  Heralded near-deterministic multi-qubit controlled phase gates with
integrated error detection have recently been proposed by Borregaard et al.
[Phys. Rev. Lett. 114, 110502 (2015)]. This protocol is based on a single
four-level atom (a heralding quartit) and $N$ three-level atoms (operational
qutrits) coupled to a single-resonator mode acting as a cavity bus. Here we
generalize this method for two distant resonators without the cavity bus
between the heralding and operational atoms. Specifically, we analyze the
two-qubit controlled-Z gate and its multi-qubit-controlled generalization
(i.e., a Toffoli-like gate) acting on the two-lowest levels of $N$ qutrits
inside one resonator, with their successful actions being heralded by an
auxiliary microwave-driven quartit inside the other resonator. Moreover, we
propose a circuit-quantum-electrodynamics realization of the protocol with flux
and phase qudits in linearly-coupled transmission-line resonators with
dissipation. These methods offer a quadratic fidelity improvement compared to
cavity-assisted deterministic gates.
