Parity problem in Bernstein Vazirani Algorithm

[Ananthan9470]

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 = x₁a₁+x₂a₂+x₃a₃(mod 2), where a is a fixed |a₁a₂a₃>. 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!

 

[Asim Riaz]

If x is α|000>+β|101>+γ|111>, then f(x) is defined as f(x) = αa1+βa2+γa3 (mod 2). In other words, the value of f(x) is determined by multiplying the coefficients of each term in x with the corresponding coefficient in a and then taking the sum of the products (mod 2). For example, if x = α|000>+β|101>+γ|111> and a = |100>, then f(x) = α(1)+β(0)+γ(0) (mod 2) = α (mod 2).