[FOM] From Elena to Kieu through Toby Ord.

[Tien D Kieu]

Dear LE,

1/ I have mentioned in my previous comments, in passing, that the energy-time 
uncertainty relation is different from that of position-momentum. Even though 
this is not related to the discussion at hand and only provides a distraction, 
I just wish to clarify here that they are indeed different. On the one hand we 
can derive the position-momentum uncertainty relation from the commutation 
relation of the momentum and positions operators; on the other hand, there is 
no time operator in quantum mechanics, so the two uncertainty relations are not 
on the same footing. Indeed, the position-momentum uncertainty relation 
concerns with the standard deviations of the measured values for position and 
momentum (as each measurement would give a different values for the same 
operator, be it position or momentum or energy). In contrast, the energy-time 
relation concerns, beside the standard deviation in energy measurement, with 
the duration of the measuring time: there is no time operator so there is no 
*uncertainty* in measuring the time, in the sense of uncertainties in measuring 
other observables in QM. 

2/ Now to the energy-time relation, I maintain that it is not detrimental to my 
algorithm. Taking the risk of repeating myself, I would like to stress that the 
energy eigenvalues to be measured are integer-valued in some unit. Thus we only 
need to obtain a standard deviation (in the energy measurement) sufficiently 
small so that the various integer-valued eigenvalues can be distinguished. That 
means finite (but small compared to one energy unit of the system) \Delta E, 
which in turns only requires a finite measuring time, in accordance with the 
energy-time uncertainty relation.
3/  With regard to some possible physical implementation of the required time-
dependent Hamiltonian, I have had in mind the use of quantum optics as 
mentioned in the Int J Theo Phys paper of mine (see below).  However, this may 
not be the only method of implementation.

4/ Of course, I would love to see the algorithm implemented by myself or 
someone. But, on the other hand, were Turing asked to implement his machines, 
he would have still been at it. Nonetheless, our failure to implement a Turing 
machine does not and cannot deny the importance of such machines. (No intention 
or implication at all to compare myself and this great genius here, I only 
bring this up to point out the unreasonableness of such request, at least at 
this stage.)
5/ I don't see what is wrong with my numerical simulation for some Diophantine 
equations, even though, as clearly stated, they are extremely simple.

6/ If my reply does not satisfy you, which I doubt that it would, then I would 
urge you to publish your arguments in a formal paper or post them on the ArXiv, 
as I have done with all of my papers, for the scrutiny of anyone interested. In 
that way, a wider community would benefit, rather than restricting them to this 
forum. This forum has a very important role in that it provides the means for 
people to discuss, without being too formal, some issues which need 
clarifications. But if the discussion leads to nowhere and if one is so 
convinced of one's own arguments, it might now be the time to formalise the 
arguments and publish them for the benefit of the many scientific communities 
(in this case, those of mathematicians, physicists and computer scientists and 
perhaps philosophers) and also for the permanent records.
7/ Some updates on the publications of mine on the topic: 
·	Contemporary Physics 44 (2003) 51- 71: “Computing the Noncomputable.” 
·	Int J  Theo Phys 42 (2003) 1461 - 1478:  “Quantum algorithms for 
Hilbert’s tenth problem.”
·	Proc Roy Soc A 460 (2004) 1535:  “A reformulation of Hilbert’s tenth 
problem through Quantum Mechanics.” 
·	in Proceedings of SPIE Vol. 5105 Quantum Information and Computation, 
edited by Eric Donkor, Andrew R. Pirich, Howard E. Brandt, (SPIE, Bellingham, 
WA, 2003), pp. 89-95, quant-ph/0304114: “Numerical simulations of a quantum 
algorithm for Hilbert’s tenth problem.” 
·	quant-ph/0310052:  “Quantum adiabatic algorithm for Hilbert’s tenth 
problem: I. The algorithm.” 

Lastly, I would like to take this opportunity to thank Toby Ord for bringing 
the various comments about my work posted on FOM to my attention (I do 
appreciate those comments very much, no one would like to see his or her work 
ignored totally, without any feedback or input from others -- such silence can 
be deafening), and for being a messenger in forwarding my comments to FOM.  (I 
have since registered with the forum and now should be able to post on the 
forum directly.)

Tien Kieu