Can the Hamiltonian be upscaled to speed up quantum gates? In quantum computing, gates are performed unitarily, i.e. $|\psi\rangle \mapsto U|\psi\rangle$, driven by some Hamiltonian, i.e. $U = \exp(-iHt)$. Consider $H$ as time independent, if it is proportionally scaled up, i.e. $H \mapsto \gamma H$, then the time for applying $U$ would become shorter, i.e., $t \mapsto \frac{t}{\gamma}$, does this speed up computation? (similar to how a classical computer can be overclocked)
 A: Yes, in the simplified model of time-independent Hamiltonian $H$ effecting a quantum gate $U=\exp(-iHt)$ multiplying $H$ by a constant factor $\gamma > 1$ does shorten the gate's duration.
In practice, physics constants, material properties, control electronics etc constrain what Hamiltonians can be engineered on any given hardware platform. For example, in architectures based superconducting qubits where engineers have a high degree of control over a large number of parameters that go into the Hamiltonian, the energy scale is generally constrained to the microwave spectrum.

Also note that while fast gates are preferable to slow gates with the same decoherence because faster gates allow us to run longer algorithms, very high gate speed becomes a mixed blessing when implementing quantum error correction because that entails the need for a classical computer running alongside the quantum device for decoding syndrome measurements. Very fast quantum gates make it more challenging to implement the classical computer to perform quantum error correction.
