Shortcut-to-Adiabatic Controlled-Phase Gate in Rydberg Atoms

TL;DR

The paper addresses the slow slowness of adiabatic CZ gates in Rydberg-atom platforms and proposes a shortcut-to-adiabaticity approach using effective counterdiabatic driving (eCD) implemented through Floquet engineering to realize an approximate counterdiabatic field H_CD(t). By designing an experimentally feasible H_eCD(t) with time-dependent amplitudes that mimic H_CD(t), the authors achieve fast, high-fidelity CZ gates across wide parameter regimes while keeping Rydberg occupation low. They provide explicit constructions of the eCD pulses and demonstrate fidelities exceeding 0.998 at sub-microsecond gate times, supported by quantum process tomography showing near-ideal chi-matrices (differences ~10^-3). An application to a minimal quantum-error-correction circuit confirms the practical advantage of the eCD gate, achieving high-fidelity recovery and outperforming the purely adiabatic approach. The work highlights a robust path toward scalable, fault-tolerant quantum computation with Rydberg atoms by combining STA concepts with Floquet-engineered controls.

Abstract

A shortcut-to-adiabatic protocol for the realization of a fast and high-fidelity controlled-phase gate in Rydberg atoms is developed. The adiabatic state transfer, driven in the high-blockade limit, is sped up by compensating nonadiabatic transitions via oscillating fields that mimic a counterdiabatic Hamiltonian. High fidelities are obtained in wide parameter regions. The implementation of the bare effective counterdiabatic field, without original adiabatic pulses, enables to bypass gate errors produced by the accumulation of blockade-dependent dynamical phases, making the protocol efficient also at low blockade values. As an application toward quantum algorithms, how the fidelity of the gate impacts the efficiency of a minimal quantum-error correction circuit is analyzed.

Figures (4)

• Figure 1: (a) Time dependence of the adiabatic pulses. In phase II the detuning is changed continuously. The explicit form of the curves is given in the text. The peak values of the pulses are $\Omega_\mathrm{max}/2\pi = 17$ MHz and $\Delta_\mathrm{max}/2\pi = 23$ MHz at a total time of $T_\mathrm{tot} = 0.594\;\mu$s. The width is given by $\tau = 0.0945\;\mu$s and the detuning in phase II takes $\approx T_\mathrm{tot}/10$. (b) Energy levels scheme and sketch of the driving sequence; the atoms are first excited to their Rydberg state (phase I), and are then de-excited (phase III); the adiabatic pulses interpolate the three phases in a continuous manner. (c) Instantaneous energy levels of the driven Hamiltonian of Eq. \ref{['eq:Ht']} as a function of time in phase I; Arrows indicate two avoided crossings between states $\ket{01}$ and $\ket{0r}$ or $\ket{10}$ and $\ket{r0}$, and states $\ket{11}$ and $\ket{1r}+\ket{r1}$, respectively.
• Figure 2: Time profile of the amplitudes $f_0(t)$ [(a)] and $f_1(t)$ [(b)] of the counterdiabatic field $H_\text{CD}$. In phase II they vanish as the Rabi pulses are exactly zero. Infidelity for the initial state $(\ket{00}+\ket{11})/\sqrt{2}$ of the adiabatic protocol (c) and of the CD protocol (d) as a function of the total protocol time $T_\mathrm{tot}$ and of the strength of the Rydberg blockade. The change of the detuning in phase II takes $\approx T_\mathrm{tot}/10$. The pulses have peak values $\Omega_\mathrm{max}/2\pi=17$ MHz and $\Delta_\mathrm{max}/2\pi=23$ MHz at $T_\mathrm{tot}=0.594\;\mu$s as in Saffman2020 and then scale as $\Omega_\mathrm{max},\;\Delta_\mathrm{max}\propto 1/T_\mathrm{tot}$ in order to maintain adiabatic evolution. The eCD frequency is chosen to be $\omega=300$ MHz. (e) QPT of the proposed eCD method: difference between the eCD $\chi$-matrix, $\chi_{\mathrm{eCD}}$, and the ideal one. The parameters are the same as in (c) and (d) with fixed $V/2\pi=500$ MHz and $T_\text{tot}=0.594\;\mu$s. The colorbar displays the phase of the matrix elements. The labels $i,x,y,z$ refer to the indices of the operator basis, while the odd (even) axes ticks refer to the upper (lower) row of $\chi$-matrix labels.
• Figure 3: Infidelity for the separable accelerated protocol for initial states $(\ket{00}+\ket{01})/\sqrt{2}$ (a), $(\ket{00}+\ket{11})/\sqrt{2}$ (b) and $\ket{11}$ (c) and relative phase error $|\phi_e|/\pi=|1-|\mathrm{arg}\braket{11|\psi_\mathrm{fin}}|/\pi|$ accumulated for initial state $\ket{11}$ (d). The arrows indicate, that the infidelities of the superposition state can be decomposed into a population error, see (c) and a relative phase error of the sensitive state $\ket{11}$, see (d). We use $H(t)$ from \ref{['eq:Ht']} with $\Omega_\mathrm{max}/2\pi=17$ MHz and $\Delta_\mathrm{max}/2\pi=23$ MHz, in combination with the approximated eCD method with $\omega=1$ GHz.
• Figure 4: Realization of a minimal quantum-error correction algorithm based on the eCD CZ gate with pulses $\Omega_\mathrm{max}/2\pi=17$ MHz, $\Delta_\mathrm{max}/2\pi=23$ MHz, protocol time $T_\mathrm{tot}=0.594\;\mu$s, blockade strength $V/2\pi=500$ MHz and eCD frequency $\omega=350$ MHz. The quantum circuit is represented in (h), while in (a)-(c) and (d)-(e) the difference of the matrix elements of the recovered density matrix for the corrected qubit is shown. The first and second row corresponds to the case in which no flip or one flip have occurred, respectively, for three different initial states, $\ket{0}$ [(a) and (d)], $\ket{1}$ [(b) and (e)] and $(\ket{0} + \ket{1})/\sqrt{2}$ [(c) and (f)]. The colorscale indicates the phase of the residual matrix elements. The infidelity of the different configurations is shown in (g) for the circuit based on the eCD gate (blue) compared to the same results using the adiabatic CZ gate in Saffman2020 (red). We excluded the initial state $\ket{11}$ without a bitflip, as this configuration is dark to the CZ gate. The dashed lines show the average over the five states. In all cases the minimum fidelities achieved by the eCD gate are larger than $0.999$.