Scalable uniform construction of highly conditional quantum gates

We present a scalable uniform technique for the construction of highly conditional multiply-controlled-not quantum gates of trapped ion qubits, such as the Toffoli gate, without using ancilla states and circuits of an exorbitant number of concatenated one- and two-qubit gates. Apart from the initial dressing of the internal qubit states with vibrational phonons and the final restoration of the phonon ground state, our technique requires the application of just a single composite pulse on the target qubit and is applicable both in and outside the Lamb-Dicke regime. We design special narrowband composite pulses, which suppress all transitions but the conditional transition of the target qubit; moreover, these composite pulses significantly improve the spatial addressing selectivity.

Figure 1

(Left) Ratios of pulse areas $A_{v,v+1}/A_{4,5}$ with $2⩽v⩽7$, vs the LD parameter $\eta$, needed for the Toffoli gate when the ion chain is initialized with $v_0=5$ phonons. (Right) Excitation profile produced by a three-component BB pulse $B_3$ with composite phases ($0,0.65\pi ,0$). Note that, for a broad range of pulse areas around $\pi$, the transition probability $p$ for $B_3$ is very close to 1, $p\gtrsim 1-{10}^{-4}$ (horizontal dashed lines).

Figure 2

Linkage diagram for a system of three ions in which the third ion is addressed with a laser tuned on the second blue sideband of a selected vibrational mode. The laser field couples states with $v$ and $v+2$ phonons with coupling strength dependent on $v$ and the LD parameter $\eta$, Eq. (1).

Figure 3

(Left) Ratios of pulse areas $A_{6,8}/A_{2,4}$ and $A_{4,6}/A_{2,4}$ vs the LD parameter $\eta$, when the ion chain is initialized with $v_0=5$ phonons. (Right) Excitation profile produced by a NB pulse $N_{13}$. The range of gate operation is $\eta \lesssim 0.3$ where the two pulse area ratios are in the vicinity of 2.

Figure 4

Simulated fidelity of the Toffoli gate with NB composite sequences from Table for initial phonon number $v_0=4$ (solid lines) and 5 (dashed lines) vs the LD parameter $\eta$ and the pulse area $A$. (Left) Standard NB sequences $N_5$ (top) and $N_9$ (bottom). (Right) Optimized NB sequences $N_5^o$ (top) and $N_9^o$ (bottom). The area is divided by $A_{\mathit{min}}=A_{v_0-3,v_0-1}$, that is, the (smallest) pulse area that has to be seen as a $\pi$ pulse by the transition $|000\rangle |v_0-3\rangle \leftrightarrow |001\rangle |v_0-1\rangle$, cf. Figs. 2 and 3.

Figure 5

Simulated fidelity of ${\mathrm{C}}^n$-not gates with $n=3\text{--}6$ with the optimized NB sequence $N_{13}^o$ from Table vs the LD parameter $\eta$ and the pulse area $A$. The area is divided by $A_{\mathit{min}}=A_{v_0-n-1,v_0-n+1}$, that is, the (smallest) pulse area that has to be seen as a $\pi$ pulse by the transition $|00\cdots 00\rangle |v_0-n-1\rangle \leftrightarrow |00\cdots 01\rangle |v_0-n+1\rangle$. Note that the same composite sequence $N_{13}^o$ implements all gates efficiently.