Quantum annealing with pairs of ${}^2\mathrm{\Sigma }$ molecules as qubits

The rotational and fine structure of open-shell molecules in a $\mathrm{\Sigma }$ electronic state gives rise to crossings between Zeeman states of different parity. These crossings become avoided in the presence of an electric field. We propose an algorithm that encodes Ising models into qubits defined by pairs of ${}^2\mathrm{\Sigma }$ molecules sharing an excitation near these avoided crossings. This can be used to realize a transverse field Ising model tunable by an external electric or magnetic field, suitable for quantum annealing applications. We perform dynamical calculations for several examples with one- and two-dimensional connectivities. Our results demonstrate that the probability of obtaining valid annealing solutions is high and can be optimized by varying the annealing times.

Figure 1

Energy levels of a $\mathrm{SrF}(X^2{\mathrm{\Sigma }}^{+}$) molecule with $B_e=0.251{\text{cm}}^{-1}$, ${\gamma }_{\mathrm{SR}}=2.49\times {10}^{-3}{\text{cm}}^{-1}$, $d=3.47\text{D}$ at $B=538\text{mT}$ as functions of the strength of a dc electric field. The avoided crossing between $\beta$ and $\gamma$ is indicated by the vertical dashed line at $E=1.18\text{kV/cm}$.

Figure 2

Coupling constants $J_{\perp }$ and $J_z$ in Eq. (6) for two SrF (top) and SrI (bottom) molecules separated by 500 nm with the intermolecular axis perpendicular to $\mathit{E}$ and $\mathit{B}$ with $B=600$ mT (top) and $B=100$ mT (bottom). The avoided crossing between $\beta$ and $\gamma$ is at $E=6.69$ kV/cm (top) and $E=0.99$ kV/cm (bottom). The shaded areas indicate the range of the parameters for the corresponding spin model, with IM denoting the Ising model.

Figure 3

Parameters of Hamiltonian (11) during QA for two side-by-side qubits (four molecules) in a rectangular configuration in the $|00\rangle$ state (top panel). Molecules in the $|\uparrow \rangle$ state are shown as shaded circles and molecules in the $|\downarrow \rangle$ state are shown as open circles. $B_a=B_b=600\text{mT}$, with intermolecular distances $r_{ij}$ set to $r_{12}=r_{34}=500$ nm and $r_{13}=r_{24}=1000$ nm. The inset shows the electric-field magnitude at both qubits during the annealing: $E\left(s\right)=6.695+0.594s$ kV/cm. $h_a=h_b=0$ in this configuration. The fields are directed along ${\mathit{r}}_{12}$.

Figure 4

Top panels: Examples of 1D (left) and 2D (right) antiferromagnetic QA setups based on pairs of molecules as qubits displayed with all qubits in the $|0\rangle$ state (antiferromagnetic refers to the sign of the couplings between the qubits). The circles show ${}^2\mathrm{\Sigma }$ molecules trapped in an optical lattice, with shaded circles indicating molecules in the $|\uparrow \rangle$ state and open circles indicating molecules in the $|\downarrow \rangle$ state, with intraqubit molecular spacing of $r_1=500$ nm and interqubit spacing of $r_2=1000$ nm in states $|\uparrow \rangle =|\beta \rangle$ and $|\downarrow \rangle =|\gamma \rangle$ depicted in Fig. 1. Middle panels: Probabilities of measuring the system and observing solution states (solid lines) and invalid states (dashed lines) during annealing with homogeneous magnetic and electric fields $B=600$ mT, $E\left(s\right)=6.695+0.594s$ kV/cm using different annealing times. The solution states for both systems are ordered antiferromagnetic configurations. Bottom panels: Final probabilities of measuring the system after annealing in 15 ms (10 ms) of the 1D (2D) configuration.

Figure 5

Probability of obtaining the solution states (solid lines) and the invalid states (dashed lines) for three configurations of qubits. One-dimensional antiferromagnetic (blue circles), 1D ferromagnetic (orange squares), and 2D antiferromagnetic (green triangles), where ferromagnetic and antiferromagnetic refer to the sign of the couplings between the qubits. For the 2D systems, we use $2\times 2$, $2\times 3$, $2\times 4$, and $3\times 3$ configurations for the corresponding number of qubits.

Figure 6

Top: A configuration of molecules leading to an Ising model with 3D connectivities. Two-dimensional layers of Fig. 4 (top right) are stacked on top of each other. Bottom: Parameters of Hamiltonian (14) during QA with intraqubit molecular spacing of $r_1=500$ nm and interqubit spacing of $r_2=1000$ nm. $B=600\text{mT}$ and $E\left(s\right)=6.695+0.594s$ kV/cm.

Figure 7

Dependence of the QA field parameters and couplings on the permanent dipole of molecules. The magnetic field is fixed at 0.6 mT and other molecular parameters of the system are fixed at values for SrF.

Figure 8

Dependence of the QA field parameters and couplings on the rotational constant of molecules ($B_e$). The magnetic field is fixed at 0.6 mT and other molecular parameters of the system are fixed at values for SrF.

Figure 9

Dependence of the QA field parameters on the spin-rotational constant of molecules (${\gamma }_{\mathrm{SR}}$). The magnetic field is fixed at 0.6 mT and other molecular parameters of the system are fixed at SrF values.

Figure 10

Coupling constants $J_{\perp }$ and $J_z$ in Eq. (6) for two molecules with spin-rotational constants of $0.0001{\text{cm}}^{-1}$ (top) and $0.01{\text{cm}}^{-1}$ (bottom), separated by 500 nm with the intermolecular axis perpendicular to $\mathit{E}$ and $\mathit{B}$ with $B=600$ mT, as functions of the electric field magnitude.