Quantum $X$-Secure $B$-Byzantine $T$-Colluding Private Information Retrieval

We consider the problems arising from the presence of Byzantine servers in a quantum private information retrieval (QPIR) setting. This is the first work to precisely define what the capabilities of Byzantine servers could be in a QPIR context. We show that quantum Byzantine servers have more capabilities than their classical counterparts due to the possibilities created by the quantum encoding procedure. We focus on quantum Byzantine servers that can apply any reversible operations on their individual qudits. In this case, the Byzantine servers can generate any error, i.e., this covers \emph{all} possible single qudit operations that can be done by the Byzantine servers on their qudits. We design a scheme that is resilient to these kinds of manipulations. We show that the scheme designed achieves superdense coding gain in all cases, i.e., $R_Q= \max \left\{0,\min\left\{1,2\left(1-\frac{X+T+2B}{N}\right)\right\}\right\}$.