Simulating open quantum dynamics on an NMR quantum processor using the Sz.-Nagy dilation algorithm

We experimentally implement the Sz.-Nagy dilation algorithm to simulate open quantum dynamics on a nuclear magnetic resonance quantum processor. The Sz.-Nagy algorithm enables the simulation of the dynamics of an $n$-qubit system using $n+1$ qubits. We experimentally simulate the action of three nonunitary processes, namely, a phase damping channel acting independently on two qubits, a two-qubit correlated amplitude damping channel, and a magnetic-field-gradient pulse acting on an ensemble of two coupled nuclear spin-$\frac{1}{2}$ particles. To evaluate the quality of the experimentally simulated quantum process, we perform convex-optimization-based full quantum process tomography to reconstruct the quantum process from the experimental data and compare it with the target quantum process to be simulated.

Figure 1

(a) Molecular structure of ${}^{13}\mathrm{C}$-labeled diethyl fluoromalonate used as the three-qubit quantum system. The spectra shown in (b), (c), and (d) correspond to the ${}^1\mathrm{H}, {}^{19}\mathrm{F}$, and ${}^{13}\mathrm{C}$ spins, respectively, obtained after applying a ${90}^{\circ }$ readout pulse on the $|000\rangle$ pseudopure state. The $J$ couplings between different nuclei are $J_{\text{HF}}=47.5$ Hz, $J_{\text{HC}}=161.5$ Hz, and $J_{\text{FC}}=-191.7$ Hz. The spin-lattice relaxation times measured for different nuclei are $T_1^{\text{H}}=3.0\pm 0.34$ s, $T_1^{\text{F}}=3.3\pm 0.15$ s, and $T_1^{\text{C}}=3.2\pm 0.38$ s, while the spin-spin relaxation times measured for different nuclei are $T_2^{\text{H}}=1.3\pm 0.24$ s, $T_2^{\text{F}}=1.4\pm 0.22$ s, and $T_2^{\text{C}}=1.2\pm 0.18$ s.

Figure 2

(a) Quantum circuit to simulate the action of the Kraus operator $A_1$ of the phase damping channel on the initial state $|00\rangle$ using the Sz.-Nagy algorithm. As required by the algorithm, the input state $|00\rangle$ is encoded as ${|000\rangle }_{\mathrm{HFC}}={|0\rangle }_{\mathrm{H}}\otimes {|00\rangle }_{\mathrm{FC}}$. The unitary dilation operator $U_{A_1}$ is realized using eight cnot gates and eight single-qubit rotation gates $R_{\varphi }^{\theta }$. (b) NMR implementation of the quantum circuit given in (a). Gray and black closed rectangles represent $\pi /2$ and $\pi$ pulses, respectively. The angles of the pulses represented by open rectangles are shown above each pulse, where ${\theta }_1=0.3737\times \frac{\pi }{2}$. The dashed rectangular blocks consist of a set of pulses which have been expanded and represented in (c). The phases are written below the corresponding pulse. The free evolution time periods are set to $\tau =0.0078$ s, ${\tau }_1=0.0105$ s, and ${\tau }_2=0.0031$ s, respectively. The measurement box is indicated by a decaying time-domain NMR signal, corresponding to tomographic measurements on all three qubits, to compute density matrix elements $\left\{{\rho }_{ij}\right\}$, with $0\le i,j\le 4$.

Figure 3

Bar plots depicting the average normalized trace distance between the experimentally obtained output Hermitian matrix ${\left(A_i{\rho }_j{A}_i^{†}\right)}_{\mathrm{expt}}$ and the theoretically expected matrix ${\left(A_i{\rho }_j{A}_i^{†}\right)}_{\mathrm{theor}}$ for the phase damping channel. Two-qubit states ${\rho }_j$ are numbered from 1 to 16 on the $x$ axis. The black closed, red cross-hatched, black horizontally lined, and blue diagonally lined bars correspond to the Kraus operators $A_1, A_2, A_3$, and $A_4$, respectively.

Figure 4

Process matrices obtained by theoretically $\left({\xi }_{\mathrm{sim}}^{\mathrm{theor}}\right)$ and experimentally $\left({\xi }_{\mathrm{sim}}^{\mathrm{expt}}\right)$ simulating phase damping channels acting independently on each qubit in a two-qubit NMR system. The bar plots in the left column represent the real part of the process matrices of the theoretically simulated [$\mathrm{Re}\left({\xi }_{\mathrm{sim}}^{\mathrm{theor}}\right)$] and the experimentally obtained phase damping channel computed using convex-optimization-based QPT [$\mathrm{Re}\left({\xi }_{\mathrm{sim}}^{\mathrm{expt}}\right)$], respectively. The bar plots in the right column represent the imaginary part of the respective process matrices. The average process fidelity of the experimentally simulated channel turns out to be $0.8883\pm 0.0375$.

