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I am doing self-study at the moment using primarily the book: Quantum Computing a Gentle Introduction by Eleanor Rieffel and Wolfgang Polak.

Getting through the earlier chapters and exercises went quite well (fortunately the earlier chapters had plenty of examples), however I got stuck on the 5th chapter on quantum circuits. Although I understand the concepts the authors present, perhaps due to a lack of examples, I have trouble applying said concepts to the exercises.

The exercises I have trouble with (and where I can't find a solution or thorough/ introductory explanation for) are the following:

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Questions:

Design a circuit for creating:
$left| W_n right> = frac1sqrtn(left| 0 dots 001 right> + left| 0 dots 010 right> + left| 0dots 100 right>) + cdots + left| 1dots 000 right>)$ from $left| 0 dots 000 right>$

And design a circuit for creating "the Hardy state":
$frac1sqrt12(3left| 00 right> + left| 01 right> + left| 10 right> + left| 11 right>)$

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Can somebody point me in the right direction or refer me to some literature/ tutorials so I can grasp these kind of exercises better?

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Perhaps a related question:
Tips and tricks for constructing circuits to generate arbitrary quantum states

As DaftWullie pointed out, the question about $W_n$ has an excellent collection of answers here.

For the Hardy state question (and a lot of other tasks like it), you can approach it as follows.

• Start with the $|0...0rangle$ state.


• Start by putting the first qubit "in the right state", which is a state $(alpha |0rangle + beta |1rangle) otimes |0...0rangle$, where $alpha$ and $beta$ are the relative weights of all basis states which start with 0 and with 1, respectively. For Hardy state specifically, two basis states start with 0: $frac1sqrt12(3left| 00 right> + left| 01 right>)$ and two basis states start with 1: $frac1sqrt12(left| 10 right> + left| 11 right>)$; their relative weights are just the sums of squares of their amplitudes: $frac912 + frac112 = frac1012$ and $frac112 + frac112 = frac212$, respectively. So you'll need to put the first qubit in the state $(sqrtfrac1012 |0rangle + sqrtfrac212 |1rangle)$ using $R_y$ gate.


• Continue by putting the second qubit in the right state, applying controlled $R_y$ gates with the first qubit as the control. To get the first two terms right, you need to convert the term $sqrtfrac1012 |0rangle otimes |0rangle$ into the term $frac1sqrt12(3left| 00 right> + left| 01 right>)$, which is the same as convert normal state $|0rangle otimes |0rangle$ into $frac1sqrt10(3left| 00 right> + left| 01 right>)$ without affecting the state $|1rangle otimes |0rangle$ (note the renormalization when switching from terms of a larger expression to standalone states!) To do this, you can do a 0-controlled $R_y$ with the first qubit as control and the second qubit as target.


• If you have more qubits, you will continue doing this, using more control qubits to make your rotations more and more specific.

You can see this paper by Shende, Bullock and Markov if you want a more formal and less ad-hoc explanation.