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In Prakash's thesis - (link to PDF), section 2.2.2 Constant size vector states:

We show that the vector state $|x\rangle$ for $x\in R^b$ can be created in time $\widetilde{O}(\log(b))$ using a specialized quantum circuit of size $O(b)$ and pre computed amplitudes. The method is useful for creating constant sized superpositions and is illustrated for a 4 dimensional state $|\phi\rangle$ in figure 2.4.

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Question

Wouldn't a sequence of conditional rotations on all nodes of a binary tree of depth $\log(b)$ result in a circuit with $O(b)$ number of conditional rotations?

For example in this circuit we need a rotation on the first qubit, then 2 conditional rotations from the first qubit on the second qubit. And if we had 3 qubits we'd have a 4 conditional rotation from the first and second qubit on the 3rd qubit.

I am having trouble understanding how an exponential speedup over a typical QRAM was achieved here.

Edit

I should clarify that I also think that the number of gates would be the same as the depth of the circuit. Here is an example of a circuit preparing a state of 4 qubits from https://arxiv.org/abs/2010.00831

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  • $\begingroup$ I'm not sure this is a duplicate but, does this answer your question? quantumcomputing.stackexchange.com/a/10282/10454 Essentially, you can perform the $2^k$ controlled rotations with $O(\log b)$ gates if I'm not mistaken, assuming you have a function $f$ that you can implement as a unitary gate that gives you the angle of rotation. If you don't understand, feel free to ping me and i'll add more details in an actual answer $\endgroup$
    – Tristan Nemoz
    May 23, 2023 at 18:17
  • $\begingroup$ @TristanNemoz Yes it looks like this question is very related to my what I'm asking. I have all the angles precomputed, I guess I'm trying to understand how can the C-R_00 and C-R_01 gate be ran in parallel. A full answer would be appreciated :) $\endgroup$ May 23, 2023 at 23:51

1 Answer 1

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As mentioned in this answer, it is possible to perform each succession of rotation in parallel.

Let us suppose that we're at depth $k$ in our tree. If we were to implement the successive rotations as depicted in the circuit you've linked, the qubit number $k$ would undergo $2^k$ controlled rotations. More precisely, each of these rotations would be controlled off of a different number. For notation's sake, let us say that we want to apply a $R_Y$ gate with angle $\theta_x$ only if the $k-1$ previous qubits are in state $|x\rangle$. That is, we want to implement this gate: $$|x\rangle|0\rangle\to|x\rangle\left(\cos\left(2\pi\theta_x\right)|0\rangle+\sin\left(2\pi\theta_x\right)|1\rangle\right)=|x\rangle R_Y\left(4\pi\theta_x\right)|0\rangle$$ (Note: using $2\pi\theta_x$ ensures that $\theta_x\in[0\,;\,1)$).

Note that what you want to do, if I understand correctly, is to load a state from a QRAM structure. In such a case, the coefficients of the vector you load could be negative, in which case some additional gates should be added, as described in Algorithm 1 of this paper. Here, I'll focus on a state with all amplitudes being positive as in your example, since it shows the gist of why we can implement such a gate efficiently.

Suppose that we have access to a function $f$ such that: $$f(x)=\theta_x$$ with $\theta_x$ being represented on $p$ qubits, $p$ translating the precision of the angle. Computing $f$ can be done efficiently thanks to the tree structure. As such, we can implement $f$ as a quantum oracle: $$U_f|x\rangle|y\rangle=|x\rangle\left|y\oplus\theta_x\right\rangle$$ We assume that this oracle can be implemented in time $T_f(k)$. Now, let's look at what happens when we apply $P$ gates on $|x\rangle\left|\theta_x\right\rangle$. Let us write $\theta_x$ in binary as: $$\theta_x=\sum_{i=1}^{p}b_{x, i}2^{-i}$$ Suppose now that we apply a $P(2\pi)$ gate on the first qubit of the second register, a $P\left(\pi\right)$ gate on the second qubit of the second register, etc... up to a $P\left(\frac{\pi}{2^{p-2}}\right)$ gate on the last qubit of the last register. As a recall, the $P(\theta)$ gates leaves $|0\rangle$ untouched and apply a $\theta$ phase on $|1\rangle$: $$P(\theta)=\begin{pmatrix}1&0\\0&\mathrm{e}^{\mathrm{i}\theta}\end{pmatrix}$$ Thus, the $P$ gate that is applied on qubit $i$ doesn't do anything if $b_{x, i}=0$. If $b_{x, i}=1$, then it adds a $\frac{2\pi}{2^{1-i}}$ phase to the state. Thus, all in all, the phase of this state is now: $$2\pi\sum_{i=1}^{p}b_{x, i}2^{1-i}=4\pi\theta_x$$ Thus, using a single layer of gates we've managed to implement the following transformation: $$\left|\theta_x\right\rangle\to\mathrm{e}^{4\mathrm{i}\pi\theta_x}\left|\theta_x\right\rangle$$ Suppose now that we replace these $P$ gates by controlled-$P$ gates controlled on the single qubit target. Then this would implement the following transformation: $$\begin{align}\left|\theta_x\right\rangle|0\rangle\to{}&\left|\theta_x\right\rangle|0\rangle\\\left|\theta_x\right\rangle|1\rangle\to{}&\mathrm{e}^{4\mathrm{i}\pi\theta_x}\left|\theta_x\right\rangle|0\rangle\end{align}$$ That is, by replacing the $P$ gates by controlled ones, we've actually implemented a $P\left(4\pi\theta_x\right)$ gate on the last qubit. Let us denote by $V$ the following gate: $$V=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-\mathrm{i}\\\mathrm{i}&-1\end{pmatrix}$$ Note that we have: $$\begin{align} VP(\theta)V^\dagger &= \frac12\begin{pmatrix}1&-\mathrm{i}\\\mathrm{i}&-1\end{pmatrix}\begin{pmatrix}1&0\\0&\mathrm{e}^{\mathrm{i}\theta}\end{pmatrix}\begin{pmatrix}1&-\mathrm{i}\\\mathrm{i}&-1\end{pmatrix} \\&=\mathrm{e}^{\mathrm{i}\frac{\theta}{2}}\begin{pmatrix}\cos\left(\frac{\theta}{2}\right)&-\sin\left(\frac{\theta}{2}\right)\\\sin\left(\frac{\theta}{2}\right)&\cos\left(\frac{\theta}{2}\right)\end{pmatrix}\\&=\mathrm{e}^{\mathrm{i}\frac{\theta}{2}}R_Y(\theta)\end{align}$$ Thus, if we apply a $V$ gate on the single target qubit before and after the $P$ gate, this transforms it into a $R_Y$ rotation (up to an unconvenvient phase).

