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I do not see any advantage in constructing very costly and imperfect physical qubit while this qubit can be simulated with using conventional computer memory (noise-free). So what is the purpose when trying to convert some atomic-like structures into very costly memory-like structures?

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The assumption underlying the question is wrong. Or rather, the phrasing is misleading.

Sure, you can easily simulate a single qubit on a classical computer, so yes, any experiment which only operates on a single qubit won't give any "computational advantage". But then again, (hopefully) nobody ever claimed to perform a single-qubit experiment and achieve a quantum advantage from the point of view of computational efficiency. That also doesn't mean that experimental demonstrations with few qubits are useless. There's a myriad of reasons why people do them. Going from building up the technology with the goal of implementing larger scale experiments, to simply testing various protocols and aspects of quantum mechanics for purposes that have nothing to do with quantum computation. That's just how physics is done in general.

But the fact that a single qubit (in ideal conditions) is trivial to simulate is completely different than saying that quantum systems are. The cost of simulating with a classical computer the dynamics of a generic $n$-qubit systems scales exponentially with $n$ (at least with any classical algorithm we know, and it is generally believed that there is no classical algorithm that can efficiently simulate quantum mechanics in the general case). Practically speaking, that means that even if you try to simulate a 10-qubit system on your computer you'll have a hard time. If you try to simulate, say, 40 qubits, you'll realise that there is simply no way for your laptop, or any other classical device, to handle it (again, of course, modulo specific situations which might be easier to simulate).

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The amount of lets say 64 bit registers that you need to represent a state of N qubits is $2^N$ registers. And you need a square of it to represent a matrix operation on it.

The reason you need this amount, is since a state of N qubits might be a superposition of all the possible combinations.In every one of them you need to store the amplitude that is multiplying it.

So if you have an aplication that is solved with quantum algoritm (Shor factoring N bits number for example, or state of a molecule with N orbitals) , you will never be able to simulate it in the limits of the memorry size of a classical computer.

So as you see, it is exponentially more costly with classical as long you are able to build a quantum conputer.

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  • $\begingroup$ In terms a concrete example, Google's Sycamore processor has 127 qubits. Assuming you could entangle them and create a superposition state across all 127 of them with only 0 or 1 coefficients for each of the $2^{127}$ product states in the computational basis, you'd need $2^{127}$ bits. For reference, there are about $2^{165}$ atoms in the Earth and about $2^{265}$ in the universe. That means if/when Google roughly doubles the size of their processor, classically simulating just a superposition state with only binary coefficient would require a bit for each atom in the universe. $\endgroup$
    – Chris E
    May 19, 2022 at 2:19
  • $\begingroup$ As for current efforts for classical simulations, I think problems involving 10 qubits are doable on most laptops, 20 qubit problems take a fairly beefy computer, 30 qubits requires a supercomputer, and good luck reaching 40 for general quantum algorithms. $\endgroup$
    – Chris E
    May 19, 2022 at 2:22
  • $\begingroup$ Why would you simulate a quantum computing performing a quantum algorithm if you can use a classical algorithm instead? In most cases, this will be much faster (there are some exceptions: "de-quantized" algorithms; but they don't simulate the QC). $\endgroup$ May 19, 2022 at 8:13
  • $\begingroup$ Because there is a family of problems that can't be solved with classical algorithm in realistic time (minimum energy of big molecula, factoring to prime numbers ...) $\endgroup$
    – Ron Cohen
    May 20, 2022 at 6:29
  • $\begingroup$ @ChrisE, *Google's Sycamore has 53 qubits, not 127. $\endgroup$ Jun 18, 2022 at 13:44
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The question is not if it can be simulated, but rather how long it takes. There are severe limitations in actually computing arbitrary properties of many-body systems in, say, the lifetime of an average human. It's not guaranteed that quantum computers can do better, but maybe sometimes they will.

Thus, at this moment, there can be no ultimate answer to your question whether there is an advantage with quantum computers. It's just that most people think there is or could be one, so they try.

EDIT: Some people have noticed the provocative nature of this post and I was asked to elaborate on the second paragraph:

The question is hard to answer precisely because it's fuzzy and depends on many aspects. There is complexity theoretic proof for sampling problems, and evidence for problems like integer factoring, and I strongly believe that there further interesting problems in BQP which are hard for classical computers. However, this discussion completely ignores the point that a machine has to be built. There's no guarantee that a scalable, fault-tolerant, decently fast quantum computer can exist. But there's also no fundamental argument why it shouldn't. I think there's consensus on this part (Gil Kalai doesn't count). Hence, we're seeing prototypes which demonstrate that it works on a small scale, but there's certainly no advantage in using these devices. We're getting there, and have to overcome technical and physical obstacles on the way. Finally, the (error-corrected!) operations on quantum computers have to be decently fast. In practice, actual runtime matters, not asymptotic scalings. If they are slow, a computational advantage will only exists for larger problem sizes, again raising the bar for the size of the quantum computer. I think these obstacles can be overcome, but I think a researcher should be aware of these aspects.

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    $\begingroup$ I don't quite understand the second paragraph. What do you mean "there can be no ultimate answer [on] whether there is an advantage with quantum computers". One can prove that specific tasks can be solved efficiently with quantum computers and not with classical ones. If you mean that there is currently no definite proof of this, I guess that's a tenable position, depending on how strict you want to be about a "definite proof". But it's not like the question is not answerable in general $\endgroup$
    – glS
    May 18, 2022 at 14:07
  • $\begingroup$ Concerning the second paragraph, if you mean QAOA or VQE algorithms or quantum annealers, then you are right but there are many algorithms with proved speed-up. $\endgroup$ May 18, 2022 at 17:31
  • $\begingroup$ @glS Well I think OP's question was fuzzy, so the answer is too. Interestingly, I didn't have complexity theory in mind at all! Whether there is an advantage or not, depends on a lot of aspects and I think nobody can honestly answer this with a "Yes, sure!" (except if they work for marketing). We're usually in an optimistic mood, because there is a lot of evidence that the answer should be "Yes" and that's good. However, it would be very naive to believe the marketing. (I have updated my answer) $\endgroup$ May 19, 2022 at 7:25
  • $\begingroup$ @MarkusHeinrich do be clear, when I said "it's not like the question is not answerable in general", I referred to the general question about quantum speed-ups, not the question by the OP. Also, I do agree with saying there is currently not a definite proof (the best we have are "almost" proofs for sampling problems) and with not being overly optimistic (I'm certainly personally not) that the answer about quantum computers allowing useful speedups is positive. My only problem with the current phrasing is "there can be no ultimate answer about (...)", which I find hardly tenable $\endgroup$
    – glS
    May 19, 2022 at 7:53
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    $\begingroup$ @glS I think I understood you correctly :) So far, I think there's no reason to believe that the "advantage question" cannot be answered in the future, and probably to the positive. Except, of course, it takes too long, quantum winter comes before, and the technology gets sacked ;) $\endgroup$ May 19, 2022 at 7:57

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