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analogy with the development of classical chaos. This
ψ¯=exp( iTnˆ2/4)exp( ikcosθ)exp( iTnˆ2/4) (11) phenomenon also has negative effects for operability of
− − − plasma traps and accelerators. In fact the very first
where nˆ = id/dθ and h¯ = 1. Here the quantum dy- Chirikov resonance-overlap criterion had been invented
−
namics simulatedon the same computer remains exactly in the pioneering work [55] for the explanation of exper-
reversible as it is shown in Fig. 9. iments on plasma confinement in open magnetic traps
The above results clearly show that there is no expo- andlater foundbroadapplicationsinacceleratorphysics
nentialinstabilityinquantummechanics[54]. This gives [56]. Since 1959 it is the only simple physical criterion
adirectindicationthattheerrorsinquantumcomputing which allows to determine the chaos border in classical
canbeefficientlycorrectedsincee.g. Shor’salgorithmre- nonlinearhamiltoniansystems[55,56,53]anditmeritsto
quires relativelysmall number of gate operations N for be marked by the Nobel symposium organizers. Indeed,
G
number factorization: N 300L3 where L is a number the interest to the quantum chaos appeared only after a
G
∼
of digits in the number to be factorized [5]. In a sense deep understanding of classical chaos had been reached.
the Shor algorithm is exponentially fast while the errors In a similar way it is possible to think that the deep
grow only as a power of time that opens broad perspec- understanding of quantum chaos in many-body systems
tives for quantum computing. This however requires a in future will allow to quantum computers operate in a
development of efficient error-correcting codes. There is better way.
also another general question related to the uncertainty The majorityofresults onquantumchaosinquantum
relation between energy and time. Indeed, the quantum computing presented in this paper were obtained with
computingduringtime∆tallowstoresolveenergylevels Bertrand Georgeot and it is my pleasure to thank him
only on the energy scale ∆E 1/∆t. This means that for the fruitful collaboration. I am also thankful to Oleg
∼
the resolution of exponentially small spacing ∆E ∆ Sushkov for valuable discussions of various questions of
n
∼
requiresexponentiallylongtimeanalogoustotheHeisen- interactionanddisorder. IthanktheIDRISinOrsayand
berg time scale in the quantum chaos t 1/∆ . Ap- the CICT in Toulouse for access to their supercomput-
H n
∼
parently only after this time scale allexponentially large ers. This research is partially done in the frame of EC
informationhiddenintheHilbertspacecanbecomeavail- programRTN1-1999-00400.
able. However,itispossiblethatonshortertimescalesa
useful information can be extracted in a way unaccessi-
ble to classical computers, e.g. Shor’s factorization. But
a question about how many such exponentially efficient
algorithms can be found remains open. ∗
http://w3-phystheo.ups-tlse.fr/∼dima
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