Experimental realization of quantum non-Markovianity through the convex mixing of Pauli semigroups on an NMR quantum processor

This experimental study aims to investigate the convex combinations of Pauli semigroups with arbitrary mixing parameters to determine whether the resulting dynamical map exhibits Markovian or non-Markovian behavior. Specifically, we consider the cases of equal as well as unequal mixing of two Pauli semigroups, and demonstrate that the resulting map is always non-Markovian. Additionally, we study three cases of three-way mixing of the three Pauli semigroups and determine the Markovianity or non-Markovianity of the resulting maps by experimentally determining the decay rates. To simulate the nonunitary dynamics of a single-qubit system with different mixing combinations of Pauli semigroups on an NMR quantum processor, we use an algorithm involving two ancillary qubits. The experimental results align with the theoretical predictions.

Figure 1

The structure of the molecule trifluoroiodoethylene with three NMR active $\mathrm{spin}-1/2{}^{19}\mathrm{F}$ nuclei acting as three qubits, along with the NMR spectra of the pseudopure state $|000\rangle$ which represents the initial state of the three-qubit system. The $x$ axis represents the frequency scale presented in parts per million (ppm). The negative values on the $x$ axis represent the frequency offset from the reference frequency (indicating upfield shifts for ${}^{19}\mathrm{F}$ nuclei).

Figure 2

(a) Schematic of the circuit used to simulate the dynamical map obtained from the convex combination of two and three Pauli dynamical maps. For both two- and three-dynamical-map mixing, the controlled operation $U$ is the same. ${\sigma }_i$ denote the Pauli matrices, with ${\sigma }_0$ being the identity matrix. The unitary operation $V$ is different for the cases of two-way and three-way mixing. The $W$ operation is equivalent to the identity operation and hence not implemented experimentally. (b) The NMR pulse sequence used to simulate the map. The rectangular shapes represent rf pulses of differing angles and phases (which are written on the top of each pulse). cnot operations between two qubits are represented by blue lines between the corresponding qubits. Step 1 corresponds to the preparation of the input state. Gradient pulses are represented by shaped green curves, while the GRAPE-optimized pulse to implement Step 2 of the circuit is represented by a large dark green curve, applied simultaneously on all three qubits. Step 3 corresponds to measurements on all three qubits.

Figure 3

Bar plots illustrating the real (Re) and imaginary (Im) components of the theoretical (Th.) and experimental (Expt.) density matrices for the specific case of unequal mixing of two Pauli semigroups at $t=0.1$ ms.

Figure 4

Convex combination of two Pauli semigroups for the case of equal mixing. (a) Comparison of the theoretical and experimental decoherence parameters $p\left(t\right)$. (b) Comparison of theoretical and experimental decay rates ${\gamma }_1\left(t\right),{\gamma }_2\left(t\right),{\gamma }_3\left(t\right)$ with mixing parameter $a=0.5$. The red dashed and blue curves represent the theoretical and the fit to the experimental parameters, respectively. Experimental data points with error bars are represented by blue dots. The decay rate ${\gamma }_1\left(t\right)$ is negative throughout, indicating non-Markovianity.

Figure 5

Convex combination of two Pauli dynamical maps for the case of inequal mixing. (a) Comparison of the theoretical and experimental decoherence parameters $p\left(t\right)$. (b) Comparison of theoretical and experimental decay rates ${\gamma }_1\left(t\right),{\gamma }_2\left(t\right),{\gamma }_3\left(t\right)$ with mixing parameter $a=0.25$. The red dashed and blue curves represent the theoretical and the fit to the experimental parameters, respectively. Experimental data points with error bars are represented by blue dots. The decay rate ${\gamma }_1\left(t\right)$ is negative throughout, indicating non-Markovianity.

Figure 6

Convex combination of three Pauli dynamical maps for the case of equal mixing. (a) Comparison of the theoretical and experimental decoherence parameters $p\left(t\right)$. (b) Comparison of theoretical and experimental decay rates ${\gamma }_1\left(t\right),{\gamma }_2\left(t\right),{\gamma }_3\left(t\right)$ with mixing parameters $x_1=x_2=x_3=0.33$. The red dashed and blue curves represent the theoretical and the fit to the experimental parameters, respectively. Experimental data points with error bars are represented by blue dots. All the decay rates are positive, indicating that the resulting map is Markovian.

Figure 7

Convex combination of three Pauli dynamical maps for the case of inequal mixing. (a) Comparison of the theoretical and experimental decoherence parameters $p\left(t\right)$. (b) Comparison of theoretical and experimental decay rates ${\gamma }_1\left(t\right),{\gamma }_2\left(t\right),{\gamma }_3\left(t\right)$ with mixing parameters $x_1=x_3=0.3,x_2=0.4$, respectively. The red dashed and blue curves represent the theoretical and the fit to the experimental parameters, respectively. Experimental data points with error bars are represented by blue dots. All the decay rates are positive, indicating that the resulting map is Markovian.

Figure 8

Convex combination of three Pauli dynamical maps for the case of inequal mixing. (a) Comparison of the theoretical and experimental decoherence parameters $p\left(t\right)$. (b)–(d) Comparison of theoretical and experimental decay rates ${\gamma }_1\left(t\right),{\gamma }_2\left(t\right),{\gamma }_3\left(t\right)$ with mixing parameters $x_1=0.2\text{and}{x}_2=x_3=0.4$, respectively. The red dashed and blue curves represent the theoretical and the fit to the experimental parameters, respectively. Experimental data points with error bars are represented by blue dots. The negativity of the decay rate ${\gamma }_1\left(t\right)$ indicates non-Markovianity of the resulting map.