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The term 'classical computer' is always used to mean standard digital computation (Turing model, Boolean circuits or just good old RAM). I have never seen it to mean other models of computation based on classical physics (such as analog computation). This is evident when papers talk about the classical complexity of a problem, when what is meant is the complexity defined given a digital model of computation.

Do we gain anything by 'classical' instead of 'digital', or is it just a shibboleth?

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I think we do gain a little bit by saying "classical" instead of "digital".

As you point out, you can certainly build classical analog computers, and in the past these were very useful. But I believe that such classical analog computer can all be efficiently simulated by digital computers - not in the sense that a digital circuit can necessarily mimic the exact physical evolution of the computer, but in the sense that they can in principle solve any given problem with a similar asymptotic runtime (possibly up to polynomial speedups or slowdowns). In other words, I think it's generally believed that the extended Church-Turing thesis holds for all physically realizable computers whose behavior does not essentially rely on quantum mechanics. (You might argue that this claim is vague or even circular, but I think that with some work you can make it both true and noncircular. Note that this claim certainly hasn't been rigorously proven, but I think it's generally accepted to be true in our world.)

So I think that referring to these computers by the broad term "classical" usefully conveys the highly nontrivial insight of the extended Church-Turing thesis: that if you just care about "macro" features like the asymptotic runtime, then it doesn't actually matter whether your computer is digital or analog - what matters is whether it can use inherently quantum phenomena like superpositions and entanglement in a controlled fashion.

Edited to add. James Wootton asks in a comment whether it's been proven that classical analog computers can be efficiently simulated. The answer is no, but I personally think that that question is making a bit of a category error. The way I see it, the extended Church-Turing thesis is not quite a sharp enough statement to be mathematically provable or falsifiable (although restricted versions may be). It's more like a general principle that evidence can accumulate to either suggest is useful or not useful. (More along the lines of a statement like "All physical regimes can be described by some type of action principle" than a mathematical proposition.)

By the way, I highly recommend Scott Aaronson's paper "NP-complete problems and physical reality" (PDF), which proposes a claim somewhat similar to the extended Church-Turing thesis - that no physically realizable process at all (classical, quantum, or whatever) can act as a computer that efficiently solves NP-complete problems.

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    $\begingroup$ (All that having been said, I think that most of the times that actually practitioners use the term "classical computer", they aren't thinking these big philosophical thoughts - they're just going along with the standard convention.) $\endgroup$ – tparker Jun 4 at 14:03
  • $\begingroup$ Has it been proved that classical analog computers can be efficiently simulated. There's a recent Nature paper that suggests not nature.com/articles/s41467-018-07327-2 $\endgroup$ – James Wootton Jun 4 at 18:16
  • $\begingroup$ @JamesWootton that paper actually does use digital simulation for the results presented. Yes, the digital simulation of ODEs incurs discretisation issues, but with a suitable choice of solver these can be kept in rigorous bounds that are not asymptotically worse than what you would also get in an analogue implementation due to shot- and thermal noise. You do get a (possibly quite large) overhead though; in particular digital computers have a substantial energy consumption due to the need to drive transistors back and forth even in steady state of the ODE. $\endgroup$ – leftaroundabout Jun 4 at 21:07
  • $\begingroup$ ...that said, I'm not sure if we couldn't get much larger speedup with analogue computers based on PDEs rather than ODEs. $\endgroup$ – leftaroundabout Jun 4 at 21:10
  • $\begingroup$ @JamesWootton Your question is substantial enough that I added to my answer instead of trying to answer in a comment. But I'll mention that respectfully, I think that Nature paper is kind of nonsense. They attempt to prove that an analog computer can do something that a digital computer can't ... by simulating it on a digital computer! They never actually do anything on a real analogue computer, so I'm highly skeptical that their result tells us anything useful at all. $\endgroup$ – tparker Jun 5 at 1:57
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I think the reason is that 'classical' is more effective.

Maybe too much. Some colleagues of mine that are not working on quantum computing, do not like the expression 'classical computing' as they think it is used with a derogatory intent, meaning 'old fashioned', 'outdated'.

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