# [FOM] Tien Kieu's reply to Elena, Lorenz, Davis and Hodges.

Toby Ord toby.ord at philosophy.oxford.ac.uk
Wed Apr 7 14:00:46 EDT 2004

Tien Kieu was quite interested to read the recent arguments appearing
here in relation to his proposed method for hypercomputation via the
Quantum Adiabatic Theorem. Since he is not on the FOM list, he has sent
me a copy of his replies to these arguments to post in his stead,

Toby.

1)-------------------------------------------------

I am grateful for the attention and time that Andrew Hodges has paid to
the algorithm.  Below are my remarks and responses, interspersed with

> "Kieu avoids the problem of demanding the running of a Turing machine
> for
> infinitely many steps by using the equivalence with the problem of
> finding
> to solutions to a Diophantine problem. Thus, he is evaluating a
> polynomial
> for infinitely many arguments instead.  I can't believe that this
> reformulation can work magic, and allow an infinite amount of
> computation to be done in a finite time.

- "Evaluating ... for infinitely many arguments" is and/or conjures up
a classical visualisation for us to comprehend the search in our
limited experience.  Unfortunately, this visualisation is not supported
by the quantum adiabatic theorem, upon which our algorithm is based.
Essentially, the theorem states that with the condition of discreteness
of the eigenvalues, continuity and differentiability of the associated
eigenfunctions and no level crossing then the initial state prepared to
be an eigenstate of the initial Hamiltonian will approach the
corresponding eigenstate of the final Hamiltonian in a *finite* time:
The slower (but still finite) the temporal rate, the higher the
approach' probability.  This theorem has been mathematically proved
for infinite dimension Hilbert spaces (see, for example, Messiah's book
on Quantum Mechanics).  I have simply exploited this amazing ability
for our algorithm to locate a single state (the ground state) among an
infinite number of states in a finite time.  Thus, the hard-to-believe
thing would be not our algorithm entirely but also the (proven) theorem
itself.

> His Hamiltonian operating on the nth occupation number state must be
> equivalent to putting n particles through a physical process, and one
> would expect this to take something like n times as long as putting
> one particle through the process.

- This expectation is once again originated from a classical way to
visualise the process, which is not supported by quantum mechanics.  In
fact, this expectation is in direct opposition to the principle of
linear superposition, which has been exploited for the parallel'
processing power of quantum computation as demonstrated in the (less
controversial) algorithm by Shor for factoring integers.

> (Kieu's method seems to make no essential use of the discreteness of
> the polynomial coefficients, only of the integer variables on which it
> is evaluated.)   The trouble with this is, as you have pointed out,
> that the coefficients require *infinite precision*. The Hamiltonian
> H=(2m^2 - n^2)^2 has infinitely many states with 'energy' 1, and none
> with 'energy' 0. But H will have infinitely many zeroes if that
> coefficient 2 is allowed to vary by any amount whatever.  Infinite
> precision is not 'effective'.  Maybe we *could* insist that the
> Hamiltonian be built entirely out of operators with discrete integer
> eigenvalues (exploiting the fact that the Diophantine polynomial has
> integer coefficients). This would get us out of the need for infinite
> precision. But in this case, it seems to me that it would be
> equivalent to performing, by quantum operations, bit-by-bit operations
> of unary arithmetic on the input integer n. Basically, this would be
> just as complicated as putting a T-machine through n steps, and we are
> back to the problem of Calude's 'operator.'
> At the very least, some argument should be supplied by Kieu to explain
> why
> in fact, despite these obvious difficulties, his Hamiltonian operator
> could
> be implemented in an effective way in a finite time.

-  The discreteness of the polynomial coefficients is essential, see
below in the discussion of \sqrt{2}.  However, I am not convinced that
in a simulation or in a physical implementation of the algorithm one
would need infinite precision just to represent an *integer* such as 2.

> More generally, there is very little discussion of the 'infinite
> limit' in
> Kieu's paper, even though infinitude is *of the essence* in this
> discussion, and finite approximations are of no real value. A lot of
> his
> theory is for two-state cases, with a rather vague assertion that this
> suffices because eventually two states will dominate.  I suspect that
> the infinite limit doesn't actually work for his theory as he thinks.
> As his theory never seems to use the fact that the Hamiltonian is a
> integer-valued polynomial, as far as I can see it would work just as
> well if it were real-valued. (Or have I missed something in the theory
> of adiabatic cooling?) Thus we could apply it to ((sqrt(2)*m - n))^2
> which as m and n range over integers, can never vanish, but attains
> arbitarily small values for sufficiently large m and n. How would his
> idea of converging to the 'ground state' apply in this case? There are
> infinitely many states of which the infimum is 0, but 0 is never
> attained. I suspect that this might show up a difficulty in the
> infinite limit.

