# Holonomic Gates in Pseudo-Hermitian Quantum Systems

## TL;DR

This paper extends holonomic quantum computation to pseudo-Hermitian systems, demonstrating how non-Abelian geometric phases can implement quantum gates in open quantum systems using a biorthogonal approach.

## Contribution

It introduces a framework for holonomic quantum gates in pseudo-Hermitian systems, showing how geometric phases lead to pseudounitary gates in open quantum systems.

## Key findings

- Non-Abelian geometric phases emerge in pseudo-Hermitian systems under adiabatic evolution.
- Pseudo-Hermitian gain/loss systems can realize arbitrary pseudo-U(2) transformations.
- The framework enables holonomic quantum computation in open quantum systems.

## Abstract

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.