[FOM] Abdul Lorenz on Kieu

[Tien D Kieu]

Dear Abdul Lorenz,

Thank you for the comments, my responses in the order given are:

1/  Even though the Hamiltonians involved are not quadratic in momentum 
operator, that does not imply that we cannot create/simulate them.  Lloyd and 
Braunstein have proposed a way to simulate Hamiltonians with arbitrary powers 
in position and momentum operators (and hence arbitrary powers in creation and 
annihilation operators) in quantum optics with linear and non-linear effects 
(including squeezing and Kerr nonlinearity).  See their paper on 
http://arxiv.org/pdf/quant-ph/9810082.

2/  Decoherence is indeed the possible killer.  The whole difficulty with the 
physical implementation of standard quantum computation (with qubits and 
quantum gates) is how to protect the computation process from decoherence 
before the process is completed.  Here with the proposed quantum adiabatic 
computation too, decoherence is important has to be taken care of in any 
physical implemtation.  It deserves further study  Some initial numerical 
investigation by Childs, Farhi and Preskill 
(http://arxiv.org/pdf/quant-ph/0108048) into unitary control errors and 
decoherence in quantum adiabatic computation for various combinatoric search 
problems has shown some inherent robustness.  This surprising robustness is 
perhaps parallel with the surprise that there exist possible quantum error 
corrections in standard quantum computation (with qubits and quantum gates).  
(Recall that initially there was strong scepticism (even from experienced 
people like Landauer) that, because of the no-cloning theorem in quantum 
mechanics, error correction was impossible in quantum computation; but the 
discovery of quantum error correction code soon afterwards was a big suprise 
and changed all that.)  At this stage, even the mere theoretical and in-
principle existence of a quantum mechanical procedure to probabilistic 
determine Hilbert's tenth problem is, at least to me, extremely interesting.

3/  I have nothing more to add to what I have already said about energy 
measurement here (except to repeat that the final spectrum is integer-valued so 
the gap is *at least* one unit in energy, which is then sufficient for 
specifying \Delta E without knowing the spectrum).  However, I would like to 
point out that beside the energy measurement we can also do with the 
measurement of occupation number (represented by the operator a^\dagger a), 
since by construction the end-point Hamiltonian (H_P in my notation) commutes 
with the number operators so these two observables (energy and occupation 
number) are compatible.  Such occupation number measurement would give us the 
positive integers n_1, n_2, etc which we can then substitute into the 
Diophantine polynomial in the places of corresponding variables to see if the 
polynomial vanishes (in which case, the equation has some positive integer 
solution) or not (no integer solution).  This point has been made in, for 
example, my paper in Contemporary Physics.

Regards,

Tien Kieu



Quoting martin at eipye.com:

> 1 - I still keep my point of view about the physical meaning of the
> squared Diophantine operator D^2. A physical Hamiltonian in Quantum
> Mechanics has dependence only on the second power of the momentum
> operator: p^2.  However, as the D^2 operator involves arbitrary powers of
> the number operator, say n^k;  this means D^2 contains terms that go as
> p^(2k)!!! In consequence, D^2 can not be interpreted as a Hamiltonian
> describing any physical system, nor the D^2 eigenvalues represent
> energies. Therefore, D^2 can not be reproduced in the laboratory.
> 
> 2.- The transition process, described by Kieu's algorithm, for passing
> from an oscillator Hamiltonian to the D^2 operator in a not well defined
> time T,  has to deal with all the decoherence processes that result from the
> interaction of the measurement and control apparatus to the coherent
> states. Decoherence would destroy the coherent state in a finite time.
> These effects are not taken into account in any way into Kieu's analysis,
> reflecting that one is assuming a completely ideal system, where a
> coherent state is to be conserved in time. A real experiment has to take
> into account that there always be dissipation and/or scattering processes
> that produce decoherence. Since decoherence grows in time, one can not
> extend the time T arbitrarily.
> 
> 3.- Regarding the required precision  one has to reach to determine
> whether there is a zero "energy"  level in the D^2 spectrum, Kieu has
> argue that it should be enough to reach a Delta E smaller than
> the space among the first two levels. In principle, given by a integer
> number multiplied by some constant that gives the dimensions of energy to
> the spectrum. But knowing this spacing is equivalent to know in advance
> the spectrum of D^2, and so, whether there is a solution to the
> Diophantine equation!!!
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