Putting everything together, we start with the following state: $$|x\rangle|0\rangle|0\rangle$$ We apply $U_f$: $$|x\rangle\left|\theta_x\right\rangle|0\rangle$$ We apply $V$ on the target qubit: $$|x\rangle\left|\theta_x\right\rangle V|0\rangle$$ We apply the $P\left(4\pi\theta_x\right)$ gate on the last qubit using the method outlined above: $$|x\rangle\left|\theta_x\right\rangle P\left(4\pi\theta_x\right)V|0\rangle$$ We then apply the $V$ gate once again: $$|x\rangle\left|\theta_x\right\rangle VP\left(4\pi\theta_x\right)V|0\rangle=|x\rangle\left|\theta_x\right\rangle \mathrm{e}^{2\mathrm{i}\pi\theta_x}R_Y\left(4\pi\theta_x\right)|0\rangle$$ We now remove the unwanted phase by applying $P(-\pi),P\left(-\frac{\pi}{2}\right),\cdots$ on the second register: $$|x\rangle\left|\theta_x\right\rangle R_Y\left(4\pi\theta_x\right)|0\rangle$$ And finally, we uncompute the second register by applying $U_f$ once again: $$|x\rangle\left|0\right\rangle R_Y\left(4\pi\theta_x\right)|0\rangle$$ We can now discard the second register as it is not entangled with the others. All in all, the depth of this circuit is $2T_f(k)+2+p+1=2T_f+p+3$, where $2T_f(k)$ comes from computing and uncomputing the second register, $p$ comes from the controlled-$P$ gates on the target qubit, $2$ comes from the two $V$ gates on the target qubit and $1$ comes from the layer of $P$ gates used to uncompute the unwanted relative phase (since they're applied on the same layer). Note that the only term that depends on $k$ is $2T_f(k)$.

$p$ has to be chosen accordingly to the desired precision. Thus, all the depth essentially comes from the $2T_f(k)$ term. I'm less confident on that part, but I think that thanks to the tree structure, $f$ can be computed in time $O(\log b)$, which in turns ensures that $U_f$ can be implemented in time $\DeclareMathOperator{\polylog}{polylog}O(\polylog b)$, hence the final complexity.

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  • $\begingroup$ Thank you this helps a lot! But it seems like we simply moved the problem to $U_f$. $f(x)$ can be completely arbitrary map (depending on the tree). So the question now becomes: How can we do an exponential number of Controlled-$\oplus f(x)$ simultaneously? $\endgroup$ May 25, 2023 at 3:44
  • $\begingroup$ To make my question more concrete using the example tree in my question: we would want $U_f$ to do the following: |00>|0> -> |00>|0.4/sqrt(2)}> and |10>|0> -> |10>|0.8/sqrt(0.8)> for 2nd level of the tree. How can we run both of these gates in parallel? $\endgroup$ May 25, 2023 at 3:47
  • $\begingroup$ @bubakazouba I think I get your point. However, the difference lies on the nature of $U_f$. As a gate implementing a classical function, $U_f$ is a permutation matrix. In fact, when building up the tree, it is possible for the user to also implement a (classical) circuit for $f$. Being given this circuit for $f$, it is possible to implement a quantum circuit that implements $U_f$ with at most a polynomial overhead. So, as long as the classical circuit for $f$ scales polynomially with the number of bits, you're fine. $\endgroup$
    – Tristan Nemoz
    May 25, 2023 at 8:01
  • $\begingroup$ @bubakazouba The operations that you describe are not two different gates. The whole point of $U_f$ is to be applied in superposition. You would have $U_f|000\rangle=|00\rangle\left|\frac{0.4}{\sqrt{2}}\right\rangle$ and $U_f|100\rangle=|10\rangle\left|\frac{0.8}{\sqrt{8}}\right\rangle$. $\endgroup$
    – Tristan Nemoz
    May 25, 2023 at 8:03
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    $\begingroup$ Thank you this makes sense! $\endgroup$ May 25, 2023 at 21:52

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