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Norbert Schuch
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Suppose we have a 3-qubit input; each bit is either 0 or 1. How to decide if there are more 1's than 0's? (OnlyOnly 1 extra qubit and single/double qubit gatesmay be used for the output. (Yes I know this can be used; you cannot use toffoliachieved using 3 Toffoli gates, nor construct the toffoli gate using single/double qubitbut can it be done without Toffoli gates?)

Now suppose we have an answer to the above question, then, how to extend the above circuit to deal with a 5-qubit input?

Suppose we have a 3-qubit input; each bit is either 0 or 1. How to decide if there are more 1's than 0's? (Only 1 extra qubit and single/double qubit gates can be used; you cannot use toffoli gates, nor construct the toffoli gate using single/double qubit gates)

Suppose we have a 3-qubit input; each bit is either 0 or 1. How to decide if there are more 1's than 0's? Only 1 extra qubit may be used for the output. (Yes I know this can be achieved using 3 Toffoli gates, but can it be done without Toffoli gates?)

Now suppose we have an answer to the above question, then, how to extend the above circuit to deal with a 5-qubit input?

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How to construct quantum circuit to count number of 0-qubits and 1-qubits

Suppose we have a 3-qubit input; each bit is either 0 or 1. How to decide if there are more 1's than 0's? (Only 1 extra qubit and single/double qubit gates can be used; you cannot use toffoli gates, nor construct the toffoli gate using single/double qubit gates)