Explicit definition of $\mathcal{P}\mathcal{T}$ symmetry for non-unitary quantum walks with gain and loss

$\mathcal{P}\mathcal{T}$ symmetry, that is, a combined parity and time-reversal symmetry is a key milestone for non-Hermite systems exhibiting entirely real eigenenergy. In the present work, motivated by a recent experiment, we study $\mathcal{P}\mathcal{T}$ symmetry of the time-evolution operator of non-unitary quantum walks. We present the explicit definition of $\mathcal{P}\mathcal{T}$ symmetry by employing a concept of symmetry time frames. We provide a necessary and sufficient condition so that the time-evolution operator of the non-unitary quantum walk retains $\mathcal{P}\mathcal{T}$ symmetry even when parameters of the model depend on position. It is also shown that there exist extra symmetries embedded in the time-evolution operator. Applying these results, we clarify that the non-unitary quantum walk in the experiment does have $\mathcal{P}\mathcal{T}$ symmetry.