Ex5 11
$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 5.11}$
Let$\ket{W_n} = \frac{1}{\sqrt{n}} (\ket{0\dots 001} + \ket{0\dots 010} + \ket{0\dots 100} + \cdots + \ket{1\dots 000}).$
Design a circuit which, upon input of $\ket{00\dots 0}$, constructs $\ket{W_n}$.
page revision: 2, last edited: 06 Jan 2022 16:51
Define the weighted Hadamard transformation
(1)Then $W_n$ can be produced from $\ket{00\ldots 0}$ by a sequence of multiply 0-controlled $H_n, H_{n-1}, \ldots, H_2, H_1$ gates like this:
Note that
(2)The first gate $H_n$ maps
(3)This is mapped by the next gate, the 0-controlled $H_{n-1}$, into the state
(4)Before the multiply 0-controlled $H_2$ gate, the state is
(5)and after $H_2 = H$ (the standard Hadamard gate), the state is
(6)And the final multiply 0-controlled $H_1 = X$ gate, will turn the state into $W_n$.