Set of unitary operators such that $U|0rangle = |psirangle$

Question

In quantum computing, one of the central areas of study is to determine efficient quantum circuits - described by unitaries - to prepare a state $|psirangle$ from the initial computational basis state $|0cdots 0rangle$.
I am currently interested the mathematical abstraction of this problem; to be specific, if we have a $d$ dimensional Hilbert space, an initial state $|0rangle$ and a final state $|psirangle$, can we say anything about the set of unitaries $U$ such that $U|0rangle = |psirangle$? In particular, I am wondering about the following two things:

Does this set of unitaries form a submanifold of $U(d)$? (It clearly does not form a Lie subgroup.)
If we know a single unitary $U_0$ such that $U_0|0rangle = |psirangle$, can we obtain other unitaries with this property from $U_0$? Can we obtain all of the unitaries with this property from $U_0$?

My guess to the first question is that this set of unitaries forms a submanifold of $U(d)$ of dimension $d^2 - d$ isomorphic to $U(d)  /  mathbb{C}^d$, but I am completely unsure about the second. Are there known answers to these questions, or in general is there anything that can be said about this class of unitaries?

user1379857 · Accepted Answer

Say your state space is $D = 2^d$ dimensions. Without loss of generality, consider an orthonormal basis of states we will label by $| i rangle$, where $i = 1 ldots D$, where $|1rangle = | psi rangle$, and $|i rangle$ for $i > 1$ are other random states.
In this basis, a few things become clear. If we have any block diagonal matrix of the form
$$
U' = begin{pmatrix} 1 &  mathbf{0}  mathbf{0} & U_{D-1}end{pmatrix}
$$
where $U_{D-1}$ is any $(D-1) times (D-1)$ matrix, then we clearly have
$$
U' |psirangle = U' |1rangle = |1 rangle = |psi rangle.
$$
Therefore, we have a whole $U(D-1)$ worth of transformations that preserve $|psi rangle$. If we have your matrix $U_0$, composing it with one of these matrices $U'$ will preserve the property that $(U' U_0) |0 ldots 0 rangle = |psi rangle$. In fact, once we have $U_0$, we can get all the other such matrices by simply multiplying it in this way $U' U_0$.

Set of unitary operators such that $U|0rangle = |psirangle$

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