Quantum chaos in small quantum networks

Abstract

We study a 2-spin quantum Turing architecture, in which discrete local rotations {α m } of the Turing head spin alternate with quantum controlled NOT-operations. We show that a single chaotic parameter input {α m } leads to a chaotic dynamics in the entire Hilbert space. The instability of periodic orbits on the Turing head and 'chaos swapping' onto the Turing tape are demonstrated explicitly as well as exponential parameter sensitivity of the Bures metric.

Quantum network architecture of tight-binding models with substitution sequences

“…For a small perturbation of the initial phase angle ¬ 0 the cumulative angles A m , B m , respectively, grow exponentially with m, and so do the deviation terms D C cf 2m … § †Ĉ cf 0 2m … § † ¡ C cfpo 2m … § † from the periodic orbits (po). Thus the total cumulative control loss induced by the perturbation can show chaotic quantum behaviour on the Turing head [9]. For the Thue± Morse control (4) we easily ® nd that for jÁ 0 iˆj ¡ 1i …S † « j ¡ 1i …1 † and nˆ8m C n … ‡ †ˆ2…¬ 1 ‡ ¬ 2 †m; C n …¡ †ˆ0; respectively, which is very similar to the result of the`regular' machine with ¬ reg…”

“…cf m is the cumulative perturbation of the angle ¬ cf m at step nˆ2m ¡ 1), we obtain an initial exponential sensitivity [9]. In the case of the present substitution sequences we observe for a small perturbation of the given ¬ 1 ;¬ 2 , no initial exponential sensitivity in the evolution of D 2 , which con® rms that any periodic orbit is stable (® gure 3 (a)).…”

“…Kim and G. Mahler Turing-head patterns 0; ¼ 2 …n †; ¼ 3 …n † f g for initial state jÁ 0 iˆj ¡ 1i …S † « j ¡ 1i …1 † under the control of substitution sequences: (a) quasi-periodic Fibonacci (qf) with ¬1ˆ2=5º, ¬2ˆ¬1 ‡ 0:0005º, (b) as in (a) but for ¬2ˆ¬1 ‡ 0:03º, (c) as in (a) but for ¬ 2ˆ¬1 ‡ 0:05º; (d) Thue± Morse (tm) control with ¬ 1ˆ2 =5º; ¬ 2ˆ¬1 ‡ 0:1001º. For each simulation the total step number is nˆ10 000. contrasted with the chaotic Fibonacci (cf) rule, ¬ cf m ‡1ˆ¬ cf m ‡ ¬ cf m¡1 (Lyapunov exponent: ln …1 ‡ 5 1=2 †=2 > 0 †, which can be interpreted as a temporal random (chaotic) analogue to one-dimensional chaotic potentials [9]: each step ¬ m is controlled by the cumulative information of the two previous steps. For a small perturbation of the initial phase angle ¬ 0 the cumulative angles A m , B m , respectively, grow exponentially with m, and so do the deviation terms D C cf 2m … § †Ĉ cf 0 2m … § † ¡ C cfpo 2m … § † from the periodic orbits (po).…”

“…This metric can be applied likewise to the total-network-state space or any subspace. In any case it is a convenient additional means to characterize various QTMs: for the regular case, ¬ reg mˆ¬ 1 (Lyapunov exponentˆ0), and any initial perturbation¯for« 0 the distance remains almost constant [9]; for the chaotic Fibonacci rule (cf), on the other hand,…”

“…Finally we display the evolution of D 2 for the total network state jÁ n i, which also shows no exponential sensitivity (® gure 3 (b), cf. ® gure 3 (c) in [9]). The respective distances for tape-spin 1 are similar to those shown.…”