Note to Self: Quantum Computing Notes: CNOT Gates
Bases:
|++> = [+ |00> + |01> + |10> + |11>]
|+-> = [+ |00> - |01> + |10> - |11>]
|-+> = [+ |00> + |01> - |10> - |11>]
|—-> = [+ |00> - |01> - |10> + |11>]
|00> = + |++> + |+-> + |-+> + |—->
|01> = + |++> - |+-> + |-+> - |—->
|10> = + |++> + |+-> - |-+> - |—->
|11> = + |++> - |+-> - |-+> + |—->
1 Control Qubit:
|10> to |11>
|11> to |10>
|++> = [+ |00> + |01> + |10> + |11>] to [+ |00> + |01> + |10> + |11>] = |++>
|+-> = [+ |00> - |01> + |10> - |11>] to [+ |00> - |01> - |10> + |11>] = |-->
|-+> = [+ |00> + |01> - |10> - |11>] to [+ |00> + |01> - |10> - |11>] = |-+>
|--> = [+ |00> - |01> - |10> + |11>] to [+ |00> - |01> + |10> - |11>] = |+->
2 Control Qubits:
|110> to |111>
|111> to |110>
|+++> = [+ |000> + |010> + |100> + |110> + |001> + |011> + |101> + |111>] to [+ |000> + |010> + |100> + |111> + |001> + |011> + |101> + |110>] = |+++>
|+-+> = [+ |000> - |010> + |100> - |110> + |001> - |011> + |101> - |111>] to [+ |000> - |010> + |100> - |111> + |001> - |011> + |101> - |110>] = |+-+>
|-++> = [+ |000> + |010> - |100> - |110> + |001> + |011> - |101> - |111>] to [+ |000> + |010> - |100> - |111> + |001> + |011> - |101> - |110>] = |-++>
|—-+> = [+ |000> - |010> - |100> + |110> + |001> - |011> - |101> + |111>] to [+ |000> - |010> - |100> + |111> + |001> - |011> - |101> + |110>] = |--+>
|++-> = [+ |000> + |010> + |100> + |110> - |001> - |011> - |101> - |111>] to [+ |000> + |010> + |100> - |110> - |001> - |011> - |101> + |111>] = (+ |++-> + |+--> + |-+-> - |—-->)
|+--> = [+ |000> - |010> + |100> - |110> - |001> + |011> - |101> + |111>] to [+ |000> - |010> + |100> + |110> - |001> + |011> - |101> - |111>] = (+ |++-> + |+--> - |-+-> + |—-->)
|-+-> = [+ |000> + |010> - |100> - |110> - |001> - |011> + |101> + |111>] to [+ |000> + |010> - |100> + |110> - |001> - |011> + |101> - |111>] = (+ |++-> - |+--> + |-+-> + |—-->)
|—--> = [+ |000> - |010> - |100> + |110> - |001> + |011> + |101> - |111>] to [+ |000> - |010> - |100> - |110> - |001> + |011> + |101> + |111>] = (- |++-> + |+--> + |-+-> + |—-->)
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