Technically, the answer is yes, but not in the way you want.
My laptop computer (given sufficient disk space) can simulate a super computer, which has about 100,000 times as many processors as my laptop computer, has communications 10x faster, and takes up about 100,000 times more space than my laptop computer.
How? Well, the supercomputer, like matter, is built up of many smaller, identical parts. And like matter, they behave in very predictable ways. And even interact with each other in very predictable ways. These rules can easily fit into the memory of my laptop, and a team of a few dozen programmers can write a good simulator.
So what's the catch? Time. It would take my laptop about 10,000 times longer to simulate the supercomputer than just running the supercomputer.
One of the things we run on the supercomputer is quantum chemistry simulations. It takes the supercomputer 30 minutes to simulate 0.01 seconds of quantum chemistry. About 180,000 times as long as the actual quantum reaction would take. And that is just a few hundred atoms, the simulation time increases exponentially as the atoms increase, like double the atoms and the runtime goes up by 100x.
Simulation cannot be real time.
We already execute your strategy in real life. I work with engineers that design processors and chips. We've developed simulators for them, to test their circuits before they spend hundreds of thousands actually building them. Our simulators are nearly perfect on the physics and chemistry, so much so that if the simulator says the circuit will work, and won't overheat, and so on, it turns out that way when it is fabricated.
So they feed in exactly the same specifications they'd send to the fabricator, we build an internal simulation of all the circuits, and they can poke it and prod it and make it run in simulation.
But it takes hours for our state-of-the-art simulator to simulate what would be microseconds of real-life operations. That's enough on a processor, microseconds of circuit simulation is enough. That is thousands of cycles.
Your problem is not whether the simulation of a bigger thing by a smaller thing is possible. Your problem is time itself, and you cannot escape it: Simulation, especially of quantum phenomena, will always take at least thousands of times longer than reality.