  A possible building block for a scalable quantum computer has recently been
demonstrated [M. Mariantoni et al., Science 334, 61 (2011)]. This architecture
consists of superconducting qubits capacitively coupled both to individual
memory resonators as well as a common bus. In this work we study a natural
primitive entangling gate for this and related resonator-based architectures,
which consists of a CZ operation between a qubit and the bus. The CZ gate is
implemented with the aid of the non-computational qubit |2> state [F. W.
Strauch et al., Phys. Rev. Lett. 91, 167005 (2003)]. Assuming phase or transmon
qubits with 300 MHz anharmonicity, we show that by using only low frequency
qubit-bias control it is possible to implement the qubit-bus CZ gate with 99.9%
(99.99%) fidelity in about 17ns (23ns) with a realistic two-parameter pulse
profile, plus two auxiliary z rotations. The fidelity measure we refer to here
is a state-averaged intrinsic process fidelity, which does not include any
effects of noise or decoherence. These results apply to a multi-qubit device
that includes strongly coupled memory resonators. We investigate the
performance of the qubit-bus CZ gate as a function of qubit anharmonicity,
indentify the dominant intrinsic error mechanism and derive an associated
fidelity estimator, quantify the pulse shape sensitivity and precision
requirements, simulate qubit-qubit CZ gates that are mediated by the bus
resonator, and also attempt a global optimization of system parameters
including resonator frequencies and couplings. Our results are relevant for a
wide range of superconducting hardware designs that incorporate resonators and
suggest that it should be possible to demonstrate a 99.9% CZ gate with existing
transmon qubits, which would constitute an important step towards the
development of an error-corrected superconducting quantum computer.
