Fast and robust quantum computation with ionic Wigner crystals

We present a detailed analysis of the modulated-carrier quantum phase gate implemented with Wigner crystals of ions confined in Penning traps. We elaborate on a recent scheme, proposed by two of the authors, to engineer two-body interactions between ions in such crystals. We analyze the situation in which the cyclotron (${\omega }_c$) and the crystal rotation (${\omega }_r$) frequencies do not fulfill the condition ${\omega }_c=2{\omega }_r$. It is shown that even in the presence of the magnetic field in the rotating frame the many-body (classical) Hamiltonian describing small oscillations from the ion equilibrium positions can be recast in canonical form. As a consequence, we are able to demonstrate that fast and robust two-qubit gates are achievable within the current experimental limitations. Moreover, we describe a realization of the state-dependent sign-changing dipole forces needed to realize the investigated quantum computing scheme.

Figure 1

Two-dimensional Coulomb crystals of ions in a Penning trap rotating at frequency ${\omega }_r$. To manipulate the internal states of the ions, laser beams can address single sites or multiple ions.

Figure 2

Ratio ${\omega }_r/{\omega }_c$ vs total angular momentum $P_{\theta }$ for ${\alpha }_z=0.7$ and $N=30$. For $P_{\theta }=0$ we retrieve the well-known limit ${\omega }_r=2{\omega }_c$, in which there is no magnetic field in the rotating frame.

Figure 3

Infidelity vs temperature for ${\alpha }_z=0.7$, $N=30$, and $P_{\theta }=4\times {10}^3{\ell }_s^{2}m{\omega }_c$. Parameters: ${\nu }_c={\omega }_c/(2\pi )=76.08$ kHz, ${\nu }_{\mathit{xy}}={\omega }_{\mathit{xy}}/(2\pi )=5.38$ kHz, ${\nu }_z={\omega }_z/(2\pi )=53.26$ kHz, and ${\nu }_r={\omega }_r/(2\pi )=32.75$ kHz. The black (solid) line corresponds to ${\tau }_g/{\tau }_r={10}^{-1}$ (${\tau }_g=3$ $\mu$s), the red (dashed) line to ${\tau }_g/{\tau }_r={10}^{-2}$ (${\tau }_g=0.3$ $\mu$s), and the blue (dashdot) line to ${\tau }_g/{\tau }_r={10}^{-3}$ (${\tau }_g=0.03$ $\mu$s), with ${\tau }_r=2\pi /{\omega }_r$. The inset (on the top left corner) provides the infidelity for ${\tau }_g/{\tau }_r={10}^{-1}$ (${\tau }_g=0.03$ $\mu$s), with ${\nu }_c=7.61$ MHz as in Ref. [28], $\nu =300$ MHz, ${\nu }_{\mathit{xy}}=537.97$ kHz, ${\nu }_z=5.33$ MHz, and ${\nu }_r=3.27$ MHz. The inset (on the bottom right corner) illustrates, for the same trapping parameters as for the former inset but with $\nu =2.4$ MHz, the infidelity for ${\tau }_g/{\tau }_r=10$ (${\tau }_g=3$ $\mu$s). Such modulation frequency $\nu$ lies within the gap among radial and $\mathrm{E⃗}\times \mathrm{B⃗}$ phonon modes (see Fig. 5).

Figure 4

Infidelity vs temperature for ${\alpha }_z=0.02$, $N=30$, and $P_{\theta }=1.3\times {10}^5{\ell }_s^{2}m{\omega }_c$. Parameters: ${\nu }_c=7.61$ MHz, ${\nu }_{\mathit{xy}}=3.80$ MHz, ${\nu }_z=152.16$ kHz, and ${\nu }_r=1.65$ kHz (see text for more details).

Figure 5

Modes for the parameters considered in Fig. 3 (top row, ${\alpha }_z=0.70$, $P_{\theta }=4\times {10}^3{\ell }_s^{2}m{\omega }_c$) and Fig. 4 (bottom row, ${\alpha }_z=0.02$, $P_{\theta }=1.3\times {10}^5{\ell }_s^{2}m{\omega }_c$).

Figure 6

Energy shifts due to the Zeeman effect vs the $m_j$ quantum number. The distances among the energy levels are not in scale.

Figure 7

Sketch of the pulse sequence in order to design the necessary dipole forces to implement the modulated-carrier phase gate. The upper lines are red detuned, whereas the lower lines are blue detuned. In the figures the time is in arbitrary units.

Figure 8

Modulated-carrier gate with the same polarization of the laser fields: ${\sigma }^{-}$ polarized (left) and ${\sigma }^{+}$ polarized (right). The distances among the energy levels are not in scale.

Figure 9

Designed laser modulation forces in order to fulfill both (37) and (36). The blue detuned laser intensity with a superimposed square wave signal (top left) and similarly for the red detuned one (top right). The resulting signals are depicted on the bottom left, whereas switching of polarization is given on the bottom right.

Figure 10

Modulated-carrier gate with different polarizations of the laser fields. The distances among the energy levels are not in scale.

Figure 11

Variant of the modulated-carrier gate with different polarizations of the laser fields. The distances among the energy levels are not in scale.