Figure 5

Effect of an independent phase damping channel on an experimentally simulated $|++\rangle$ state using the Sz.-Nagy algorithm. Represented is the damping of the real part of the off-diagonal density matrix elements (a) ${\rho }_{12}$ and (b) ${\rho }_{13}$. Time $t$ is given in seconds on the $x$ axis. The black closed and red cross-hatched bars correspond to theoretical and experimental values, respectively.

Figure 6

Process matrices obtained by theoretically $\left({\xi }_{\mathrm{sim}}^{\mathrm{theor}}\right)$ and experimentally $\left({\xi }_{\mathrm{sim}}^{\mathrm{expt}}\right)$ simulating the two-qubit fully correlated amplitude damping channel on the two-qubit NMR system. The bar plots in the left column represent the real part of the process matrices of the theoretically simulated [$\mathrm{Re}\left({\xi }_{\mathrm{sim}}^{\mathrm{theor}}\right)$] and the experimentally obtained fully correlated amplitude damping channel computed using convex-optimization-based QPT [$\mathrm{Re}\left({\xi }_{\mathrm{sim}}^{\mathrm{expt}}\right)$], respectively. The bar plots in the right column represent the imaginary parts of the respective process matrices. The average process fidelity of the experimentally simulated channel turns out to be $0.9216\pm 0.0017$.

Figure 7

Bar plots corresponding to the average normalized trace distance between the experimentally obtained output Hermitian matrix ${\left(A_i^{\text{CAD}}{\rho }_j{A_i^{\text{CAD}}}^{†}\right)}_{\mathrm{expt}}$ and the theoretically expected matrix ${\left(A_i^{\text{CAD}}{\rho }_j{A_i^{\text{CAD}}}^{†}\right)}_{\mathrm{theor}}$ for the correlated amplitude damping channel. Two-qubit states ${\rho }_j$ are numbered from 1 to 16 on the $x$ axis. The black closed and red cross-hatched bars correspond to Kraus operators $A_1^{\text{CAD}}$ and $A_2^{\text{CAD}}$, respectively.

Figure 8

Effect of a correlated amplitude damping channel on an experimentally simulated $|11\rangle$ state using the Sz.-Nagy algorithm: (a) the ground-state population $P_{00}$ and (b) the excited-state population $P_{11}$. The values of the damping parameter $p$ are represented on the $x$ axis. The black closed and red cross-hatched bars denote theoretical and experimental values, respectively.

Figure 9

Bar plots corresponding to the average normalized trace distance between the experimentally simulated output Hermitian matrix ${\left(A_i{\rho }_j{A}_i^{†}\right)}_{\mathrm{sim}}^{\mathrm{expt}}$ using the Sz.-Nagy algorithm and the experimentally obtained matrix ${\left(A_i{\rho }_j{A}_i^{†}\right)}_{\mathrm{QPT}}^{\mathrm{expt}}$ via quantum process tomography of MFGP implemented on two qubits. Two-qubit states ${\rho }_j$ are numbered from 1 to 16 on the $x$ axis. The black closed, red cross-hatched, black horizontally lined, and blue diagonally lined bars correspond to Kraus operators $A_1, A_2, A_3$, and $A_4$, respectively.

Figure 10

In the row $\mathrm{Re}\left({\eta }_{\mathrm{QPT}}^{\mathrm{expt}}\right)$ and $\mathrm{Im}\left({\eta }_{\mathrm{QPT}}^{\mathrm{expt}}\right)$ denote the real and imaginary parts of the experimental process matrix obtained by performing quantum process tomography of the shaped MFGP applied on two qubits. In the bottom row $\mathrm{Re}\left({\eta }_{\mathrm{sim}}^{\mathrm{expt}}\right)$ and $\mathrm{Im}\left({\eta }_{\mathrm{sim}}^{\mathrm{expt}}\right)$ represent the real and imaginary parts of the process matrix of the same MFGP experimentally simulated using the Sz.-Nagy algorithm. The process fidelity between ${\eta }_{\mathrm{QPT}}^{\mathrm{expt}}$ and ${\eta }_{\mathrm{sim}}^{\mathrm{expt}}$ turns out to be $0.8687\pm 0.0192$.