$N$-Body Interactions between Trapped Ion Qubits via Spin-Dependent Squeezing

We describe a simple protocol for the single-step generation of $N$-body entangling interactions between trapped atomic ion qubits. We show that qubit state-dependent squeezing operations and displacement forces on the collective atomic motion can generate full $N$-body interactions. Similar to the Mølmer-Sørensen two-body Ising interaction at the core of most trapped ion quantum computers and simulators, the proposed operation is relatively insensitive to the state of motion. We show how this $N$-body gate operation allows for the single-step implementation of a family of $N$-bit gate operations such as the powerful $N$-Toffoli gate, which flips a single qubit if and only if all other $N\text{−}1$ qubits are in a particular state.

Figure 1

Quantum phase gates with trapped ions. (a) A chain of trapped ions whose many-body spin state $|\psi ⟩$ is decoupled from the motional state $|n{⟩}_m$ of a single harmonic phonon mode $m$ represented by vibrational integer index $n\ge 0$. Ions are addressed with bichromatic laser fields with carrier spin-flip Rabi rates ${\mathrm{\Omega }}_i$. (b) Motional sideband transitions driven by the laser field. Tuning the laser field on resonance with the first red and blue sideband transitions at frequency $\pm {\omega }_m$ from the carrier [47] generates a spin-dependent force which is mediated by simultaneous absorption and of emission phonons. Tuning the tones at the second red and blue sidebands at $\pm 2{\omega }_m$ from the carrier generates spin-dependent squeezing by absorption and emission of pairs of phonons. (c) Displacing the motion of mode $m$ in a closed loop of phase space adds a phase $\widehat{\mathrm{\Phi }}$ to the quantum state that is given by the area of the enclosed contour. (d) The MS gate using spin-dependent displacements results in a spin-dependent phase linear in the spin operators ${\widehat{\sigma }}_{{\varphi }_i}^{(i)}\equiv {\widehat{\sigma }}_x^{(i)}\cos {\varphi }_i+{\widehat{\sigma }}_y^{(i)}\sin {\varphi }_i$ of ion $i$. When applied to multiple ions, the resulting phase $\widehat{\mathrm{\Phi }}$ is thus quadratic in the spin operators, corresponding to two-body MS interaction [37, 48]. (e) Motional squeezing shrinks one direction in phase space but expands the other to conserve the phase-space element area. (f) $N$-body entangling gate. Synchronized spin-dependent squeezing (cross symbols) applied in between displacements produces squeezing of the motion along the momentum axis. The phase $\widehat{\mathrm{\Phi }}$ now depends exponentially on the spins, and therefore contains products of $N$ spin operators. The phase-space axes are displayed with dimensionless units ${\tilde{x}}_m={\widehat{x}}_m/(2x_m^0)$ and ${\tilde{p}}_m={\widehat{p}}_m/(2p_m^0)$, with the convention $[{\tilde{x}}_m,{\tilde{p}}_m]=i/2$.

Figure 2

Three-body entangling gates. (a) Phase-space evolution for three spins, following the sequence of alternating spin-independent displacements and spin-dependent squeezing operations [c.f. Fig. 1 and Eq. (5)]. Each ion squeezes (antisqueezes) the momentum quadrature of the $m$th motional mode by a factor of $e^{-\overline{\xi }}$ ($e^{\overline{\xi }}$) if its spin points downward (upward). The state-dependent phase-space area ${\widehat{\mathrm{\Phi }}}_{\mathrm{seq}}$ accumulated in the evolution generates the gate $U_{\mathrm{seq}}=e^{-i{\widehat{\mathrm{\Phi }}}_{\mathrm{seq}}}$ with a maximal squeezing of the oscillator mode by a factor of $e^{N\overline{\xi }}$ when all spins are aligned. (b) Overlap between the proposed many-body gate in Eq. (8) (accompanied by one-qubit rotations on the third qubit as described in the main text) and the $N$-Toffoli gate $T_N^{(N)}$ depending on the maximal squeezing of phase-space coordinates in dB [$10{\log }_{10}(e^{N\overline{\xi }})$] for $N=3$, 4. Inset: ideal $T_3^{(3)}$ operator in the computational basis.