Figure 1

The solid curve in Fig. 1 depicts the shortest path connecting the north pole $(1,0,0)$ to a point $(0,\cos \varphi ,\sin \varphi )$ under the metric $g$. The dashed curve is the geodesic under the standard metric and represents a segment of a great circle.Reuse & Permissions

Figure 2

(A) Coupling topology where the first qubit $\left(I_1\right)$ and third qubit $\left(I_3\right)$ are coupled only indirectly via the second qubit $\left(I_2\right)$ with coupling constants $J_{12}=J_{23}=J$. (B) Schematic energy-level diagram for the spin system in a static magnetic field in the $z$ direction, which determines the quantization axis. The Zeeman energy of a state $\mid b_1{b}_2{b}_3⟩$ is ${\sum }_{k=1}^3(\frac{1}{2}-b_k){\omega }_i$, where ${\omega }_k$ is the Larmor frequency of spin $I_k$; $b_k=1$ and 0 correspond to the low- and high-energy eigenstates of the angular momentum operator along the $z$ direction. The schematic representation in (B), corresponds to the case ${\omega }_1\approx {\omega }_3$. The coupling term [Eq. (2)] results in an additional shift of the states $\mid 000⟩$ and $\mid 111⟩$ by $\pi J$ and of the states $\mid 010⟩$ and $\mid 101⟩$ by $-\pi J$ (not visible in the figure because $\mid J\mid \ll \mid {\omega }_k\mid$. The effect of the CNOT(1, 3) unitary transformation is indicated by arrows.Reuse & Permissions

Figure 3

Efficient pulse sequences based on sub-Riemannian geodesics for the implementation of ${\mathcal{U}}_{13}=\exp \{-i\frac{\pi }{2}2I_{1z}{I}_{3z}\}$ (A), $\sqrt{{\mathcal{U}}_{13}}=\exp \{-i\frac{\pi }{4}2I_{1z}{I}_{3z}\}$ (B), simulating coupling evolution by angles $\frac{\pi }{2}$ (A) and $\frac{\pi }{4}$ (B) between indirectly coupled qubits, and of a Toffoli gate (C). Qubits $I_1$, $I_2$, and $I_3$ are assumed to be on-resonance in their respective rotating frames. Narrow and wide vertical bars correspond to hard pulses with flip angles $\pi ∕2$ and $\pi$, respectively, if no other flip angle is indicated. Rotations around the $z$ axis are represented by dashed bars. The unitary operator ${\mathcal{U}}_{13}$, which is locally equivalent to the CNOT(1,3) gate, is synthesized by sequence (A) in a total time $T_C^{*}=2\tau =1.253J^{-1}$. The amplitude of the weak pulses (represented by gray boxes) with a duration of $\tau =0.627J^{-1}$ is ${\nu }_a=uJ∕2=0.52J$. The hard-pulse flip angles $\theta =31.4°$ and $\alpha =180°-\theta =148.6°$. Sequence (B) of total duration $(4+\sqrt{7})∕4J^{-1}=1.66J^{-1}$ synthesizes the propagator $\sqrt{{\mathcal{U}}_{13}}$. The amplitude of the weak pulse (gray box) with a duration of $\sqrt{7}∕4J^{-1}=0.661J^{-1}$ is ${\nu }_w=3J∕\sqrt{7}=1.134J$. Pulse sequence (C) realizes the Toffoli gate in a total time $(6+\sqrt{7})∕4J^{-1}=2.16J^{-1}$. The sequence is based on the sequence for $\sqrt{{\mathcal{U}}_{13}}$ and a weak pulse with the same amplitude and duration as in sequence (B).Reuse & Permissions

Figure 4

Simulated (left) and experimental (right) ${}^1\mathrm{H}$ spectra of the amino moiety of ${}^{15}\mathrm{N}$ acetamide with $J_{12}=-87.3\mathrm{Hz}$, $J_{23}=-88.8\mathrm{Hz}$, and $J_{13}=2.9\mathrm{Hz}$. Starting from thermal equilibrium, in all experiments the state ${\rho }_A=I_{1x}$ was prepared by saturating spins $I_2$ and $I_3$ and applying a ${90}_y^{°}$ pulse to spin $I_1$, where ${\rho }_A$ is the traceless part of the density operator [18]. (A) Spectrum corresponding to ${\rho }_A=I_{1x}$, (B) spectrum obtained after applying the propagator ${\mathcal{U}}_{13}=\exp \{-i\frac{\pi }{2}2I_{1z}{I}_{3z}\}$ to ${\rho }_A$, (C) resulting spectrum after applying the propagator $\sqrt{{\mathcal{U}}_{13}}=\exp \{-i\frac{\pi }{4}2I_{1z}{I}_{3z}\}$ to ${\rho }_A$, (D) spectrum after applying the Toffoli gate to ${\rho }_A$.Reuse & Permissions

