We report detailed dc and ac magnetic susceptibilities, specific heat, and thermal conductivity measurements on the frustrated magnet ZnCr$_2$Se$_4$. At low temperatures, with increasing magnetic field, this spinel material goes through a series of spin state transitions from the helix spin state to the spiral spin state and then to the fully polarized state. Our results indicate a direct quantum phase transition from the spiral spin state to the fully polarized state. As the system approaches the quantum criticality, we find strong quantum fluctuations of the spins with the behaviors such as an unconventional $T^2$-dependent specific heat and temperature independent mean free path for the thermal transport. We complete the full phase diagram of ZnCr$_2$Se$_4$ under the external magnetic field and propose the possibility of frustrated quantum criticality with extended densities of critical modes to account for the unusual low-energy excitations in the vicinity of the criticality. Our results reveal that ZnCr$_2$Se$_4$ is a rare example of 3D magnet exhibiting a field-driven quantum criticality with unconventional properties.
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