Compressed sensing quantum process tomography for superconducting quantum gates

An important challenge in quantum information science and quantum computing is theexperimental realization of high-fidelity quantum operations on multi-qubit systems. Quantum processtomography (QPT) is a procedure devised to fully characterize a quantum operation. We firstpresent the results of the estimation of the process matrix for superconducting multi-qubit quantumgates using the full data set employing various methods: linear inversion, maximum likelihood, andleast-squares. To alleviate the problem of exponential resource scaling needed to characterize amulti-qubit system, we next investigate a compressed sensing (CS) method for QPT of two-qubitand three-qubit quantum gates. Using experimental data for two-qubit controlled-Z gates, takenwith both Xmon and superconducting phase qubits, we obtain estimates for the process matriceswith reasonably high fidelities compared to full QPT, despite using significantly reduced sets ofinitial states and measurement configurations. We show that the CS method still works when theamount of data is so small that the standard QPT would have an underdetermined system of equations.We also apply the CS method to the analysis of the three-qubit Toffoli gate with simulatednoise, and similarly show that the method works well for a substantially reduced set of data. For the CS calculations we use two different bases in which the process matrix is approximately sparse (thePauli-error basis and the singular value decomposition basis), and show that the resulting estimatesof the process matrices match with reasonably high fidelity. For both two-qubit and three-qubitgates, we characterize the quantum process by its process matrix and average state fidelity, as wellas by the corresponding standard deviation defined via the variation of the state fidelity for differentinitial states. We calculate the standard deviation of the average state fidelity both analyticallyand numerically, using a Monte Carlo method. Overall, we show that CS QPT offers a significantreduction in the needed amount of experimental data for two-qubit and three-qubit quantum gates.