What you say is true. And I also agree with Greg that it then becomes essential that real amplitudes can be negative, such that interference can still occur; this point is discussed by Feynman at some length in his 1982 article Simulating physics with computers.

But AFAICT, doing quantum computations (or more broadly, computing quantum simulations for large-scale systems) is made much easier when the state space is “complexified”.

There seem to be at least two mathematical reasons for this (and maybe people can suggest still more reasons than these two). The first is that complexification creates a powerful invariance, namely, the unitary invariance associated with the ambiguity of the unraveling of simulated trajectories. The second is that when we do model order reduction of a large-scale quantum system, the geometry of the complexified state space is a Kahler geometry — much nicer than a Riemannian geometry.

But it has everything to do with the fact that the amplitudes don’t have to be positive. Or even more to the point, that quantum probability violates positivity properties of classical probability. Qauntum computing affords a more powerful kind of statistical sampling than classical randomized computing. That is what I think is the heart of the matter.

I hear this a lot, but I don’t think it makes a lot of sense. The power of quantum computing has nothing to do with the amplitudes being complex; nothing changes (substantially) if we restrict ourselves to real valued amplitudes.

With regard to the need for care in speaking, e.g., “physics as physicists have understood it for 80 years”, I am pretty confident that we mainly agree.

There may be a difference in emphasis, though. Being young, optimistic, and in a hurry, younger people tend to focus upon breakthroughs as the primary path to new understanding. Breakthrough stories are what journalists prefer too, because they’re easier to write about.

There is also, however, a slower path of change—equally important IMHO—in which the entire perspective of a community shifts, and the main emphasis is upon synthesis, especially engineering synthesis.

E.g., it would be hard to argue that magnetic resonance imaging represented a breakthrough in “new physics” or “new mathematics”. Rather the breakthrough was one of a new perspective upon imaging, and also, the engineering synthesis of many established ideas into a unified practical technology.

Obviously, both paths—breakthrough and synthesis—are essential to the health of the science and technology ecosystem. They are directly related, one to the other.

IMHO, one of the many respects in which your blog serves the community well, is as a forum where both points of view are discussed, and from which both benefit. For this, thank you very much!

Second, precisely because of issues like the ones you mention, I was very careful to refer not to quantum mechanics, but to quantum mechanics as physicists have understood it for 80 years. If noise were enough to shift us from one theory of computation to another, to me that would change our entire picture of QM — in a way that picture decidedly isn’t changed by the impracticality of a specific technology like atomic-resolution microscopy.

Because, don’t we have to be wary of a non-excluded middle here?

There is a historical precedent: the belief in the 1940s and 1950s of Pauling, von Neumann, Weiner, and Feynman, the atomic-resolution biomicroscopy was achievable by electron microscopy (or other high-energy means). Basically, they argued that “Either atomic-resolution microscopy is possible, or wave mechanics (at least as physicists have understood it for 30 years) is wrong.”

Their generation was reluctant to embrace the non-excluded middle “… or else radiation damage is a show-stopper.” This falsely-excluded middle was a very convenient belief for their generation — surely so mundane a mechanism as radiation damage would not permanently obstruct such a noble scientific enterprise? Yet so it proved.

Had their generation taken the problem of radiation damage more seriously, and looked systematically for a way around it, they might have invented magnetic resonance imaging almost 50 years earlier. They had all the required physics and mathematics in-hand, as early as 1935. And so a great opportunity was overlooked for a long time.

Personally, I admire and have great respect for the formalism of quantum error correction. However, history teaches us to have equal or greater respect for the ingenuity of Mother Nature’s decoherence mechanisms!

Just as the physicists of the 1920s-1980s did not have a good grasp on the potentialities of microscopy—in part because they were blinded by a natural reluctance to embrace “the middle”—it is IMHO completely possible that we do not yet have a good grasp on the potentialities of quantum computation and simulation, and that the “middle” we are reluctant to embrace is noise.

See, I’ve never had any emotional commitment to quantum computing being possible. For me, the key point is that either quantum computing is possible, or quantum mechanics (at least as physicists have understood it for 80 years) is wrong. And while I’m betting on the former possibility, I’m hoping for the latter, since it would be much more scientifically interesting.