[FOM] Abdul Lorenz on Kieu
Tien D Kieu
kieu at swin.edu.au
Sun Apr 25 16:36:01 EDT 2004
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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