Parametrized process characterization with reduced resource requirements

Quantum process tomography (QPT) is a powerful tool to characterize quantum operations, but it requires considerable resources, making it impractical for more than two-qubit systems. This work proposes an alternative approach that requires significantly fewer resources for unitary process characterization with a built-in method for state preparation and measurement error mitigation. By measuring the quantum process as rotated through the $X$ and $Y$ axes on the Bloch sphere, we can acquire enough information to reconstruct the quantum process matrix $\chi$ and measure its fidelity. We test the algorithm's performance against standard QPT using simulated and physical experiments on several IBM quantum processors and compare the resulting process matrices. We demonstrate with numerical experiments that the method can improve gate fidelity via a noise reduction in the imaginary part of the process matrix, along with a stark decrease in the number of experiments needed to perform the characterization.

Figure 1

Number of quantum circuit executions for quantum process characterization: here, we compare the resources used by different QPT techniques against the resources used by PPC. For different numbers of rotations ($N_{\theta }=31,51$, and 71 lines), standard QPT surpasses PPC in the number of resources required for its execution for $n>2$, and state-of-the-art techniques like ACTQPT and BKD surpass PPC for $n>6$.

Figure 2

Diagrammatic sketch of PPC in the Bloch sphere for one-qubit systems. With the information obtained from rotation on states $\mathcal{U}|0\rangle$ and $\mathcal{U}|1\rangle$ we get information about the quantum process represented by $\mathcal{U}$. The dotted lines represent the rotations of states $\mathcal{U}|0\rangle$ and $\mathcal{U}|1\rangle$ around the $y$ axis. In order to clarify this procedure, the inset displays the mean value ${\langle {\sigma }_z\rangle }_{{\theta }_s}$ (the experiment's output) as a function of ${\theta }_s$. There, the behavior of ${\langle {\sigma }_z\rangle }_{{\theta }_s}$ for different values of ${\theta }_s$ gives information about the observables ${\langle {\sigma }_z\rangle }_{\phi }$ and ${\langle {\sigma }_x\rangle }_{\phi }$ that conform the tomography information of the state $|\phi \rangle =\mathcal{U}|{\phi }_0\rangle$.

Figure 3

Comparison between tomography results for the one-qubit system. We present the $\chi$-matrix representation in the Pauli matrices for the Hadamard gate $H$ using QPT and PPC. In (a) and (b), we depict the $\chi$-matrix using QPT and PPC, respectively, by running the experiment in a noiseless simulator. In (c) and (d), we show the resulting $\chi$ matrices for QPT and PPC, respectively, by running the experiment on the zeroth qubit on the IBMQ Bogota back end. In both cases, we considered $N=5000$ shots and least-squares estimation for QPT and $N=5000$ shots and $N_{\theta }=51$ for PPC. There is good agreement in the $\chi$ matrix's real part but a difference in the imaginary part, even when using the noiseless simulator. In the noiseless simulator, the imaginary part is statistically irrelevant since the statistical error is on the order of $N^{-1/2}/2\sim 7\times {10}^{-3}$ [40]. On the other hand, in the QPT experiments, the imaginary part is statistically relevant due to SPAM errors. At the bottom, we present the $d_{\infty }(·)$ distance between the resulting process matrices.

Figure 4

$\chi$ matrix for the cx gate: Here, we depict the $\chi$-matrix representation in Pauli matrices of the cnot gate obtained using the QPT and PPC methods. In (a) and (b) we use a noiseless simulator to run QPT and PPC, respectively. In (c) and (d), we use qubits 0 and 1 on the IBMQ Bogota back end to run QPT and PPC, respectively. In the experiments we used $N=5000$ shots and least-squares estimation for the QPT configuration and $N=5000$ shots and $N_{\theta }=51$ for the PPC configuration. We obtained results similar to those of the one-qubit case; there is good agreement with the real part but a slight difference in the imaginary ones. Again, some of the imaginary values stem from an inherent error introduced by the estimation procedure in the QPT method. In this case, the imaginary part produced by PPC and QPT in the numerical experiments is still statistically non-significant since we can consider the same level of statistical error used in the one-qubit case [40], i.e., $\sim 7\times {10}^{-3}$. For the physical experiments, the imaginary part overpasses the standard deviations (${\sigma }_{\mathrm{PPA}}\sim 2\times {10}^{-3}$ and ${\sigma }_{\mathrm{QPT}}\sim 6\times {10}^{-3}$), reveling imperfections in the quantum gate. In this case, the distance $d_{\infty }(·,·)$ in the simulation is similar to that of the experimental result.

Figure 5

(a) Preparation and (b) measurement sets of quantum circuits for one-qubit quantum process tomography.

Figure 6

Fitting parameters: Conditional probabilities and initial phase (a)–(c) using $Y_{\theta }$ rotation and (d)–(f) using the $X_{\theta }$ rotation. The error bars represent the standard deviation of each parameter in the fitting process. The shadow regions indicate the optimal values for $N_{\theta }$ and $N$.

Figure 7

Back-end specifications: $n_q$ stands for the number of qubits, ${\overline{\mathcal{E}}}_r$ stands for the average readout error rate, ${\overline{\mathcal{E}}}_{SX}$ stands for the average $SX$ error rate, and ${\overline{\mathcal{E}}}_{CX}$ stands for the average $CX$ error rate.