Time-optimal control of independent spin-1/2 systems under simultaneous control

We derive the explicit solution of the problem of time-optimal control by a common magnetic field for two independent spin-$\frac{1}{2}$ particles. Our approach is based on the Pontryagin maximum principle and a symmetry reduction technique. We experimentally implement the optimal control using zero-field nuclear magnetic resonance. This reveals an average gate error of $1\%$ and a 70–$80\%$ decrease in the experiment duration as compared to existing methods. This analytical solution and experimental demonstration of time-optimal control in such a system provides a route to achieve time-optimal control in more general quantum systems.

Figure 1

Representation of the orbit space $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)/{\sim }_{\times }$. The point $A$ corresponds to the equivalence class $\left[({\mathbf{1}}_2,{\mathbf{1}}_2)\right]$; The point $B$ corresponds to the equivalence class $\left[(\mathrm{Phase},{\mathbf{1}}_{\mathbf{2}})\right]$ with $\mathrm{Phase}$ gate defined in (A12) and $C$ corresponds to the equivalence class $\left[(\mathrm{Phase},-{\mathbf{1}}_{\mathbf{2}})\right]$. Trajectories are depicted in the orbit space joining $A$ and $B$, $A$ and $C$.

Figure 2

Flow chart of the procedure to find the optimal control $X=X_1\otimes {\mathbf{1}}_2+\gamma \mathbf{1}\otimes X_1$. For brevity, we use the notation $E_{\gamma }(\omega ,a,b,t):=e^{i\omega {\sigma }_{z}t}{e}^{i(\gamma a{\sigma }_z-\gamma b{\sigma }_y-\omega {\sigma }_z)t}$ and $J(\omega ,a,b,t):=e^{i\omega {\sigma }_{z}t}(ia{\sigma }_z-ib{\sigma }_y)e^{-i\omega {\sigma }_{z}t}$.

Figure 3

Schematic representation of the control via a spatially uniform magnetic field $\vec{B}$ of the ${}^1\mathrm{H}\text{−}{}^{13}\mathrm{C}$ system.

Figure 4

(a) Comparison of time costs for single-spin rotation $R_{\mathbf{n}}^H\left(\theta \right):=e^{-i\mathbf{n}·\vec{\sigma }(\theta /2)}\otimes {\mathbf{1}}_2$ on ${}^1\mathrm{H}$ in the ${}^1\mathrm{H}\text{−}{}^{13}\mathrm{C}$ system between TOC and the composite-pulse sequence [16, 17] implemented via $R_{\mathbf{n}}^H\left(\theta \right)=R_{{\mathbf{n}}_{\perp }}^C\left(\pi \right)e^{-iH\left({\vec{u}}_0\right)\tau }{R}_{{\mathbf{n}}_{\perp }}^C(-\pi )e^{-iH\left({\vec{u}}_0\right)\tau }$. Here ${\vec{u}}_0$ is a constant field along $\mathbf{n}$ and $\tau =\frac{\theta }{2{\gamma }_1\left|{\vec{u}}_0\right|}$. (b) Time-optimal fields and corresponding trajectories on the Bloch sphere for realizing $R_y^{\mathrm{H}}\left(\pi \right):=e^{-i{\sigma }_y(\pi /2)}$.

Figure 5

TOC pulses in different $\gamma$ cases. (a) TOC $\pi$ pulse on ${}^{13}\mathrm{C}$ along the $y$ axis in ${}^1\mathrm{H}\text{−}{}^{13}\mathrm{C}$. (b) TOC $\pi /2$ pulse on ${}^1\mathrm{H}$ along the $y$ axis in ${}^1\mathrm{H}\text{−}{}^{31}\mathrm{P}$.

Figure 6

Schematic diagram of zero-field NMR spectrometer. The NMR sample, contained in a 5-mm NMR tube, is detected with an atomic magnetometer with a ${}^{87}\mathrm{Rb}$ vapor cell operating at $155{}^{\circ }\mathrm{C}$.

Figure 7

Pulse generation circuit. The antiparallel diodes are placed in series with the pulse coils to isolate the noise of amplifier. The antiparallel diodes introduce a voltage drop, which is measured in calibration experiments.

Figure 8

Dependence of signal amplitude on the dc pulse amplitude for pulses of magnetic field in the $x$ (a), $y$ (b), and $z$ (c) directions. The solid curves overlaying the data are fits to curves described in the text.

Figure 9

Experimental result of measuring TOC $\pi$ pulse. TOC $\pi$ pulse is measured by acquiring the load voltage of the resistor in series with the pulse coils.

Figure 10

Randomized benchmarking (RB) experimental results. (a) RB pulse sequences. (b) RB results for ${}^1\mathrm{H}$ single-spin TOC control. Each point $\overline{F}$ is an average over 32 random sequences of $r$ Clifford gates, and the error bars indicate the standard error of the mean. The $y$ axis is in log scale. A single exponential decay (solid line) fits the fidelity decay and reveals an average gate fidelity of 0.99.

Figure 11

Robustness tests of TOC and composite pulse scheme with respect to pulse imperfections. The amplitude distortion ratio in $k=x,y,z$ is defined as ${\eta }_k=\frac{\mathrm{amplitude}\mathrm{distortion}\mathrm{in}k}{\mathrm{ideal}\mathrm{value}\mathrm{in}k}$. Lower panels in (a) and (b) show the isosurface of $\mathrm{fidelity}=0.99$ for TOC and composite pulse, respectively. Upper panels in (a) and (b) are the fidelity contour maps for these two methods. The better fidelity of TOC is indicated by the larger area with a fidelity higher than 0.99.