‘t Hooft and quantum computation


Gerard ‘t Hooft is one of greatest living physicists, one of the main contributors to the Standard Model. He has been awarded the Nobel prize in physics on 1999. I have had the opportunity to meet him in Piombino (Italy) at a conference on 2006 where he was there to talk about his view on foundations of quantum mechanics. He is trying to understand the layer behind quantum mechanics and this question has been a source of discussions here where he tried to find a fair audience to defend his view. Physics StackExchange, differently from MathOverflow for mathematicians,  has not reached the critical mass with most of the community, Fields medalists included for the latter, where the stars of the physics community take time to contribute. The reason is in the different approaches of these communities that can make a hard life for a Nobel winner while mathematicians’ approach appears polite and often very helpful. This can also be seen with the recent passing away of the great mathematician William Thurston (see here).

‘t Hooft’s thesis received an answer from Peter Shor that is one of the masters of quantum computation. What makes interesting the matter is that two authoritative persons discussed about foundations. Shor made clear, as you can read, that, out of three possibilities, a failure of quantum computation could give strong support to ‘t Hooft’s view and this is the case ‘t Hooft chose. Today, there is no large scale quantum computer notwithstanding large efforts by research and industry. Large scale quantum computers is what we need to turn this idea into a meaningful device. The main reason is that, as you enlarge your device, environment driven decoherence changes this into a classical system losing its computational capabilities. But, in principle, there is nothing else to prevent a large scale quantum computer from working. So, if you are so able to remove external disturbances, your toy will be turned into a powerful computational machine.

People working on this research area rely heavily on the idea that, in principle, there is nothing from preventing a quantum device to become a large scale computing device. ‘t Hooft contends that this is not true and that, at some stage, a classical computer will always outperform the quantum computer. Today, this has not been contradicted from experiment as we have not a large scale quantum computer yet.

‘t Hooft’s idea has a support from mathematical theorems. I tried to point out this in the comments below Shor’s answer and I received the standard answer:

and how does your interpretation of this theorem permit the existence of (say) superconductors in large condensed matter systems? How about four-dimensional self-correcting topological memory?

This is a refrain you will always hear as, when some mathematical theorems seem to contradict someone pet theory, immediately theoretical physicists become skeptical about mathematics. Of course, as everybody can see, most of the many body quantum systems turn into classical systems and this is a lesson we learn from decoherence but few many-body systems do not. So, rather to think that these are peculiar we prefer to think that a mathematical theorem is attacking our pet theory and, rather to do some minimal effort to understand, one attacks with standard arguments as this can change a mathematical truth into some false statement that does not apply to our case.

There are a couple of mathematical theorems that support the view that increasing the number of elements into a quantum system makes it unstable and this turns into a classical system. The first is here and the second is this. These theorems are about quantum mechanics, the authors use laws of quantum mechanics consistently and are very well-known, if not famous, mathematical physicists. The consequences of these theorems are that, increasing the numbers of components of a quantum system, in almost all cases, the system will turn into a classical system making this a principle to impede for a general working of a large scale quantum computer. These theorems strongly support ‘t Hooft’s idea. Apparently, they clash with the urban legend about superconductors and all that. Of course, this is not true and the Shor’s fears can be easily driven away (even if I was punished for this). What makes a system unstable with respect to the number of components can make it, in some cases, unstable with respect to quantum perturbations that could be amplified to a macroscopic level. So, these theorems are just saying that in almost all cases a system will be turned into a classical one but there exists an non-empty set, providing a class of macroscopic quantum systems, that can usefully contain a quantum computer. This set is not large as one could hope but there is no reason to despair whatsoever.

Laws of physics are just conceived to generate a classical world.

M. Hartmann, G. Mahler, & O. Hess (2003). Gaussian quantum fluctuations in interacting many particle systems Lett. Math. Phys. 68, 103-112 (2004) arXiv: math-ph/0312045v2

Elliott H. Lieb,, & Barry Simon (1973). Thomas-Fermi Theory Revisited Phys. Rev. Lett. 31, 681–683 (1973) DOI: 10.1103/PhysRevLett.31.681


7 Responses to ‘t Hooft and quantum computation

  1. aramharrow says:

    Those papers seem to say that in some cases, quantum systems appear to be classical. But it doesn’t follow that quantum systems _always_ appear classical. Thus, I don’t see how they support ‘t Hooft’s idea, or any other skeptical take on quantum computing.

  2. mfrasca says:

    You misunderstood both the papers and my post. Of course, I never said that a quantum many-body system becomes _always_ classical. But if you read the post with attention you will find that, in some cases, quantum fluctuations are amplified rather than damped and macroscopic quantum behavior happens (one can just think about quantum phase transitions where quantum fluctuations can extend to all the system). The papers just say that classical behavior is really generic for quantum many-body systems and most of them becomes classical systems as the number of components become increasingly large. This makes smaller the set of possible quantum systems that one can think of to build up a quantum computer and it means that a principle exists that bounds the realization of a quantum computer. The proof of existence relies always on people, very smart indeed, that is working in the field. This supports ‘t Hooft’s view but does not prove him right yet.

    • Peter Shor says:

      The problem is that ‘t Hooft’s view implies that all quantum many-body systems are classically computable in the limit of very large systems. Theorems that say some but not all quantum systems behave classically in the limit don’t support ‘t Hooft’s theory.

      • mfrasca says:

        Dear Prof. Shor,

        Now I see your point. No, I cannot claim that any many-body quantum system becomes a classical system (I did not do it anywhere), on the basis of these theorems, but that a quite generic effect exists in the limit of an increasing numbers of particles. This is also stated in my post but I think it should made clearer. My take is that such an effect is real, we seem to agree on this, and goes somehow in the direction ‘t Hooft took. But it does not prove him right either for such an extremist take. Maybe, his theory is in need for further work.


  3. David says:

    Marco Frasca: have you read any of ‘t Hoofts most recent papers ? I ask because I’d love to know what you think of his general hypothesis.

    • mfrasca says:

      Dear David,

      my view on questions about quantum mechanics and relativity is similar to the one given by Steven Weinberg: We incurred in the fundamental laws of nature entailing exactness. So, I personally avoid to try to find what is behind them but rather I like a reformulation work as I did quite recently in http://arxiv.org/abs/1201.5091. Given this, I respect the work of colleagues that think differently from me but, of course, I would appreciate to see a couple of conditions to be satisfied:

      1) Concrete models.

      2) Experimental predictions to check the idea.

      Given these, I could admit that a search for a deeper layer to quantum mechanics or relativity should prove to be viable. So far, I have never seen both conditions met.


  4. […] overlooked in such discussions, defending ‘t Hoof ideas and so, generating some fuss (see here and the discussion I have had with Peter Shor and Aram Harrow). Indeed, we finally stipulated that […]

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