- The case of \sqrt{2} is not applicable for our algorithm because of
the two following reasons, one mathematical and the other operational.
Mathematically, the quantum adiabatic theorem, on which we base our
algorithm, requires the assumption of all the eigenvalues being
*discrete*.   With \sqrt{2}, on the other hand, the final Hamiltonian
would have zero as the limiting point for its eigenvalues:  Given an
arbitrarily small \epsilon > 0, there always exists countably many
eigenvalues which are less than \epsilon (and greater than zero).
Thus, it is not obviously clear whether quantum adiabatic theorem is
still valid with \sqrt{2} or not.  Operationally, furthermore, we have
to be able to classically set up the computing system before starting
any quantum (adiabatic) computation.  With \sqrt{2} we would have
needed (i) infinite precision in the sense that *all* the digits of
\sqrt{2} are readily available (not necessarily all digits at once but
could be one by one); and (ii) *infinite time* to classically "read in"
*all* the infinitely many digits of \sqrt{2} before the quantum
computation starts.  (The "read in" is a classical act, either through
measurement or through the supply from a Turing machine in a simulation
of the quantum algorithm, and thus would require an infinite amount of
time to process all the digits - in contrast to quantum processes in
which many can be processed at once in parallel as embodied in the
superposition principle.) These two reasons separate the irrational
coefficients of general equations from the rational coefficients of
Diophantine equations, and perhaps limit our algorithm to the latter -
thus providing a way out to the "counter example" of \sqrt{2}.

> There is certainly something very odd about his assertions on page 6
> 'truncating' the number of states according to which Diophantine
> equation
> one is looking at. The trivial example of H=(2m^2 - n^2)^2 shows that
> the
> lowest energy states (H=1) can continue forever. How then can any
> 'truncation' be valid? More deeply, his remark goes against the whole
> spirit of a method which must to applicable to any and every
> Diophantine
> equation, as it suggests taking some first step which depends on
> inspecting
> the equation to be considered. But worse, we know from computability
> theory
> that there is no algorithm which, supplied with a general Diophantine
> equation, can give you a bound on its solutions. So any such inspection
> would be a waste of time. I confess I have not been able to see whether
> this discussion of 'truncation' is essential to his argument about
> taking
> the infinite limit. But if it is not, I cannot see why he brings it
> up, and
> if it is, then it must render his argument invalid.

-  You are right, truncation is not essential to the algorithm at all.
The only reason that I mentioned truncation was that I wanted to
anticipate and prepare the ground for a discussion on simulations, on
classical computers, of the algorithm in the next paper.  Such
simulations would require some *controllable* truncation. (With
hindsights, I should not, perhaps, have mentioned truncation in the
present paper.)

> In the concluding section, Kieu claims that there is no conflict with
> the
> classical insolubility of the problem. Kieu claims that probability
> makes
> all the difference. But the classical (Turing) argument shows the
> non-existence of a halt-testing machine by applying it to itself. If,
> as
> Kieu suggests, his quantum-theoretic test could be made into a
> classical
> procedure (by simulating the solution of the Schroedinger equation on a
> computer), then it looks as if you could still get a similar
> by applying it to itself - and getting a probability which is both more
> than 1/2 and less than 1/2. Anyway, I do not see that probability *in
> itself* makes all the difference. Thus this section does not convince
> me of
> the lack of conflict and I am more inclined to suspect something is
> wrong
> with the theory as applied to infinitely many states.

- I am sorry that I have failed in convincing a reader in this aspect.

> If there *is* a solution to a Diophantine equation, there is no
> problem -
> you can do it trivially just by ordinary computation over all integer
> values until you reach the solution in a finite time. In this case Kieu
> offers no improvement on the obvious. But his positive claim is that
> for
> each insoluble equation there is a time T, after which you know that
> there
> is near-certainty of there being *no* solution for any value, however
> large. This suggests that after time T, the system has 'sensed' the
> non-existence of any zero at or above 2^(2^(2^(2^.....[as many as you
> like].....2^T)))...

-  Yes, such "suggestion" is correct, even though it is
counterintuitive (as counterintuitive as with many other quantum
mechanical results).  This is a result implied by the quantum adiabatic
theorem, which states that after certain time (this time here is the
inverse of the rate of change of the time-dependent Hamiltonian) the
probability to obtain the final ground state is not less than a given
real number (between 0 and 1).  The required time is of course a
function of this real number, even though the theorem does not specify
this function (otherwise, our task of deciding solutions for
Diophantine equations would have been somewhat simpler).  Disputing
this "suggestion" would thus entail disputing the quantum adiabatic
theorem itself, and not just our algorithm.

> I find this hard to believe. The adiabatic cooling method which is the
> basis of Kieu's idea, is claimed only to give a square-root speed-up in
> search routines, the same speed-up as comes from more conventional
> quantum-computational schemes. How is it then that it can speed up an
> infinite search *more than exponentially* - and so arrive at a finite
> time?

