  High-fidelity single- and two-qubit gates are essential building blocks for a
fault-tolerant quantum computer. While there has been much progress in
suppressing single-qubit gate errors in superconducting qubit systems,
two-qubit gates still suffer from error rates that are orders of magnitude
higher. One limiting factor is the residual ZZ-interaction, which originates
from a coupling between computational states and higher-energy states. While
this interaction is usually viewed as a nuisance, here we experimentally
demonstrate that it can be exploited to produce a universal set of fast single-
and two-qubit entangling gates in a coupled transmon qubit system. To implement
arbitrary single-qubit rotations, we design a new protocol called the two-axis
gate that is based on a three-part composite pulse. It rotates a single qubit
independently of the state of the other qubit despite the strong ZZ-coupling.
We achieve single-qubit gate fidelities as high as 99.1% from randomized
benchmarking measurements. We then demonstrate both a CZ gate and a CNOT gate.
Because the system has a strong ZZ-interaction, a CZ gate can be achieved by
letting the system freely evolve for a gate time $t_g=53.8$ ns. To design the
CNOT gate, we utilize an analytical microwave pulse shape based on the SWIPHT
protocol for realizing fast, low-leakage gates. We obtain fidelities of 94.6%
and 97.8% for the CNOT and CZ gates respectively from quantum progress
tomography.
