Pulse-Gated Quantum-Dot Hybrid Qubit

A quantum-dot hybrid qubit formed from three electrons in a double quantum dot has the potential for great speed, due to the presence of level crossings where the qubit becomes chargelike. Here, we show how to exploit the level crossings to implement fast pulsed gating. We develop one- and two-qubit dc quantum gates that are simpler than the previously proposed ac gates. We obtain closed-form solutions for the control sequences and show that the gates are fast (subnanosecond) and can achieve high fidelities.

Figure 1

(a) Schematic of the quantum-dot hybrid qubit and of the physics underlying gate operations. The logical qubit states are $|0{⟩}_L=|S⟩|\downarrow ⟩$ and $|1{⟩}_L=\sqrt{1/3}|T_0⟩|\downarrow ⟩-\sqrt{2/3}|T_{-}⟩|\uparrow ⟩$, where $|S⟩$, $|T_{-}⟩$, and $|T_0⟩$ are two-particle singlet ($S$) and triplet ($T$) states in the left dot, and $|\uparrow ⟩$ ($|\downarrow ⟩$) denotes a spin-up (spin-down) electron in the right dot. Introducing tunneling amplitudes $t_1$ and $t_2$ to a (1,2) excited state $|E⟩$ induces transitions between $|0{⟩}_L$ and $|1{⟩}_L$. (b) Energies of $|0{⟩}_L$, $|1{⟩}_L$, and $|E⟩$ versus detuning $\epsilon$ between the two dots. The ground state has charge occupation (2,1) when $\epsilon <{\epsilon }_A$ and (1,2) when $\epsilon >{\epsilon }_A$; the qubit operates mainly in the regime $\epsilon \le {\epsilon }_A$. The energy difference between the qubit states $|0{⟩}_L$ and $|1{⟩}_L$ is large for all values of $\epsilon$, but there is an avoided crossing between $|0{⟩}_L$ and $|E⟩$ at detuning ${\epsilon }_A$ (blue box), and another avoided crossing between $|1{⟩}_L$ and $|E⟩$ at ${\epsilon }_B$ (dotted magenta box). Pulse-gate transitions between $|0{⟩}_L$ and $|1{⟩}_L$ can be performed by using both avoided crossings. Pulses to the detuning ${\epsilon }_P$ are used to induce phase differences between the three states. A gating sequence to perform arbitrary rotations in the qubit Hilbert space is indicated with arrows; along the detuning axis, the pulse sequence is ${\epsilon }_i\to {\epsilon }_B\to {\epsilon }_P\to {\epsilon }_A\to {\epsilon }_P\to {\epsilon }_B\to {\epsilon }_f$. (c) The corresponding circuit diagram for the gate sequence, with time progressing from left to right. Gates $P_1$, $A$, and $P_2$ are tunable, with the control parameters ${\varphi }_1$, $\theta$, and ${\varphi }_2$ given in Eqs. (2, 3, 4).

Figure 2

Numerically calculated infidelity (1-fidelity) caused by dephasing during qubit rotations without use of dynamical decoupling [35], as a function of the interdot tunnel coupling in a double quantum dot. Solid red circles show the results for a hybrid qubit, and open triangles, diamonds, and squares show the results for the exchange gate of singlet-triplet ($ST$) qubits [16, 20] for GaAs, natural Si and isotopically purified Si, with ${}^{28}\mathrm{Si}$ exhibiting the best overall gate fidelity. The two curves for the Si hybrid qubit represent different values of the singlet-triplet splitting (0.05 meV for the upper curve and 0.5 meV for the lower curve).

Figure 3

Pulse-gating protocol for a two-qubit controlled-phase gate. (a) The detunings of the avoided crossings $A$ and $B$ of the target qubit shift by $\delta \epsilon$ (horizontal arrows on the figure) when the state of the control qubit is changed from $|0{⟩}_L$ to $|E⟩$. (b) Realistic device geometry for a top-gated $\mathrm{Si}/\mathrm{SiGe}$ heterostructure, described in Ref. 24. 2D Thomas-Fermi modeling [34] of this device yields $\delta \epsilon \gtrsim 0.1\mathrm{meV}$, ample for operation of a conditional gate. (c) Gate sequence for a conditional rotation of the phase of the target qubit. First, a $B$ gate is applied to the control qubit, which leaves $|0{⟩}_L$ unchanged and transforms $|1{⟩}_L\to |E⟩$. The target qubit is then pulsed to anticrossing $B^{\prime }$, then to $P^{\prime }$, and then again to $B^{\prime }$, which changes the phase of the target qubit only when it starts in state $|1{⟩}_L$ and the control qubit is in state $|E⟩$. Finally, applying a $B$ gate to the control qubit converts it back to a spin qubit. The gray shading denotes that the operations on the target qubit are conditional between application of the two $B$ gates to the control qubit. This gate sequence changes the phase of the target qubit only when the control qubit starts in state $|1{⟩}_L$. The operations also perform a conditional phase rotation on the control qubit, which can be adjusted to be a multiple of $2\pi$ by appropriate choice of pulse parameters.