{11}(t)&=& \rho_{11} e^{-4at}, \\ \rho_{12}(t)&=& \frac{1}{2}\rho_{12} (e^{-4at}+e^{-2at})+\frac{1}{2}\rho_{13} (e^{-4at}-e^{-2at}), \\ \rho_{13}(t)&=& \frac{1}{2}\rho_{12} (e^{-4at}-e^{-2at})+\frac{1}{2}\rho_{13} (e^{-4at}+e^{-2at}), \\ \rho_{14}(t)&=& \rho_{14} e^{-2at}, \\ \rho_{22}(t)&=& 2\rho_{11} e^{-4at}at+\frac{e^{-4at}}{4}(\rho_{22}+\rho_{23}+\rho_{32}+\rho_{33})\\ &&+\frac{e^{-2at}}{2}(\rho_{22}-\rho_{33})+\frac{1}{4}(\rho_{22}-\rho_{23} - \rho_{32}+\rho_{33}), \\ \rho_{23}(t)&=& 2\rho_{11} e^{-4at}at+\frac{e^{-4at}}{4}(\rho_{22}+\rho_{23}+\rho_{32}+\rho_{33})\\ &&+\frac{e^{-2at}}{2}(\rho_{23}-\rho_{32}), \\ \rho_{24}(t)&=& -e^{-4at}(\rho_{12}+\rho_{13})+\frac{e^{-2at}}{2}\\ &&\times(2\rho_{12}+2\rho_{13}+\rho_{24}+\rho_{34})+\frac{1}{2}(\rho_{24}-\rho_{34}), \\ \rho_{33}(t)&=& 2\rho_{11} e^{-4at}at+\frac{e^{-4at}}{4}(\rho_{22}+\rho_{23}+\rho_{32}+\rho_{33})\\ &&-\frac{e^{-2at}}{2}(\rho_{22}-\rho_{33})+\frac{1}{4}(\rho_{22}-\rho_{23} - \rho_{32}+\rho_{33}), \\ \rho_{34}(t)&=&-e^{-4at}(\rho_{12}+\rho_{13})+\frac{e^{-2at}}{2}\\ &&\times(2\rho_{12}+2\rho_{13}+\rho_{24}+\rho_{34})-\frac{1}{2}(\rho_{24}-\rho_{34}), \end{array} $$

$$ \begin{array}{@{}rcl@{}} \rho_{44}(t)&=& \rho_{11}(1-4e^{-4at}at)-e^{-4at}\\ &&\times (\rho_{22}+\rho_{23}+\rho_{32}+\rho_{33})\\ && +\frac{1}{2}(2\rho_{11}+\rho_{22}+\rho_{23}+\rho_{32}+\rho_{33}+2\rho_{44}). \end{array} $$

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where $a=\frac {\Gamma }{2} \cosh ^{2}r$. Note that $\rho _{ji}(t)= \rho _{ij}^{*}(t)$ and also ρ₁₁(t) + ρ₂₂(t) + ρ₃₃(t) + ρ₄₄(t) = 1.