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Quantum computers provide exponential speedup relative to classical computers. However, it is empirical fact that increasing of number of qubits makes the the computer harder and harder to keep working (decoherence, decreasing energy level separation, longer relaxation times ... ). To do that you have to go to lower temperatures and better shielding form environment perturbation and higher precision of manufacturing.

I would like to know if there are some fundamental rules of nature which connects this parameters with number of qubits, a least for adiabatic quantum computers and quantum annealing, which seems to be most practical.

  1. Eg. if required temperature (reps. $\beta=1/T$ ) would scale also exponentially with number of qubits ( because energy difference between levels would scale exponentially as well ), it would effectively make the computer performance scaling just linearly with advances in cooling and shielding technology.
  2. perhaps, there can be fundamental limit which at the end connect relaxation time to number of qubits, in such a way that increasing number of qubits would require exponential increase of relaxation time ( i.e. if energy level separation is exponentially decreased than period of quantum beating between them takes proportionally longer). This would mean that processing performance measured in number of examined qubit configurations per time would be actually limited by some constant.

there is some interesting movie about practical adiabatic quantum computing which can be good introduction into practical issues.

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  • $\begingroup$ Every relationship in physics eventually breaks down in some limit of the relevant physical parameter range and quantum computing is no exception. That quantum effects like decoherence require ever deeper temperatures to suppress is not necessarily correct, as the example of superconductivity proves, which is perfectly stable even at finite temperature. However, adiabatic QC does seem to suffer from the defect that you describe, which makes this particular implementation a dead end. I do agree with you that the main physics problem in QC is why it won't work, in the end. $\endgroup$ – CuriousOne May 10 '15 at 13:09
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Only a partial answer here, since I am not a real expert into the field (especially not adiabatic quantum computation):

If such limitations were known, these particular lines of research would be pretty much dead. Since they are not, I conclude that no such limits are known.

If we take all possible implementations into account, I do not know a single accepted result that gives fundamental bounds on how large a quantum computer can be or whether it can exist at all, although there are people that expect such a result.

However, many fundamental limitations are known. There is an overview here (ArXiv), which focusses more on classical computers, but it seems that the fundamental limits for analogue computers might also apply to adiabatic quantum computation and might put upper bounds on their scalability.

Another experimental problem is discussed in this PRL paper (ArXiv): It seems that the coherence time is upper bounded in certain systems. This might pose a problem for error-correction, since the speed of for example quantum gates might be lower bounded by some other phenomenon. However, I do not know, what people thought about the ideas.

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