Experimental Time-Optimal Universal Control of Spin Qubits in Solids

Quantum control of systems plays an important role in modern science and technology. The ultimate goal of quantum control is to achieve high-fidelity universal control in a time-optimal way. Although high-fidelity universal control has been reported in various quantum systems, experimental implementation of time-optimal universal control remains elusive. Here, we report the experimental realization of time-optimal universal control of spin qubits in diamond. By generalizing a recent method for solving quantum brachistochrone equations [X. Wang et al., Phys. Rev. Lett. 114, 170501 (2015)], we obtained accurate minimum-time protocols for multiple qubits with fixed qubit interactions and a constrained control field. Single- and two-qubit time-optimal gates are experimentally implemented with fidelities of 99% obtained via quantum process tomography. Our work provides a time-optimal route to achieve accurate quantum control and unlocks new capabilities for the emerging field of time-optimal control in general quantum systems.

Figure 1

Schematic representation of quantum TOC. Blue lines represent paths of quantum evolution in the $\mathrm{SU}(2^n)$ operator space, where $n$ stands for the number of qubits. To realize a target evolution operator $U_F$ at $t=T$ starting from the identity operator $I$ at $t=0$, there are several choices of evolution path ${\mathrm{\Gamma }}_i(i=1,2,\dots )$. The goal of TOC is to figure out which evolution costs the minimum time $T$.

Figure 2

Comparison on time costs for target gate operator $R(\widehat{\mathbf{z}},\theta )$ between the derived TOC and the Euler rotations. The parameters are set to be $\delta =0$ and ${\nu }_1=5\mathrm{MHz}$. (a) Theoretical comparison on time with $\theta \in (0,\pi ]$. (b) Comparison of experimental gate time for $\theta =\pi /8$, $\pi /4$, $\pi /2$, and $\pi$. The gate time for TOC is considerably shorter than that for Euler rotation. (c) State evolutions during $R(\widehat{\mathbf{z}},\pi )$ with TOC and Euler rotation. The initial state is $(|m_s=0⟩+i|m_s=-1⟩)/\sqrt{2}$. The gate time of TOC is 26.8 ns shorter than that of Euler rotation.

Figure 3

State trajectories under the two-qubit controlled-u gate by TOC with initial states (a) $|0,1⟩$ and (b) $|0,0⟩$. The left panels show the state evolutions of the nuclear and electron spins on the Bloch spheres. When the nuclear spin qubit is in the state $|1⟩$ ($|0⟩$) labeled by the blue arrows, the electron spin qubit undergoes the paths labeled by red lines to the state $|0⟩$ ($|-1⟩$). The right panels show the experimental dynamics of state populations $P_{0,m_I}$ (black circles) and $P_{-1,m_I}$ (grey diamonds), and lines are theoretical predictions of the populations. The error bars on the data points are the standard deviations from the mean.

Figure 4

Quantum process tomography for the controlled-u gate by TOC. The left and right panels are the real and imaginary parts of the reconstructed process matrix $\chi$. The error bar of each point is about 0.01 due to the statistics of photon counts. An average gate fidelity of 0.99(1) can be obtained from the process matrix.