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