Is it possible to run grover's diffusion step on a subset of the possible input space?
By this, I mean is it possible to do the diffusion process with a state space isn't in a total superposition of all states.
Let's say I have a 2 qubit system in a equal superposition of states $\left|00\right>$, $\left|01\right>$ and $\left|10\right>$, but have exactly a 0 probability of measuring a $\left|11\right>$. Also, let's say I have a function $f$ that only outputs a 1 for the input 00. Is there a modification to Grover's where I can do the diffusion only on the $\left|00\right>$, $\left|01\right>$ and $\left|10\right>$ states and leave the $\left|11\right>$ state unchanged?
If that is possible, does it speedup the diffusion? Say I have a state with $n$ qubits but $2^x$ non-zero states where $x < n$, does this modified grover's algorithm still run in $\mathcal O(2^{n/2})$, or can it run in $\mathcal O(2^{x/2})$?