David Bohm was one of the more interesting researchers in the field of the foundations of quantum theory. While a graduate student at Berkeley under Oppenheimer (after a short stay a Caltech where he was unhappy), he wrote, with David Pines, some early very influential papers on plasma physics. His textbook on quantum theory is a model for almost all of our modern quantum textbooks. And then, of course, there is the Aharonov-Bohm effect which shows how the vector potential can affect results even when the particles involved transverse regions where the
vector potential magnetic field vanishes. It was supposedly while writing this textbook on quantum theory that he began to question the foundations of quantum theory and which led him to develop a non-local hidden variable theory for nonrelativistic quantum mechanics-a task whose supposed impossibility in turn led John Bell to formulate his famous inequality. Bohm, however, had consorted with the Oppenheimer crowd at Berkeley and got pulled into the whole “are you a communist” political mess and thus could not obtain a job in the United States, instead obtaining a job first in Brazil (on the recommendation of no one less than Einstein) and then in the United Kingdom where he continued his work in foundations and consorted with an assortment of interesting characters, including the Dali Lama and Jiddu Krishnamurti.
As you can see, I’ve always been fascinated by Bohm’s life. But what do I think of his hidden variable theory? It must be said right off the bat that Bohm constructed his theory more as a proof of principle than as the final solution to the foundational problems as he saw them (In fact it is probably best to say that Bohm did not believe there was a “final” solution as far as I can tell.) Well at various times in my life I have felt that Bohm’s theory was an intriguing direction towards understanding quantum theory. But as I’ve learned more and more from quantum compting, I’ve begun to think less and less of this theory. Indeed, I would say that quantum computing has taught me that there is something radical missing from approaches along the lines of Bohm’s non-local hidden variable theory. Quantum computing ruined Bohmian mechanics for me.
“Bah!” you say. What can quantum computing, which is obsensibly founded along the mantra “do not question quantum theory…accept it and rejoice at the splended information processing you can now do!” have to do with questions from the foundations of quantum theory, who are questions more associated with philosophy than constructive technological advance (yes this was a low blow to philosophy…sorry couldn’t resist)?
Well I would say it has pretty much everything to do with interpretations of quantum theory! Take you favorite interpretation of quantum theory. Now ask the question, how does this interprettion help explain to me why quantm computing is more powerful than classical computing. Whenever I do this for any of the interpretations, I find that I walk away even more appreciative of the weaknesses of each of the interpretations of quantum theory. For example, back to Bohmian mechanics. Now how does the idea that there is a pilot wave, or such, guiding the trajectory of a particle give us insight into why quantum computers can efficiently factor integers? Sure it seems reasonable that non-local hidden variable theories can be more powerful than local hidden variable theories, but why does the particular implementation of a non-local theory, as advocated by the Bohmian interpretation crowd, give us any extra insight into the power of quantum computing? Indeed, this is the crux of my problem: the more I learn quantum computing, the more I see it conected to the theory of computation. And the more I see it connected to the theory of computation, the less satisfying I find explanations such as “well it’s just a non-local theory”. Explanations such as that are like saying BQP is in PSPACE, so the power of quantum computing is obviously that of PSPACE. This leads to further weaknesses, I think, like the extreme wastefulness of non-local hidden variable theories in terms of their representation of the flow of classical inormation. I mean one of the astounding result of quantum computing is not that you can factor integers, but that you can’t also do everything in, say, NP. Why this theory with Goldilocks like power, able to solve problems not so difficult so as to rearrange our theory of tractible computation, but at the same time able to solve problems widely thought to be intractable on a classical computer?
Of course, you will object that I am asking too much of an interpretation. The interpretation is only supposed to make you feel good at night, when you crawl into bed with your copy of Cohen-Tannoudji et al, not to actually be useful (sorry, another low blow.) But I believe that an interepretation of quantum theory, which is obsensibly about resolving our conflicting feelings about the classical world we think we know and the quantum world, will only satisfy me if it comes along with an equal solution to resolving the conflicting feelings about why quantum computers are of the intermediate power we widely suspect them to be. Maybe, indeed this also offers an explanation for why there is little agreement over interpretations: the problem is related to a problem in computational compelxity, BQP?=BPP, whose resolution would represent a major insight in long standing difficult problems in computational complexity.