Quantum Computing Asked on June 26, 2021
I had a question earlier about generating the superposition of all the possible states: Here. In that case, we could apply $H^{otimes n}$ to the state $|0rangle^{otimes n}$, and each state has the same amplitude in the superposition: $|0rangle^{otimes n} to dfrac{1}{sqrt{2^n}}sum_{i=0}^{2^n-1} |irangle $. However, it is possible for us to tune the amplitude of certain states in the superposition? Say if I have 4 qubits and 4 Hadamard gates (one on each), that would generate a superposition of 16 states. Can I add some additional procedures to increase the amplitude of $|0110rangle$ and $|1001rangle$ and the rest states have the same and reduced amplitude?
Thanks!!
Yes indeed you can!
First a simple example: if you want to increase all the amplitudes of the all the states that look like $|cdot cdot cdot 1rangle$ then you just need to apply an $R_y$ gate on the final qubit.
If on the other hand you want to increase the amplitudes of the specific states $|0110rangle$ and $|1001rangle$ by specific amounts, then you need to apply a series of controlled $R_y$ gates which isn't so trivial, and generically takes an exponential depth. If you want to increase their amplitudes, but you don't care by how much specifically, you can use Grover's algorithm as mentioned by Bertrand Einstein IV.
Correct answer by Rajiv Krishnakumar on June 26, 2021
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