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By now, there has been enough grasp on quantum chaos for systems with a small number of degrees of freedom. The major tool used is periodic orbit theory to approximate the spectral distribution. Is there any suitable generalization to a quantum field theory with one spatial dimension whose classical analog exhibits spatiotemporal chaos? Periodic orbit theory breaks down because we can always combine prime cycles at spatially adjacent regions to form another prime cycle with an order far greater than any of the original prime cycles. The prime cycles become too dense with a combinatorial explosion.

On a related note, what is the qualitative distribution of a one dimensional quantum cellular automata which is Turing complete?

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I'm not quite sure what you mean by "the qualitative distribution" of a one dimensional cellular automata. Could you clarify this? Two alternating species are sufficient for Turing completeness, given an appropriate update rule. – Joe Fitzsimons Oct 18 '11 at 10:23

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