  Quasiparticle poisoning, expected to arise during the measurement of Majorana
zero mode state, poses a fundamental problem towards the realization of
Majorana-based quantum computation. Parafermions, a natural generalization of
Majorana fermions, can encode topological qudits immune to quasiparticle
poisoning. While parafermions are expected to emerge in superconducting
fractional quantum Hall systems, they are not yet attainable with current
technology. To bypass this problem, we employ a photonic quantum simulator to
experimentally demonstrate the key components of parafermion-based universal
quantum computation. Our contributions in this article are twofold. First, by
manipulating the photonic states, we realize Clifford operator Berry phases
that correspond to braiding statistics of parafermions. Second, we investigate
the quantum contextuality in a topological system for the first time by
demonstrating the contextuality of parafermion encoded qudit states.
Importantly, we find that the topologically-encoded contextuality opens the way
to magic state distillation, while both the contextuality and the
braiding-induced Clifford gates are resilient against local noise. By
introducing contextuality, our photonic quantum simulation provides the first
step towards a physically robust methodology for realizing topological quantum
computation.
