Optimizing entangling quantum gates for physical systems

Optimal control theory is a versatile tool that presents a route to significantly improving figures of merit for quantum information tasks. We combine it here with the geometric theory for local equivalence classes of two-qubit operations to derive an optimization algorithm that determines the best entangling two-qubit gate for a given physical setting. We demonstrate the power of this approach for trapped polar molecules and neutral atoms.

Figure 1

Optimization of nonlocal gates in the Weyl chamber. Local invariants of the B gate (orange) and cnot (red) are approached iteratively (each violet/blue point corresponds to one step in the optimization of the effective spin-spin Hamiltonian, cf. Sec. 3a).

Figure 2

Gate error vs iteration step for direct (black dashed lines) and local invariant (red dash-dotted lines) optimization of the cnot and B gates with time evolution generated by Eq. (16). The blue dotted line shows the gate error for direct optimization of a specific instance of the local equivalence class [cnot] (see text). Red solid curves display the value of the local invariants functional $J_T^{LI}$ (for the direct optimizations the gate error is equal to the value of the functional).

Figure 3

Optimal gate error as a function of the gate duration for Rydberg gate optimization based on local equivalence classes ($J_T^{LI}$) and on direct optimization ($J_T^D$) of two specific two-qubit gates [$\pi$-phase $\equiv$ diag$(-1,1,1,1)$ and cnot] without (a) and with (b) an additional constraint suppressing population in the intermediate state $|i\rangle =|5p_{1/2}\rangle$. Dashed lines with open symbols (solid lines with filled symbols): simulation including (neglecting) spontaneous emission from $|i\rangle$. Large filled symbols: calculations with trapping potential kept on during the gate.

Figure 4

Left: optimal Rabi frequencies without (a) and with (b) spontaneous emission from $|i\rangle$. Black dotted line represents the initial “guess” pulse form. Right: corresponding dynamics of the two-qubit basis states in the complex plane.