[doubleunplussed]

I've heard the following called "Aaronson's trilemma": Either the extended Church-Turing thesis is false, or quantum computers are impossible, or...there exists a classical polynomial factoring algorithm that runs in polynomial time.

One of these things must be true, and debates around quantum computing usually focus on the first two. But as argued, we don't have great reasons to believe factoring in polynomial time is impossible. We certainly don't have a proof that no such algorithm exists.

[bsder]

Factoring is one of the few quantum algorithms that is odd man out in terms of noise/implementation/scaling/etc. This should be indicating something about the structure of factoring that we don't understand.

Out of all the "classical" problems that we might find another algorithm for, "factoring" would be my bet for the one that we are missing a better algorithm.

[wbl]

What is the extended Church-Turing thesis? We already know quantum computers give speedups beyond classical lower bounds.

[daveFNbuck]

It's basically that BPP captures all realistic polynomial-time computations. The speedup would have to be sufficient to show something like a problem in BQP that isn't in BPP. I don't think anyone has been able to show that unconditionally yet.

You'd also need to accept that Quantum computers are realistic, which is why Aaronson's trilemma includes quantum computers being impossible.

[wbl]

That's true but Grover search shows that quantum computers can give big speedups so I don't know why I would believe BPP=BQP.

[nabla9]

Grover's algorithm still takes exponentially many operations to solve a search problem that can be brute-forced in exponentially many steps. Much faster, but still in EXPTIME. No jump from exponential to polynomial complexity class.

[scottlocklin]

This sort of statement is exactly why serious people shouldn't take "quantum complexity theory" seriously. The complexity class BQP is bullshit: there are no quantum computers, the end.

Feel free to prove me wrong by building one which does useful calculations. No time limit, until you die, in which case "time's up."

Edit add for downvoters: the strong Church Turing thesis is also almost certainly, and very obviously bullshit. How does that make you feel?

[inflatableDodo]

I'm betting on n. Where n stands for us not fully understanding how causality works yet, or a lot about what the hell is going on in general. I'm also rooting for something interesting emerging unexpectedly from Umbral Moonshine, because who can't help but love the Monster Group? https://www.quantamagazine.org/mathematicians-chase-moonshin...

[thechao]

You’re merely goalposting Scott “Quantum Computing Fists of Fury” Aaronson’s trilemma. If you have proof that quantum computers are physically unrealizable, I’ll happily help you publish that paper. There’s no way my brother could belittle my lack of education if I had a Turing Award or Nobel Prize, or two.

[daveFNbuck]

Which statement are you saying demonstrates why people shouldn't take quantum complexity theory seriously?

[scottlocklin]

BQP doesn't exist: there are no quantum computers. Saying BQP is like saying "magic fairy dust perfect unitary transformers that effectively encode a perfect complex number in the same sense a protractor theoretically can solve NP-complete problems by encoding real numbers."

[doubleunplussed]

The qbits and unitaries don't have to be perfect - it's about how things scale when you add qbits. Quantum error correction shows that you can add more qbits to make up for imperfection, and still be left with a quantum computer, i. e. the imperfection hasn't demoted the thing to a classical computer.

Of course they haven't built one yet, but none of the difficulties encountered so far have involved discovering new physics, which is what you would need to do to rule out quantum computers since the laws of physics as currently understood permit them.

[daveFNbuck]

So we shouldn't take research into things that don't exist yet seriously?

[nabla9]

The problem here is that you just state something that looks like strong personal opinion. You vaguely refer to serious people.

[nabla9]

The original Church-Turing thesis says nothing about computational complexity. Extended Church-Turing thesis says that all Turing machine equivalent computers can compute the same problems withing polynomial time.

Quantum computers are not known to be capable of solving NP-Complete problems in polynomial time.

[mokus]

Do we know that quantum effects will actually scale large enough to realize those speed ups for arbitrarily-large problems though?

[Khoth]

To paraphrase stuff from Scott Aaronson that I can't find right now: The currently known laws of physics say yes, and nobody's proposed any new laws of physics consistent with current observations which say no and don't allow even more computing power.

[heavenlyblue]

Yes - is there any extended reading available re. subj.?