The cross-correlation study of the unresolved $\gamma$-ray background (UGRB)<br />
with galaxy clusters has a potential to reveal the nature of the UGRB. In this<br />
paper, we perform a cross-correlation analysis between $\gamma$-ray data by the<br />
Fermi Large Area Telescope (Fermi-LAT) and a galaxy cluster catalogue from the<br />
Subaru Hyper Suprime-Cam (HSC) survey. The Subaru HSC cluster catalogue<br />
provides a wide and homogeneous large-scale structure distribution out to the<br />
high redshift at $z=1.1$, which has not been accessible in previous<br />
cross-correlation studies. We conduct the cross-correlation analysis not only<br />
for clusters in the all redshift range ($0.1 &lt; z &lt; 1.1$) of the survey, but<br />
also for subsamples of clusters divided into redshift bins, the low redshift<br />
bin ($0.1 &lt; z &lt; 0.6$) and the high redshift bin ($0.6 &lt; z &lt; 1.1$), to utilize<br />
the wide redshift coverage of the cluster catalogue. We find the evidence of<br />
the cross-correlation signals with the significance of 2.0-2.3$\sigma$ for all<br />
redshift and low-redshift cluster samples. On the other hand, for high-redshift<br />
clusters, we find the signal with weaker significance level (1.6-1.9$\sigma$).<br />
We also compare the observed cross-correlation functions with predictions of a<br />
theoretical model in which the UGRB originates from $\gamma$-ray emitters such<br />
as blazars, star-forming galaxies and radio galaxies. We find that the detected<br />
signal is consistent with the model prediction.