  Cross-resonance (CR) gate has emerged as a promising scheme for
fault-tolerant quantum computation with fixed-frequency qubits. We
experimentally implement entangling CR gate by using a microwave-only control
in a tunable coupling superconducting circuit, where the tunable coupler
provides extra degrees of freedom to verify optimal condition for constructing
CR gate. By developing three-qubit CR Hamiltonian tomography protocol, we
systematically investigate the dependency of gate fidelities on spurious qubit
interactions and present the first experimental approach to the evaluation of
the perturbation impact arising from spectator qubits. Our results reveal that
the spectator qubits lead to reductions in CR gate fidelity dependent on ZZ
interaction and particular frequency detunings between spectator and gate
qubits, demonstrating a more serious impact from the target spectator than from
the control spectator. Our experiments uncover optimal CR operation regime and
provide insight into an improvement of the CR gate by suppression of unwanted
qubit interactions.
