# Holonomic Gates in Pseudo-Hermitian Quantum Systems. (arXiv:1906.12058v2 [quant-ph] UPDATED)

The time-dependent pseudo-Hermitian formulation of quantum mechanics allows
to study open system dynamics in analogy to Hermitian quantum systems. In this
setting, we show that the notion of holonomic quantum computation can equally
be formulated for pseudo-Hermitian systems. Starting from a degenerate
pseudo-Hermitian Hamiltonian we show that, in the adiabatic limit, a
non-Abelian geometric phase emerges which realizes a pseudounitary quantum
gate. We illustrate our findings by studying a pseudo-Hermitian gain/loss
system which can be written in the form of a tripod Hamiltonian by using the
biorthogonal representation. It is shown that this system allows for arbitrary
pseudo-$\mathrm{U}(2)$ transformations acting on the dark subspace of the
system.