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Ananthan9470

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I don't understand the following aspect of the parity problem and if someone could please explain it to me, I would be grateful.

In the given quantum circuit, the output f(x) is defined to be x.a = x

I hope I got till this much correct. My question is, what if x is something like α|000>+β|101>+γ|111>? Then how is f(x) defined? Any help will be appreciated. Thanks!

In the given quantum circuit, the output f(x) is defined to be x.a = x

_{1}a_{1}+x_{2}a_{2}+x_{3}a_{3}(mod 2), where a is a fixed |a_{1}a_{2}a_{3}>. For example, if x=|101> and a=|100>, x.a = 1+0+0(mod 2) = 1.I hope I got till this much correct. My question is, what if x is something like α|000>+β|101>+γ|111>? Then how is f(x) defined? Any help will be appreciated. Thanks!