Kernel-based quantum regressor models learning non-Markovianity

Quantum machine learning is a growing research field that aims to perform machine learning tasks assisted by a quantum computer. Kernel-based quantum machine learning models are paradigmatic examples where the kernel involves quantum states, and the Gram matrix is calculated from the overlap between these states. With the kernel at hand, a regular machine learning model is used for the learning process. In this paper we investigate the quantum support vector machine and quantum kernel ridge models to predict the degree of non-Markovianity of a quantum system. We perform digital quantum simulation of amplitude damping and phase damping channels to create our quantum data set. We elaborate on different kernel functions to map the data and kernel circuits to compute the overlap between quantum states. We show that our models deliver accurate predictions that are comparable with the fully classical models.

Figure 1

Quantum circuits compute the overlap between two quantum states in the kernel function to calculate the Gram matrix. For the inversion test $U$ represents either the amplitude damping or phase damping channel depicted in Fig. 2. For the ancilla-based algorithm (ABA) $U=T^{†}H$ [60].

Figure 2

Quantum circuits for simulating AD and PD channels.

Figure 3

Expectation values delivered by the noisy qasm_simulator exhibit small dispersion after 8000 shots, in contrast to the ideal statevector_simulator. We only observe correlations in the plane defined by $\langle {\sigma }_x\rangle$ and $\langle {\sigma }_z\rangle$.

Figure 4

QSVM prediction of non-Markovianity as a function of the rotation angle $\theta$ for different kernel circuits. The inversion test outperforms the others. We considered the amplitude damping channel, the exponential kernel function, and the hyperparameters $\{C=0.5,\epsilon =0.01\}$.

Figure 5

Both QSVM and QKRR deliver accurate predictions of the degree of non-Markovianity, based on the mean squared error score. For a small training data set QSVM performs better (not shown here). For a sufficiently large number of points QKRR provides a smaller mean squared error.

Figure 6

Exponential kernel function delivers the best prediction of non-Markovianity.