Scalable quantum search using trapped ions

We propose a scalable implementation of Grover’s quantum search algorithm in a trapped-ion quantum information processor. The system is initialized in an entangled Dicke state by using adiabatic techniques. The inversion-about-average and oracle operators take the form of single off-resonant laser pulses. This is made possible by utilizing the physical symmetries of the trapped-ion linear crystal. The physical realization of the algorithm represents a dramatic simplification: each logical iteration (oracle and inversion about average) requires only two physical interaction steps, in contrast to the large number of concatenated gates required by previous approaches. This not only facilitates the implementation but also increases the overall fidelity of the algorithm.

Figure 1

The MS basis states for $N=6$ ions with three excitations form a series of independent chainwise linkages. Since the total number of excitations is half the number of ions, then from Eq. (9) it follows that states with $m_j>0$ are inaccessible and are therefore not shown. The states that make up the longest ladder are all symmetric Dicke states [31]. The number of motional $n_p$ and ionic $n_i$ excitations for each level can be inferred from $m_j$ [Eq. (9)].

Figure 2

Simulation of the Grover search algorithm with $N$ ions and $N/2$ excitations (register dimension $C_{N/2}^N$), for $N=6$ (upper panel), $N=8$ (middle panel), and $N=10$ (lower panel). The system of ions is assumed to be initialized in the symmetric Dicke state $|W_{N/2}^N\rangle$. The laser pulses have a Gaussian shape, $g(t)=g_0{e}^{-t^2/T^2}$. The thin vertical lines display the times when each logical mathematical step (an implementation of the Grover operator) is completed. The computed numerically scaled detunings and peak Rabi frequencies for the oracle and the inversion-about-average operators, respectively, are given in Table . The marked-state population (the fidelity) of around $99\%$ is obtained after $n_g=3$, 6, and 12 steps, respectively, in exact agreement with Grover’s value [Eq. (5)]. The numerical simulation includes all off-resonant transitions to states with $m_j\ne 0$.