Figure 5

Experimental two-dimensional ${}^{15}\mathrm{N}$ spectrum of ${}^{15}\mathrm{N}$ acetamide representing the effect of the CNOT(1,3) gate. The resonance of ${}^{15}\mathrm{N}$ is split in three lines due to coupling to spin $I_1$ and $I_3$ as shown on top of the figure. The label $(1,0)$ represents the resonance line of spin $I_2$ when spin $I_1$ is in state $1$ and spin $I_3$ is in the state $0$. In the experiment shown in the figure, the spin $I_2$ precesses for time $t_1$ followed by application of the CNOT(1,3) gate, followed by recording the precession of spin $I_2$ as a function of time real time $t_2$. The experiment is repeated by incrementing $t_1$. The two-dimensional signal $s(t_1,t_2)$ is then Fourier transformed and the Larmor frequency of various resonances of spin $I_2$ during the time $t_1$ and $t_2$ are plotted along vertical and horizontal axis, respectively. Using the relation $\mathrm{CNOT}(1,3)=\exp \{-i\frac{\pi }{2}(I_{1z}-I_{3z}\left)\right\}\exp \{-i\frac{\pi }{2}{I}_{3x}\}{\mathcal{U}}_{13}\exp \{-i\frac{\pi }{2}{I}_{3y}\}$, the experimental pulse sequence was based on the implementation of the propagator ${\mathcal{U}}_{13}$ shown in Fig. 3A.Reuse & Permissions

Figure 6

(A) Broadband version of the ideal $U_{13}$ sequence shown in Fig. 3A, which is robust with respect to frequency offsets of the spins. Positive coupling constants $J_{12}=J_{23}=J>0$ (with $J_{13}=0$) and hard spin-selective pulses are assumed. The delay $\Delta$ is $\tau ∕\left(4m\right)$ and the flip angle $\delta$ is $2\pi {\nu }_a\tau ∕\left(4m\right)=0.5119∕m$ (corresponding to $29.33°∕m$). (B) Experimentally implemented pulse sequence synthesizing ${\mathcal{U}}_{13}$ for the spin system of ${}^{15}\mathrm{N}$ acetamide with $J({}^1\mathrm{H},{}^{15}\mathrm{N})\approx -88\mathrm{Hz}$, $m=2$, flip angles $\alpha =148.6°$, $\theta =31.4°$, and $\delta =14.66°$ and delay $\Delta =890.2\mathrm{\mu }\mathrm{s}$. Narrow and wide bars correspond to $90°$ and $180°$ pulses, respectively, if no other flip angle is indicated, $z$ rotations are represented by dashed bars.Reuse & Permissions

Figure 7

(A) Broadband version of the ideal $\sqrt{{\mathcal{U}}_{13}}$ sequence shown in Fig. 3B, which is robust with respect to frequency offsets of the spins. Positive coupling constants $J_{12}=J_{23}=J>0$ (with $J_{13}=0$) and hard spin-selective pulses are assumed. The delay $\Delta$ is $\sqrt{7}∕\left(16mJ\right)=0.1654∕\left(mJ\right)$ and the flip angle $\delta$ is $3\pi ∕\left(8m\right)$ (corresponding to $67.5°∕m$). (B) Experimentally implemented pulse sequence synthesizing $\sqrt{{\mathcal{U}}_{13}}$ for the spin system of ${}^{15}\mathrm{N}$ acetamide with $J({}^1\mathrm{H},{}^{15}\mathrm{N})\approx -88\mathrm{Hz}$, $\exp \{-i(\pi ∕2\left)I_{1z}{I}_{3z}\right\}$ for $J({}^1\mathrm{H},{}^{15}\mathrm{N})\approx -88\mathrm{Hz}$ with $m=2$, $\delta =33.75°$, $\Delta =\sqrt{7}∕[16m\mid J({}^1\mathrm{H},{}^{15}\mathrm{N})\mid ]=939.5\mathrm{\mu }\mathrm{s}$, ${\Delta }_1=1∕\left(4\mathrm{\Delta }{\nu }_{13}\right)=806.5\mathrm{\mu }\mathrm{s}$, and ${\Delta }_2=1∕[2\mid J({}^1\mathrm{H},{}^{15}\mathrm{N})\mid ]=5.68\mathrm{ms}$.Reuse & Permissions

Figure 8

(A) Broadband version of the ideal Toffoli sequence shown in Fig. 3C, which is robust with respect to frequency offsets of the spins. Positive coupling constants $J_{12}=J_{23}=J>0$ (with $J_{13}=0$) and hard spin-selective pulses are assumed. (B) Experimentally implemented pulse sequence synthesizing a Toffoli gate for the spin system of ${}^{15}\mathrm{N}$ acetamide. Delays $\Delta$, ${\Delta }_1$, ${\Delta }_2$, and the small flip angle $\delta$ are defined in Fig. 7.Reuse & Permissions