Accurate Quantum Logic Gates by Spin Echo in Rydberg Atoms

Scalable quantum computing is based on realizable accurate quantum gates. For neutral atoms, it is an outstanding challenge to design a high-fidelity two-qubit entangling gate without resorting to difficult techniques such as shaping laser pulses or cooling atoms to motional ground states. By using spin echo to suppress the blockade error, we propose an easily realizable controlled-phase Rydberg quantum gate of high intrinsic fidelity. In the context of spin echo, we show that the fundamental blockade error of the traditional Rydberg gate, of the order of $ϵ\sim {10}^{-3}$, actually results from two “clockwise” rotations of Rabi frequencies ${\overline{\mathrm{\Omega }}}_{\pm }=V\pm \sqrt{V^2+{\mathrm{\Omega }}^2}$. In our “echo” sequence, such an error can be suppressed to the order of ${ϵ}^2$ by adding two “anticlockwise” rotations with frequencies $-{\overline{\mathrm{\Omega }}}_{\pm }$. With the blockade error effectively removed, the error caused by Rydberg-state decay becomes the final fundamental limit to the gate accuracy, which, in principle, can be reduced beyond the level of ${10}^{-5}$. Furthermore, due to the small population involved in the “echo” process, the spin-echo gate is robust against the variation of Rydberg blockade caused by the drift of the qubits, so that it can still be much more accurate than that of a traditional Rydberg gate even for qubits cooled only to the submillikelvin regime.

Figure 1

The refocusing of a two-qubit state $|r_{c}0⟩$ in a three-step time-reversal sequence. The state evolution of the target qubit is illustrated by using atomic energy levels in (a)–(c), and by using the Bloch sphere in (d), respectively. (a) [(c)] A $\pi$-pulse rotation between the ground state $|0⟩$ and the Rydberg state $|r_{0(1)}⟩$ of the target, where the dashed line highlights the fact that the state $|r_{0(1)}⟩$ is no longer resonant because of the positive (negative) energy shift $V_0(-V_1)$ in the state $|r_c{r}_{0(1)}⟩$. (b) The transition $|r_0⟩\to |r_1⟩$ via a large microwave Rabi frequency $i{\mathrm{\Omega }}_{\mu }$. The laser Rabi frequencies in (a) and (b) satisfy the condition ${\mathrm{\Omega }}_{t4}=-{\mathrm{\Omega }}_{t2}{V}_1/V_0$, so that the phase accumulations for different irrational Rabi oscillations during the first and last $\pi$ pulses cancel each other out.

Figure 2

The sequence of the state excitation in the spin-echo controlled-phase gate. Horizontal lines denote atomic eigenstates, vertical arrows denote $\pi$ rotations between atomic states, and each number inside a circle shows the order of the $\pi$ pulse that it accompanies. There is a wait duration $T=\phi /V_1$ between the third and fourth pulses. When $\phi =\pi$, a $C_Z$ gate is realized. The Rabi frequency in the third pulse is much larger than the energy shift $|V_{0(1)}|$ of the state $|r_c{r}_{0(1)}⟩$, but those in the second and fourth pulses are much smaller than $|V_{0(1)}|$, so as to satisfy the blockade condition.

Figure 3

The population and phase dynamics of the three components $|r_{c}0⟩,|r_c{r}_0⟩$, and $|r_c{r}_1⟩$ in the wave function $|\psi ⟩$ during pulses 2–4 of the gate sequence when the input state is $|00⟩$. The calculation is performed by using ${\hat{H}}_{\mathrm{v}}$ and ${\hat{H}}_t$ in Eq. (2) on the the initial state $-i|r_{c}0⟩$, with parameters (see Sec. 3) $[C_6(r_c{r}_0),C_6(r_c{r}_1)]/2\pi =[56.2,-25.6]$ THz $\mu m^6$, $L=8\mu \mathrm{m}$, $({\mathrm{\Omega }}_{t2},{\mathrm{\Omega }}_{t3},{\mathrm{\Omega }}_{t4})=V_{+}(1/\eta ,\eta ,|\varkappa |/\eta )$, and $\eta =18$. Here, $\arg (\cdot )$ gives the argument of a complex variable. Because the population involved in the transfer and the deviation of the argument of $⟨r_{c}0|\psi ⟩$ from $-\pi /2$ are tiny, we use the common logarithm to show the population transfer and phase dynamics for the state coefficient of $|r_{c}0⟩$. Similarly, the populations in $|r_c{r}_0⟩$ and $|r_c{r}_1⟩$ are also small and are shown with the common logarithm. The periodic population and phase evolution during pulses 2 (4) has a period of $2\pi /{\overline{\mathrm{\Omega }}}_{t2(t4)}$ because $|{ϵ}_{+}-{ϵ}_{-}|={\overline{\mathrm{\Omega }}}_{t2(t4)}$ [see Eq. (5)]. The values of $\arg ⟨r_c{r}_1|\psi ⟩$ at the end of pulse 3 and at the beginning of pulse 4 differ by $V_{1}T$ [see Eq. (7)].

Figure 4

The population and phase dynamics of the state component $|r_{c}0⟩$ in the wave function from pulse 2 to pulse 4 (a) of the spin-echo gate and (b) during the second pulse of the traditional gate. (c),(d) The gate sequences for the spin-echo gate and the traditional gate, respectively. The Rabi frequency for the second pulse in (b) [or (d)] is equal to that for pulse 2 in (a) [or (c)]. Other parameters are the same as those in Fig. 3. Compared to (a), the population and phase error at the right edge of (b) are much larger. Note that there is a duration of pulse 3 in (a) that is short compared to those of pulses 2 and 4.

Figure 5

(a) [(b)] The error of the spin-echo $C_z$ gate scaled up by ${10}^5$ when the trap depth $U$ is $20[5]$ mK. For comparison, the solid and dashed curves in (c) and (d) show the gate errors scaled up by ${10}^3$ for the spin-echo protocol and the traditional protocol, respectively. The parameters used in the numerical calculation are the same as those in Figs. 3 and 4, except that here we assume a thermal distribution of the atomic location in the dipole trap and consider drift of the atoms during the gate sequence.

Figure 6

The time-reversed dynamics of many-body observables in a four-atom system of Eq. (A1): (a) [(b)] works in the dressing [near-resonant] regime, and shows the average population in the state $|\uparrow ⟩\equiv (|0⟩+|1⟩)/\sqrt{2}$ [$|1⟩$] where each atom is initialized. The forward (backward) state evolution occurs during $(0,0.4]\mu \mathrm{s}$ and $(0.4,0.827]\mu \mathrm{s}$, respectively, so that the many-body observables $\mathcal{M}$ and $\mathcal{P}$ also return to the initial values at the end of the spin-echo sequence. We ignore the duration of the transition $|r_0⟩\to |r_1⟩$ at $0.4\mu \mathrm{s}$. See the text for parameters. Note that not only $\mathcal{M}$ and $\mathcal{P}$, but any physical observable should follow time-reversed dynamics similar to those shown here.