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Math Help - Quantum Computation Problem

  1. #1
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    Sep 2008
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    Quantum Computation Problem

    I'm stuck on the following problem:

    Suppose we are given a qubit |q> in an unknown state, but we know it is either |0> or
    1/sqrt(2)(|0> + |1>). We would like to devise a circuit to determine, with certainty,
    which of the two is the case, using any number of unitary and measurement
    operations.

    Either explain how to construct such a circuit, or prove that no such circuit is
    possible.

    Can somebody give me a suggestion for tackling this? My intuition tells me that it shouldn't be possbile and I'd like to prove by contradiction, but I haven't yet been able to come up with anything.

    Thanks
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  2. #2
    Member
    Joined
    Sep 2008
    Posts
    95

    Re: Quantum Computation Problem

    Well, after a lot of googling for a push in the right direction (I didn't
    find a proof but an assertion), it turns out to be impossible! I think this
    is why:

    Assume that there is such a circuit described by the unitary operator A.
    Then given two input vectors |S1> and |S2>, without loss of generalisation
    let:

    | A>|S1> -> U|A>|s1> = |A1>|S1>
    | A>|S2> -> U|A>|s2> = |A2>|S2>

    Now since U is unitary, it is norm preserving so that

    <A,S1|A,S2> = <A1,S1|A2,S2> but <A,S1|A,S2> = <A|A><S1|S2> = <S1|S2> and
    <A1,S1|A2,S2> = <A1|A2><S1|S2>

    so we have <S1|S2> = <A1|A2><S1|S2>

    Clearly either A1=A2 or <S1|S2> = 0

    But by assumption, our circuit differentiates between S1 and S2, so we must
    have A1!=A2. It follows that <S1|S2>=0. Clearly the vectors in the question
    are non-orthogonal, so no such circuit exists.
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