  Engineering deterministic photonic gates with simple resources is one of the
long-standing challenges in photonic quantum computing. Here, we design a
passive conditional gate between co-propagating photons using an array of only
two-level emitters. The key resource is to harness the effective photon-photon
interaction induced by the chiral coupling of the emitter array to two
waveguide modes with different resonant momenta at the emitter's transition
frequency. By studying the system's multi-photon scattering response, we
demonstrate that, in certain limits, this configuration induces a non-linear
$\pi$-phase shift between the polariton eigenstates of the system without
distorting spectrally the wavepackets. Then, we show how to harness this
non-linear phase shift to engineer a conditional, deterministic photonic gate
in different qubit encodings, with a fidelity arbitrarily close to 1 in the
limit of large number of emitters and coupling efficiency. Our configuration
can be implemented in topological photonic platforms with multiple chiral edge
modes, opening their use for quantum information processing, or in other setups
where such chiral multi-mode waveguide scenario can be obtained, e.g., in
spin-orbit coupled optical fibers or photonic crystal waveguides.
