Figure 6

Simulation results of evaluating the trace of the Hessian matrix for the circuit shown in Fig. 2(b) with random assignments of the parameters and \(O = Z^{\otimes 4}\). The simulations are performed by sampling 2000 random parameter vectors \(\{\boldsymbol{\theta}_{m}\}_{m=1}^{2000}\) with \(\theta _{i} \sim \operatorname{Unif}[0,2\pi [\) and then evaluating the trace of the corresponding Hessian matrix \(\operatorname{Tr} [H(\boldsymbol{\theta}_{m})]\). These values are used to build the histogram showing the frequency distribution of \(\operatorname{Tr} [H]\). The length of the arrows are, respectively: “Numerical 2σ” (black solid line) twice the numerical standard deviation, “Approximation” (dashed red) twice the square root of the approximation in Eq. (31), “Bound” (dashed-dotted green) twice the square root of the upper bound in Eq. (30)