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I am reading John Watrous' notes from his course CPSC 519 on quantum computing. In a pre-discussion before presenting Deutsch's algorithm to determine whether a function is constant or not, the author presents a function $B_f |x rangle |y rangle = |x rangle |y oplus f(x) rangle $, and the diagram:

The inital state is $|0 rangle |1 rangle$, and after the first two Hadamard transforms, will be $$big(frac 1 sqrt 2 |0rangle +frac 1 sqrt 2 |1ranglebig)big(frac 1 sqrt 2 |0rangle-frac 1 sqrt 2 |1ranglebig) .$$

Up to this far I understand. The author then writes: "After performing the $B_f$ operation the state is transformed to:

$$frac 1 2 |0 rangle big(|0 oplus f(0)rangle - |1 oplus f(0)ranglebig) + frac 1 2 |1 rangle big(|0 oplus f(1)rangle) - |1 oplus f(1) ranglebig).$$

I am not sure how this was obtained, from what I understand, the operation should be
$$frac 1 sqrt 2 big( |0rangle + |1ranglebig) otimes big|(frac 1 sqrt 2 |0rangle +frac 1 sqrt 2 |1rangle) oplus f(frac 1 sqrt 2 |0rangle +frac 1 sqrt 2 |1rangle) bigrangle$$ (simply subbing in $x,y$ to $B_f$). Any insights appreciated as this subject is completely new to me, although I have a decent mathematics and computer science background.

Remember that when you define the oracle effect as $B_f |x rangle |y rangle = |x rangle |y oplus f(x) rangle $, $f(x)$ is a classical function of a classical 1-bit argument, so you do not have a way to compute $f(frac 1 sqrt 2 |0rangle +frac 1 sqrt 2 |1rangle)$ (a function of a quantum state).

The quantum oracles that implement classical functions are defined as follows:

1. Define the effect of the oracle on all basis states for $|xrangle$ and $|yrangle$: $B_f |x rangle |y rangle = |x rangle |y oplus f(x) rangle $.




2. This will automatically define the effect of the oracle on all superposition states: the oracle is a quantum operation and has to be linear in the state on which it acts. So if you start with a state $frac12 (|00rangle + |10rangle - |01rangle - |11rangle)$ (which is the state after applying Hadamard gates) and apply the oracle, you need to apply oracle to each basis state separately. You'll get

$$B_f frac12 (|00rangle + |10rangle - |01rangle - |11rangle) = frac12 (B_f|00rangle + B_f|10rangle - B_f|01rangle - B_f|11rangle) =$$

$$ = frac12 (|0rangle|0 oplus f(0)rangle + |1rangle|0 oplus f(1)rangle - |0rangle|1 oplus f(0)rangle - |1rangle|1 oplus f(1)rangle)$$

Which is the same as the expression in the notes, up to a different grouping or terms.

The part about the oracles being defined by their effect on basis states is implicit in a lot of sources I've seen, and is a frequent source of confusion. If you need more mathematical details on this, we ended up writing it up here.