# Tailored Quantum Alternating Operator Ansatzes for Circuit Fault Diagnostics
## 6 Performance with Different Parameter Optimizations  
Using the family of CFD instances from the previous section, we consider the performance of QAOA strategies with different methods of parameter optimization: BRUTE, INTERP, LINANOOPT, and LINCOEFOPT. Each of the first three requires the expected cost of the final wave function to update the angle parameters. For implementation on a quantum computer, the expected cost of the final wave function has to be sampled with repeated measurements, and there will be a trade-off between repeated runs to accurately compute this value and run the protocol with updated parameters based on the approximation [6]. The angles were selected from the interval \([-\pi,\pi]\). For BRUTE, we chose 100 random seeds and ran the Nelder-Mead algorithm with 100 iterations to optimize each choice, using the final Hamiltonian as the cost function and selecting the choice that minimized the cost function the most. For INTERP, we used the interpolation function called INTERP in [6], which showed good performance from this method in their results. We used the Nelder-Mead algorithm (with 100 random seeds and 100 iterations each) on \(p=2\) and then for \(p>2\), beginning with the linear interpolation from \(p-1\) and optimizing with 100 iterations for this result. For LINANOOPT, we started the protocol with a linear ramp for each angle (i.e., for round \(k\in[1,p]\), \(\alpha_{k}=k\,\pi/p\) and \(\beta_{k}=\pi-k\,\pi/p\)) and then optimized the angles (with 200 iterations). For LINCOEFOPT, we used a coefficient \(\Gamma\in\{9/10,1/2,1/4,1/8,1/16\}\) and used the same linear ramp but replaced \(\alpha_{k},\beta_{k}\) with \(\alpha_{k}\Gamma,\beta_{k}\Gamma\) and then optimized \(\Gamma\) over 20 iterations before selecting the best performing \(\Gamma\). In each situation described, Approach 3 (with an initial state cost function) is different from the other approaches, since it has three angles to select for a single round \(k\). As the initial starting point for LINANOOPT and LINCOEFOPT (with \(\Gamma\)), it begins with \(\gamma_{k}=\pi-k\,\pi/p\), \(\alpha_{k}=\Theta(k\leq p)(2\,k\,\pi/p)+\Theta(k>p)(\pi-2\,k\,\pi/p)\) (reaching a maximum around \(p/2\) and minimums at 0 and \(p\)), and \(\beta_{k}=k\,\pi/p\). Note that Approach 3 was also considered without an initial state cost function, and the results support that this seemed to benefit its performance. The results from [6] suggest that the Nelder-Mead algorithm can perform as well as other optimization algorithms, such as the Broyden-Fletcher-Goldfarb-Shanno algorithm. Utilizing other optimizers for these tailored ansatzes is an interesting area of future study [21; 32; 33]. We report the results with \(p\) set to 2 and 3 for BRUTE, 5 for INTERP, 5 and 10 for LINANGOPT, and 50 for LINCOEFOPT. The relative simulation range capable on a workstation grows depending on the size of the relevant constrained space for the problem and so the more constrained approaches have an increased simulation range, as such we report results on larger instances for these approaches for each parameter selection. Figure 4 details the performance of each ansatz with each parameter selection strategy. The results support that ansatzes which exploit a more constrained space by tailoring the operators used to the problem structure generally perform better in the CFD problem and therefore are important methods for improving the practical applicability of QAOAs and other variational quantum algorithms. Indeed, Approaches 4 and 5 consistently performed the best, and Approach 3 performed comparably with some parameter selections for certain sizes. Surprisingly though, Approach 1 performed quite strongly for LINCOEFOPT with \(p=50\). The performance across these different metrics generally supports that while a linear tramp function might be well suited for certain approaches, such as Approaches 1 and 4, it is interesting to consider different functions to parameterize the angles for other approaches.