Extended Data Fig. 1: Gate optimization procedure and variational Hamiltonian Engineering.
From: Programmable simulations of molecules and materials with reconfigurable quantum processors
(a) To find smooth gates we perform the GrAPE optimization procedure in two steps. In the first step, we penalize rapid changes in the pulse profile by introducing an extra term in the cost-function. In this case, the resulting relationship between noise-free gate-error vs. time (yellow circles) saturates around 10−6. For the second step we initialize the search with the smooth gates found in the previous step, which are re-optimized by removing the smooth penalty. This substantially reduces the noise-free gate error (orange). The data are shown for a N = 4 qubit cluster. Since the first step already confined the problem into a subspace of the search space with smooth gates, the resultant pulses also remain smooth after the second step. On the right we show an example smoothened pulse profile for the hyperfine angle θ and the Rydberg phase ϕ (see Methods) with an noise-free gate error rate \({\mathcal{E}}=1{0}^{-3}\). (b) Higher order interaction terms can be controllably engineered via a simple modification of the K = 6 Floquet projection sequence. For example, reducing the second time-step and increasing the fourth time-step by the same amount, δ preserves the target Hamiltonian at leading order. (c) By tuning τ and δ, for a system of two interacting spin-3/2’s, a very large family of coefficients J1, J2, J3 in the general Hamiltonian \({H}_{T}={J}_{1}({{\hat{\bf{S}}}}_{1}\cdot {{\hat{\bf{S}}}}_{2})+{J}_{2}{({{\hat{\bf{S}}}}_{1}\cdot {{\hat{\bf{S}}}}_{2})}^{2}+{J}_{3}{({{\hat{\bf{S}}}}_{1}\cdot {{\hat{\bf{S}}}}_{2})}^{3}\) can be engineered. In particular, the roughly horizontal gray lines correspond to constant δ grid lines, and roughly vertical gray lines correspond to values with constant τ. We see that changing δ primarily modifies J2 and J3, while changing τ primarily modifies J1, consistent with our analysis that δ picks out certain higher-order terms. We further note that the especially interesting AKLT family, with J2/J1 = 116/243 and J3/J1 = 16/243 lies among the family of efficiently engineerable interactions.
