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Is ist possible to do the grover amplification while excluding states with amplitude values of zero? I was testing this with the classic grover and two diffrent superpositions. In the first I used the normal superposition created through h-gates. In the second I used a superposition in which the state |00> has the amplitude 0.

Classic Grover:

Grover circuit with h-gate superposition

With Amplitude:

qsphere of above grover circuit

Grover whith null amplitude in state |00>:

Grover circuit with null amplitude

With Amplitude:

qsphere of above grover circuit

I want to ignore the null amplitude and have a result in which all amplitudes besides |11> are also 0.

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  • $\begingroup$ What do you mean by exclude? If you do not mark a basis state with an oracle, its amplitude will be in the end equal (almost) zero. There is no need to exclude some states ex ante. States that are not marked with oracle will be excluded by operations of Grover algorithm. $\endgroup$
    – Martin Vesely
    Jul 21, 2022 at 16:11
  • $\begingroup$ I want to mark the state |011>. The amplitude of every other state should be 0 after I used my grover circuit once. This works easily if the initial state is a Superposition achieved though hadarmad gates. The Problem is for other initial states. As seen in the second circuit the state |000> has already an Amplitude of zero bevor the gorver circuit is applied. The other states all have the same amplitude. After the grover diffuser Step the Amplitudes of the states are mirrored by the mean of all amplitudes. But since state |000> has an amplitude of 0 the mean is not the same as in circuit 1. $\endgroup$
    – MrWasabi
    Jul 25, 2022 at 14:22
  • $\begingroup$ Just few notes. 1) Your circuit is designed for marking two-qubit states and you want to mark three-qubit state. 2) Ancilla qubit ($q_2$) should be initialized with gate $X$ followed by Hadamard gate to get init state $|-\rangle$. 3) Qubits $q_0$ and $q_1$ should be initialized to state $|+\rangle$ with Hadamard gate. 4) The oracle is a little bit strange, first decide which state you want to mark and then implement the oracle with Toffoli gate and $X$ gates. $\endgroup$
    – Martin Vesely
    Jul 25, 2022 at 14:32
  • $\begingroup$ You are right of course, but that is not the acutal Problem I have. Even with the circuit you describe, unmarked states, that already have an amplitude of 0 bevor the diffuser is applied will influence the end sate. Basic example: amplitudes of the 4 state |x1x2> are: 0, 0.577, 0.577, -0.577. The Diffuser will mirror them on their mean: 0.14425. Basically I want a circuit which will thake the mean of only (0.577 + 0.577 - 0.577) / 3 instead of (0 + 0.577 + 0.577 - 0.577) / 4. $\endgroup$
    – MrWasabi
    Jul 25, 2022 at 14:51
  • $\begingroup$ I see...I would say that the Grover algorithm will not work under this setting. If you have a look at the derivation of its action, there is an assumption that input state is uniformly distributed superposition, i.e. all basis states are present in the init state. $\endgroup$
    – Martin Vesely
    Jul 25, 2022 at 14:53

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