## How to implement such a gate where one input bit is zero and the other is one and the output should be one?

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I have created 10 qubits, where I have set 2 in superposition by applying Hadamard gate to each of the 2 qubits. So the 2 qubits state can be $$|00\rangle$$, $$|01\rangle$$, $$|10\rangle$$, $$|11\rangle$$. I need to assign values based on the values present in 2 superpositioned qubits. For eg, if the 2 qubits are in $$|00\rangle$$ I need to set the other qubits as $$|00011110\rangle$$, if the 2 qubits are in $$|01\rangle$$ states, I need to set the other qubits as $$|10101010\rangle$$ and if the 2 qubits are in $$|11\rangle$$ I need to set the other qubits as $$|11110000\rangle$$. So in the broad sense after applying H gate to 2 qubits when they have totally 4 states for each state I need to assign qubits other values. I saw I can apply the Toffoli gate if the state is |11⟩. However, if the state is $$|01\rangle$$ or $$|00\rangle$$ then what is the way?

Hi. Welcome to Quantum Computing SE! It is preferable that you use MathJax to typeset your posts. Review How to write a good question?. I've ed it on your behalf this time.

– Sanchayan Dutta – 2019-03-14T17:12:37.433

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I believe the question boils down to "How to apply a controlled version of the X gate if the control qubits should be in states other than $$|11\rangle$$?"

Toffoli gate CCNOT applies an X gate to the target qubit if both control qubits are in the $$|1\rangle$$ state. If you need to apply an X gate if the control qubits are in some different state, you can do the following:

1. Convert the control qubits to $$|11\rangle$$ state using X gates on qubits which have to be in $$|0\rangle$$ state.
2. Apply CCNOT gate.
3. Convert the control qubits back to their original state, using the same sequence of X gates as in step 1.

For example, if you need to apply an X gate to target if control qubits are in $$|01\rangle$$ state, you can do

X(control[0]);
CCNOT(control[0], control[1], target);
X(control[0]);


You can use the same approach when implementing gates controlled on any pattern of control qubit states.

This question looks very similar to problem A2 from the recent Q# contest. If you're looking to implement this in Q#, check ControlledOnBitString library function which does exactly this. You can find the full code for the problem in the Superposition kata.