Fig. 3: Numerical results for short-time second-order product-formula simulations.
a Number of exponentials needed to achieve simulation precision ϵ = 10−3 at t = 0.1 under an MFI Hamiltonian with J = 1, h = 0.5, g = 1.2 as in Eq. (11). The gate counts for local (green) and global (yellow) obs (short for observables), Z1 and \(\frac{1}{n-1}\mathop{\sum }_{j = 1}^{n-1}{Z}_{j}{Z}_{j+1}\) are obtained from different theoretical and empirical error estimations. b Simulation of dynamical quantum phase transition with k = 3 local observable as in Eq. (12). We use a 12-qubit TFI Hamiltonian with J = 0.2 and h = 1 as in Eq. (13). By fixing the precision ϵ = 0.05 and a gate budget of 500, the guaranteed simulation times are t = 1.80 from Thm. 1 and \({t}^{{\prime} }=1.15\) from worst-case analysis in ref. 33, with step lengths δt = 0.12 and \(\delta {t}^{{\prime} }=0.08\). In the inset, we validate the local approximation of the rate function λn(t) by λk(t).
