There are a few proven ways to reduce sound transmission, but none of them include using silicon mats. As noted above, soundproof is all but unattainable in practice, but a notable reduction in sound transmission is more than doable.
Insulation in the wall and ceiling cavity helps absorb high and mid frequency sounds and will provide a noticeable difference. This is the easiest and most common approach, but lacks in effectiveness for low frequency sounds and impact noise from bass, footsteps, etc.
To address these sounds, you need sheer mass in the wall materials which is most often accomplished through the use of two layers of xheavy fire rated drywall with offset seams. Sounds transmission is significantly reduced as it attempts to pass through heavy mass. Green glue can be used between layers but is a waste of time and money, though fun to use.
For impact noises like footsteps, they travel through framing, mechanical connections, and building materials. The only way to address this is by decoupling your drywall from the framing through the use of steel studs and genie clips. I've never tried this approach but there is plenty of info on the web to research.
All this assumes you are currently building the room and have access. And note all the best efforts can be easily defeated if the room has hvac connections to other parts of the house that allow sound to bypass your efforts
