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There is nothing in standard quantum mechanics that says a quantum computer might not work. In fact, the very concept of quantum calculations was pioneered by Richard Feynman as the consequences of the "standard" Copenhagen interpretation of quantum mechanics, and not the "many worlds interpretation". Therefore any interpretation which accurately predicts regular quantum experiments will also accurately predict the workings of a quantum computer; I'm afraid there will be no "interpretation breakthrough" if a quantum computer is actually built.

Of course, having a working quantum computer might lead to new philisophical issues which might hint at a "proper" interpretation of QM -- but only on philosophical, not physical grounds. With a quantum computer, huge numbers could be factored with seeming ease. But "where" are the computations happening? According to the many-worlds theory, they would be happening in parallel universes, which can interfere with our own on quantum scales. In contrast, the Copenhagen interpretation has these calculations happening in the wavefunction itself -- and this idea might make less sense to some people.

But there are philosophical problems with the many-worlds interpretations as well. Why is consciousness limited to only "one" universe? Why isn't the superposition of all of our minds able to perceive the superposition of all universes in which we exist? At exactly what point does the actual universe-branching occur? (Many-worlds theory has several different answers to this last question, none of them very satisfying.) And finally, the very foundation of quantum mechanics is based on probabilities, but it's impossible to accurately measure probabilities if we're in one universe branching off from others -- because we can't measure what proportion of other universes branch off from any given event.

So what other theory is a contender? My personal favorite is John Cramer's Transactional Interpretation of quantum mechanics, originally published in 1986. According to this interpretation (and other similar theories), the mysterious nature of quantum mechanics is a natural result of time-symmetry of the microscopic world. In particular, the future can affect the past on a microscopic scale, so long as no one has already made a measurement of those past events. This doesn't just prevent time-travel paradoxes -- it also explains why measurements appear to "collapse" quantum wavefunctions in the first place. Basically, this interpretation is a subset of "hidden variable" interpretations of quantum mechanics that aren't ruled out by Bell's theorem ; particles now have "hidden information" about their own future. The main reason this theory isn't more popular (in my opinion) is that people aren't very good at considering backwards causality (we never experience it, after all!), and incorrectly think that this type of theory doesn't allow for free will.

With regards to the philosophical questions of "where" the quantum computer's calculations are happening, the Transactional Interpretation would say that the computer is borrowing information from the future -- in a sense, "checking" its calculations before the calculations are even finished!

Of course, these are mainly philosophical issues, not physical ones. As soon as two interpretations have different testable predictions of nature, the debate over interpretations of quantum mechanics will become grounded in experiment -- which is probably the only way this is ever going to be resolved.

Here are some technical papers about quantum computing, and here is Oxford's quantum computing page.

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