- Firstly, the claim of square-root speed-up in search routines is
derived (by Roland and Cerf, PRA 65 (2002) 042308) only for
*particular* quantum adiabatic processes, with particular initial state
and with particular class of the time-dependent Hamiltonians (governing
the quantum adiabatic processes).  This claim is certainly *not*
universal for all searching Hamiltonians as the search time should be
dependent on the path connecting the initial to the final Hamiltonians
(see Farhi, Goldstone and Gutmann in
http://arxiv.org/abs/quant-ph/0208135 ).  As a matter of fact, there
exists certain class of Hamiltonians (see Bae and Kwon in
http://arxiv.org/abs/quant-ph/0205048, Eryigit and Eryigit in
http://arxiv.org/quant-ph/0309021, and also Das, Kobes and Kunstatter
in J Phys A 36 (2003) 2839) that results in a finite and constant
searching time, *independent* of the size of the database to be search!
Secondly, it should be pointed out here that the time-dependent
Hamiltonian in our algorithm is not the same as the one employed in
Grover's search."

I sincerely thank Martin Davis and Andrew Hodges for their interest in
and their questions and comments on the algorithm.  I do believe that
the problem under consideration is too important to not have a thorough
discussion and investigation.  I would thus be grateful for any further
comment.

2)-------------------------------------------------

In the long and critical writings of Elena and Lorenz about the
proposed algorithm, I can only identify the main objection that the
algorithm cannot be realised physically because of (thanks to) the
energy-time uncertainty principle.

I myself, with obvious vested interests in the problem, would love to
find out why such proposal cannot be implemented physically.  However,
unfortunately, the reason given above by Elena and Lorenz is flawed
because of a serious misunderstanding of the uncertainty principle as
applied to the algorithm.

On the one hand, we can understand the uncertainty principle as
follows.  The system to be measured is prepared in a state, then is let
to evolve in time subjecting to certain Hamiltonian, then its energy is
measured to obtain some value E_I (which, by quantum mechanical
postulate, is an eigenstate of the Hamiltonian at the time of
measurement).  We repeat the whole cycle (of preparing the system in
the *same* state, letting it evolve in the *same* Hamiltonian, before a
final measurement of its energy) for N times, N being large.  Out of
that large N, we would get the value E_1 for the measured energy for
N_1 times, say, and the value E_2 for N_2 times, etc.  The ratio N_i/N
will tend to the probability that we get E_i in a measurement.  With
these probabilities we then can deduce the average energy \bar{E} and
its standard variance \Delta E.  The energy-time uncertainty principle
states that the product of this energy variance and the measuring time
(taken to be the same in each repetitive measurement) has to be greater
than \hbar divided by 2.  That is, in order to reduce the energy
variance/spread we would have to increase the measuring time
accordingly (such measuring time is the interaction time between the
system being measured and the measuring apparatus).  That is all to the
uncertainty principle (which is quite different from the other
uncertainty principle concerning position and momentum).

Seen in this light, the energy-time uncertainty principle has nothing
to do with our algorithm, as the algorithm does not concern with the
spread/variance of the energy measured.  (The measuring time involved
in the uncertainty principle is NOT the evolution time of our
algorithm.) The algorithm only refers to the probabilities (as
indicated by the ratio N_i/N above).  If in any *single* measurement we
obtain E_k = 0 then we can immediately stop the whole thing and declare
that the Diophantine equation under consideration has a solution.  It
is only the case of no solution that we would need to carry out the
repetition above to obtain the probabilities in order to identify the
ground state as the one that corresponds to a probability of more than
one-half.  (After each measurement, we then go back and prepare our
system in the coherent state and repeat the cycle again - thus, I am
quite mystified by the remark by Elena that measurement would affect
our process?  In which way?)

In summary, the uncertainty principle cannot be invoked here to rule
out the possibility of some physical implementation of our algorithm.
Up to now, we cannot find any physical principle that prohibits such a
physical implementation.  On the other hand, physical implementation
aside, we have indicated that a simulation of the algorithm on
classical computers is indeed also possible.

3)-----------------------------------------------

I would like to add two further (but related) points applicable to both
the sections referring to Elena-Lorenz and Davis-Hodges:

a)  With integer coefficients, the final Hamiltonian has integer-valued
eigenvalues; that is, the separation between any two eigenvalues is *at
least* ONE unit.  This is to be contrast to the case of \sqrt{2}, say,
in which there is no non-zero lower bound for the gap between
eigenvalues and thus might require infinite precision measurement.

b)  Thanks to this finite gap (of at least one unit) with Diophantine
equations, the precision in energy measurement, \Delta E, is thus only
required to be (substantially) less than one unit in order to
distinguish between the integer-valued eigenvalues.  That is, according
to the energy-time uncertainty, \Delta E \Delta t > \hbar/2, the
*measuring time* \Delta t need only be *finite* for such a *non-zero*
\Delta E.

Sincerely,

Tien Kieu