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Could a quantum computer calculate the values of the Riemann zeta function that are currently out of reach with classical computers?

Any counterexamples to the RH would be somewhere in the range that we couldn't calculate the value in a neighborhood of the region in any time to get the result. Is there any way a quantum computer could do this and report back the value thru some sort of measurement process?

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Could a quantum computer calculate the values of the Riemann zeta function that are currently out of reach with classical computers?

This is generally true of any computer that has more processing capacity than existing computers whether it is a quantum computer or not.

Any counterexamples to the RH would be somewhere in the range that we couldn't calculate the value

True.

in a neighborhood of the region in any time to get the result.

Not true. The Riemann zeta function extends to infinity and an exception could be at any point arbitrarily beyond our ability to calculate it numerically.

Some patterns that hold true for a very long time numerically, ultimately fail. Professor John Baez has some notable examples at this blog post. For example, he discusses one that fails at approximately $1.397*10^{316}$.

By comparison, the Universe is about $4.32*10^{17}$ seconds old, and the fastest process in the Standard Model is the decay of the W and Z bosons which have a mean lifetime of about $3*10^{-25}$ seconds. So, at a rate of one computation per W boson decay for the lifetime of the universe, you would get about $10^{42}$ calculations, in a Universe that has fewer than $10^{90}$ particles in it (including neutrinos and dark matter particles, if they exist). This wouldn't come even close to finding the exception to the series identified by Professor Baez. Your quantum computer would need to do $10^{194}$ calculations per W boson decay time period, per particle in the Universe, for the life of the Universe, to test all of the possibilities numerically. Of course, if you can't time travel, you have far fewer seconds available to you to do that calculation than the age of the Universe, and you can't actually include every single particle in the universe in your quantum computer.

But, even that rate of calculation would be no guarantee of a result that would definitively find or rule out an RH exception.

In short, this problem would be impossible to be certain that you could solve via brute force numerical methods of any kind.

Is there any way a quantum computer could do this and report back the value thru some sort of measurement process?

It can search all sorts of numbers, but it can't check all of them in an infinite series. You might get lucky, or you might not.

This isn't to say that a quantum computer couldn't be useful.

For example, it could use machine learning to look for near misses for counterexamples and a pattern might develop that would help you decide where it would be fruitful to look numerically.

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  • $\begingroup$ "This is generally true of any computer that has more processing capacity than existing computers whether it is a quantum computer or not." I think the question was whether quantum computers indeed have more processing capacity when it comes to these kinds of problems. $\endgroup$ – Al Nejati Oct 9 '18 at 3:09
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    $\begingroup$ @AlNejati That would be an unanswerable question. One could in theory build a quantum computer that has much more processing capacity than a current state of the art computer, but this hasn't been done yet. But, any way you cut it, the processing power of a quantum computer would still be finite. $\endgroup$ – ohwilleke Oct 9 '18 at 3:16
  • $\begingroup$ I don't even think it's been proven yet that quantum computers are asymptotically faster than classical ones, even assuming you could have a large number of qubits. The most that can be said is that some quantum algorithms are known that give speedup but it's possible that classical algorithms with similar efficiency exist. I could be wrong about this though. $\endgroup$ – Al Nejati Oct 9 '18 at 3:25
  • $\begingroup$ @AlNejati The OP is basically asking if this is possible, and so it is fair to make the assumption that there exist quantum algorithms that are faster than any currently known classical algorithms, but it doesn't really matter, because so long as your algorithms take finite time to do a certain number of calculations, the result is the same. All computers fail at the task of testing all possible terms in an infinite series. $\endgroup$ – ohwilleke Oct 9 '18 at 3:29
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    $\begingroup$ +1 Especially for the link to the blog post about patterns that ultimately fail. The Borwein Integrals were something of a revelation to me and a remarkable example of the danger physicists face when making assumptions that seem reasonable in mathematics, but don't work. I wonder how many physicists would have happily assumed the $\frac \pi 2$ result held all the time ? I guess we can't close the mathematics department after all. :-) $\endgroup$ – StephenG Oct 9 '18 at 5:06

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