Multi-qubit Rydberg gates between distant atoms

TL;DR

The paper presents a protocol for fast, high-fidelity multi-qubit C_kZ gates in neutral-atom platforms by globally addressing atoms arranged in a star-graph and adiabatically steering between ground states and MIS-like Rydberg configurations. A parity-dependent geometric phase φ_g = π ν_q is imprinted while dynamical phases cancel, enabling C_kZ up to local single-qubit corrections; the scheme extends to distant-qubit gates via a quantum bus of auxiliary atoms. The authors provide a detailed analysis of the adiabatic spectrum, non-adiabatic leakage, and decay/thermal errors, and they show how pulse shaping and STIRAP-based sign changes of the interaction improve fidelity. These results point to enhanced connectivity and reduced gate depth in neutral-atom quantum processors, with fidelities in the few ×10^{−3} regime for small to moderate k and feasible extensions to larger graphs via buses.

Abstract

We propose an efficient protocol to realize multi-qubit gates in arrays of neutral atoms. The atoms encode qubits in the long-lived hyperfine sublevels of the ground electronic state. To realize the gate, we apply a global laser pulse to transfer the atoms to a Rydberg state with strong blockade interaction that suppresses simultaneous excitation of neighboring atoms arranged in a star-graph configuration. The number of Rydberg excitations, and thereby the parity of the resulting state, depends on the multiqubit input state. Upon changing the sign of the interaction and de-exciting the atoms with an identical laser pulse, the system acquires a geometric phase that depends only on the parity of the excited state, while the dynamical phase is completely canceled. Using single qubit rotations, this transformation can be converted to the C$_k$Z or C$_k$NOT quantum gate for $k+1$ atoms. We also present extensions of the scheme to implement quantum gates between distant atomic qubits connected by a quantum bus consisting of a chain of atoms.

Figures (14)

• Figure 1: Schematic illustration of a two-dimensional array of atoms irradiated by lasers (shaded blue) to implement the C$_k$Z quantum gates between neighboring atomic qubits in star-graph configurations, or between distant atomic qubits connected by auxiliary atoms prepared initially in state $\ket{1}$. Red transparent circle around a Rydberg excited atom corresponds to blockade range. Inset shows the level scheme of atoms involving the qubit encoding ground state subleveles $\ket{0},\ket{1}$ and the strongly interacting Rydberg state $\ket{r}$.
• Figure 2: (a) Star-graph configuration of $N=k+1$ atoms. Half-filled circles represent atomic qubits in either state $\ket{0}$ or $\ket{1}$. A global laser field with Rabi frequency $\Omega$ and detuning $\Delta$ couples the atomic states $\ket{1}$ and $\ket{r}$ while atoms in state $\ket{0}$ are decoupled from the laser (inset). Atoms in Rydberg state $\ket{r}$ interact with each other. Graph edges denote Rydberg blockade interactions between the connected atoms, $B= B_{0j}\gg\max(\Delta,\Omega)$, while absence thereof implies weak-interactions, $B_{ij} \ll\max(\Omega)$ for $i,j>0$. (b) Schematic representation of the computational basis states of the atoms $\ket{\mathbf{q}}$, which is the ground state of $\mathcal{H}$ for $-\Delta \gg \Omega$, whereas the corresponding MIS configurations $\ket{R_{\mathbf{q}}}$ is the ground state for $\Delta \gg \Omega$.
• Figure 3: Circuit implementing the C$_k$Z gate. Applying the Hadamard gates on the target qubit before and after the C$_k$Z gate will result in C$_k$NOT, which is the Toffoli gate for $k=2$.
• Figure 4: Energy eigenvalues $\mathcal{E}_n$ of Hamiltonian (\ref{['Eq:Hamiltonian']}) for $N=3,4,5$ atoms (left, center, right panels) arranged in symmetric star-graph configurations ($B_{0j}= B, \, B_{jj+1}= \mathrm{const} \, \forall \, j \geq 1$) vs detuning $\Delta$. The parameters are $\Omega=\Omega_0 \, \forall \, \Delta \in [-\Delta_0, \Delta_0]$; $\Delta_0=2.4\Omega_0$, $|B|=6\Omega_0$ ($B\gtrless 0$ for upper/lower panels) for $N=3,4$; and $\Delta_0=3.2\Omega_0$, $|B|=5.6\Omega_0$ for $N=5$. In step I (upper panels) we implement the transformation $\mathcal{U}_{\mathrm{I}}$: Starting in state $\ket{\mathbf{q}} = \ket{\alpha_1}$ for $\Delta=-\Delta_0$, the system adiabatically follows the eigenstate $\ket{\alpha_1}$ with lowest energy $\mathcal{E}_1$ (solid blue line) reaching state $\ket{R_\mathbf{q}}$ for $\Delta=\Delta_0$. In step II (lower panels) we implement the transformation $\mathcal{U}_{\mathrm{II}}$: Now starting in state $\ket{R_\mathbf{q}} = \ket{\alpha_m}$ for $\Delta=-\Delta_0$, the system adiabatically follows the eigenstate $\ket{\alpha_m}$ with highest energy $\mathcal{E}_m$ (solid blue line) returning to state $\ket{\mathbf{q}}$ at $\Delta=\Delta_0$ and acquiring the sign change $(-1)^{\nu_{\mathbf{q}}}$. During steps I and II, the non-adiabatic transitions away from states $\ket{\alpha_{l=1,m}}$ to other instantaneous eigenstates $\ket{\alpha_n}$ are quantified by the dimensionless parameter $\eta_{ln}=|\langle \alpha_l|\partial_t|\alpha_n\rangle|^2 \tau/\Delta_0$ (color depth of the solid lines for the corresponding $\mathcal{E}_n$). Eigenenergies of "dark" eigenstates, decoupled from the laser for all $\Delta$, are not shown.
• Figure 5: (a) Time dependence of the Rabi frequency $\Omega(t)$ (right vertical axis) and detuning $\Delta(t)$ (left vertical axis) during step I with $B>0$ (a1) and step II with $B<0$ (a2) as per Eqs. (\ref{['eq:PulseI']}) and (\ref{['eq:OmegaDelta']}), with parameters $\Omega_0,\Delta_0,B$ as in Fig. \ref{['Fig:Spectrum']}. (b) Dynamics of populations of the initial/final states $\ket{\mathbf{q}}=\ket{11\ldots 1}$ and the corresponding intermediate MIS states $\ket{R_{\mathbf{q}}}=\ket{1r\ldots r}$ with $\nu_{\mathbf{q}} = k$ Rydberg excitations for $N=k+1=3,4,5$ atom graphs (top, middle, bottom panels) with $\tau=16\pi/\Omega_0$. Inset in each panel shows the total phase $\phi(t) = \arg \braket{\Psi(0)}{\Psi(t)}$ of the state $\ket{\Psi}$ of the system.