Introduction to Quantum Physics and Information Processing

136 Introduction to Quantum Physics and Information Processing

Then we can construct

U = e

iθ

V σ

x

V

†

W σ

x

W

†

= e

iθ

(cos γ + i sin γ ˆn · ~σ)

= e

iθ

e

iγ ˆn·~σ

,

which is a valid representation for a unitary operator! If we identify

V = A, V

†

W = B and W

†

= C,

then we have the requisite representation for U.

We can implement C-V σ

x

V

†

W σ

x

W

†

by the circuit of Figure (7.16).

|xi

• •

|yi

W

†

W

V

†

V

FIGURE 7.16: Circuit to evaluate C-U up to the phase factor

It is straightforward to see that when x = 0, the output is V V

†

W W

†

y = y,

and when x = 1, the output is V σ

x

V

†

W σ

x

W

†

y = Uy up to the phase. So

this gives C-U up to the phase factor. We need to additionally implement the

con ...