Trapped Ion Chain as a Neural Network: Error Resistant Quantum Computation

We demonstrate the possibility of realizing a neural network in a chain of trapped ions with induced long range interactions. Such models permit one to store information distributed over the whole system. The storage capacity of such a network, which depends on the phonon spectrum of the system, can be controlled by changing the external trapping potential. We analyze the implementation of error resistant universal quantum information processing in such systems.

Figure 1

(a) Ratio between the frequencies of the second and first vibrational modes as a function of the exponent of the trapping potential for 20 ${\mathrm{Ca}}^{+}$ ions. (b) Final overlap averaged over 500 initial configurations as a function of the initial overlap, for the patterns associated with the two lowest vibrational modes of 40 ions in a potential $V=\rho |x{|}^{0.5}$ with $\rho =6.6\times {10}^{-20}\mathrm{J}/{\mathrm{m}}^{1/2}$. The black squares (triangles) correspond to the first (second) pattern.

Figure 2

Fidelities of the $\mathcal{H}$ gate (left) and the Bell gate (right), with respect to time ($r_2=0.95r_1$). The classical fidelity bounds are $2/3$ and $2/5$, respectively (horizontal dashed lines). The inset in gray shows the fields $A\lambda$ and $B_1\lambda ={10}^{-5}B\lambda$ and $B_2\lambda ={10}^{-6}B\lambda$ with respect to time. For adiabaticity, the chosen fields require $T\gg 7\times {10}^6\hslash /\lambda$. The fidelities are largely independent of the actual dynamics of the fields.