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First, I would like to state that I went through many excellent sources of information (among them - Grover's original paper, several QCSE posts like 1 2, and many more sources) - And yet I couldn't find a satisfying answer, so there must be something I don't understand.

I am trying to settle my mind with the known fact that Grover's algorithm can retrieve a specific value $w$ from an unsorted list in $O(\sqrt{N})$ steps, while $N$ is the size of the list. It is understood that $\frac{\pi}{4}\sqrt{N}$ iterations over Grover's iterator (Grover's oracle + diffuser operator) are needed, which may implies upon an overall computational complexity of $O(\sqrt{N})$.

enter image description here Grover's algorithm with n = 2, N = 4, one iteration over Grover's iterator.

But since Grover's orcale + diffuser are nested inside the iterator, as I understand it - We should be able to implement both of them independently of $N$, such that in each iteration both the oracle and the diffuser should contribute $O(1)$ steps to the overall complexity - If we want to achieve an overall complexity of $O(\sqrt{N})$.

There are several ways to implement the oracle and the diffuser, but as far as I understand - a multi-controlled gate over the $n = log(N)$ qubits in the counting register is unavoidable, in both:

Implementation of the diffuser

Implemention of the diffuser for n = 4 qubits with an mcz gate.

Implementation of the oracle

Implemention of the oracle for the $w = 1001$ with an mcx gate.

Decompositon of such multi-controlled gates would produce a circuit depth dependent of $n = log(N)$. So if in each iteration at least $O(log(N))$ steps being performed, then it seems to me that the overall complexity of the algorithm should be at least $O(\sqrt{N} \ log(N))$.

What am I missing? How is the overall complexity of $O(\sqrt{N})$ is being achieved after all?

Thanks!

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1 Answer 1

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The complexity for an oracle-based algorithm (e.g. in the case of search) is only counted in terms of the number of calls to the oracle. Yes, if you tried to implement a given oracle, it could be hugely costly (that's the whole point of the oracle-based counting), and that cost could grow with the system size. But that aspect is not taken into account.

If you want to view it another way, we're counting the number of oracle calls. The oracle has to be made out of a circuit. So, at worst, your running time is (number of calls to oracle)$\times$(circuit size of oracle). BUT, this overhead is true both for the quantum search and the classical search. Since the quantum computer could just implement the classical algorithm, it's going to be no worse (although there's some chance there could be a better quantum algorithm). Thus, the gap between quantum/classical is evident from the number of calls to the oracle.

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  • $\begingroup$ @ DaftWullie thanks for the answer. I get the point that the oracle is being called $O(\sqrt{N})$ times insteaf of $N$ calling to $f(x)$ classically. However, what's the good in that if the oracle can't be implemented independently of $N$? I was thinking that by assuming that the oracle acts in one step, there's a a basis of $O(1)$ possible implementation of the oracle. In addition - It also looks that any implementation of the diffuser also depends on $N$, so what's the reasoning for that? Thanks a lot! $\endgroup$
    – Ohad
    Aug 26, 2022 at 3:10
  • $\begingroup$ I would say that the oracle is always going to be a computation involving all the bits, so it's going to take a time at least $O(n)$ (with n=\log N$). In fact, you'd be pretty amazed if that's all it was. On the other hand, you're solving an NP problem, so you're guaranteed to be able to recognise a solution efficiently. This means your oracle runs in a time polynomial in n. $\endgroup$
    – DaftWullie
    Aug 26, 2022 at 7:02
  • $\begingroup$ Thus, the running time of an oracle, in practice, is going to be vastly higher than all the gates surrounding it, i.e. the diffuser. So, we don't worry about the cost of the diffuser. But the whole point is that $\sqrt{2^n}\text{poly}(n)$ is so much better than $2^n\text{poly(n)}$ (given that they're the same polynomial). $\endgroup$
    – DaftWullie
    Aug 26, 2022 at 7:02
  • $\begingroup$ Still don't get it. Let's assume I accept the claim that the oracle is a magical black box, I don't care what's inside and considering it as 1 step. Still, the diffuser is nested inside the iterator, so if we want to claim for $O(\sqrt{N})$ overall complexity for Grover's algorithm, we must prove that we can implement the diffuser in $O(1)$ steps. I am focusing my wonders on the claim that Grover's algorithm achieves a complexity of $O(\sqrt{N})$ - Regardless the comparison to the classical case. $\endgroup$
    – Ohad
    Aug 27, 2022 at 12:30
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    $\begingroup$ Exactly! The key is that both the quantum and classical oracle will have the same dependence on $n$. I don't know that technically an "efficient" algorithm has to mean that it's $poly(n)$, I think it's just something that is commonly used. Grover's algorithm is provably the best you can do in terms of an unstructured search (arxiv.org/abs/quant-ph/9605034), so in some sense it is actually the most efficient search algorithm you can come up with! On another note, the speed improvement is asymptotic because the constant quantum overheads in reality are brutal $\endgroup$ Aug 31, 2022 at 17:49

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