Circuit for controlled rotations conditioned on a $d$ bit precision estimate of $\theta$ for $O(d)$ gates

Question

Suppose I have a $d$-bit estimate of $\theta \in (0, 2 \pi]$, say a register of qubits $|\tilde{\theta} \rangle = |z_1 \rangle \ldots |z_d \rangle$ for $|z_i \rangle \in \{|0 \rangle, |1 \rangle \}$ where $\theta \approx \sum_{i=1}^{d} z_i \pi /2^{i}$, and I wish to implement a transformation $|\tilde{\theta} \rangle |0 \rangle \mapsto |\tilde{\theta} \rangle (cos(\theta) |0 \rangle + sin(\theta)|1 \rangle)$ using a quantum circuit consisting of two-qubit gates which can be represented by the controlled unitary transformation

$U_{c_i} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos(\pi / 2^i) & -sin(\pi / 2^i) \\ 0 & 0 & sin(\pi / 2^i) & cos(\pi / 2^i) \\ \end{bmatrix}$

I am trying to come up with a formula for how a sequence of these controlled operations would be applied to the ancilla qubit $|0 \rangle$ (maybe more than a single qubit?) to result in an overall rotation by $\approx \theta $ but I am not sure how to proceed. Any insights appreciated.

Edited the matrix as in the answer.

Answer 1

$U_{c_i}$ should not have $z_i$ in it. That information is on the input. It is not in the gate.

So instead

$$ U_{c_i} = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&\cos(\pi/2^i)&-\sin(\pi/2^i)\\ 0&0&\sin(\pi/2^i)&\cos(\pi/2^i)\\ \end{pmatrix} $$

If this gate acts on the $i$'th qubit of $| \tilde{\theta} \rangle$ and the target, this will give the appropriate controlled rotation.

To do all of $\tilde{\theta}$ just do all these $U_{c_i}$ on the appropriate pairs as described above. This is thanks to the fact that all of these operations are rotating along the same axis. You are just building up

$$\prod_{i=1}^d R_z (\frac{z_i}{2^i}\pi) = R_z (\sum_{i=1}^d \frac{z_i \pi}{2^i}) = R_z (\tilde{\theta})$$