# Improving the Solving of Optimization Problems:
## 5 Quantum Circuit Model-Based Solvers
### Vqe
Many ansatz structures are suitable for this goal, and the quality of the approximation also depends on this choice. Therefore, the structure employed can be considered as a hyperparameter of the problem. The optimal choice typically depends on the problem characteristics. However, some features of an ansatz can be evaluated to identify the best fit for the problem at hand [63]:  
Figure 3: VQE example. * Expressibility: It is the span of possible quantum states that the ansatz can reach in the Hilbert space, or how well it can explore the space of the possible states. It can be quantitatively assessed by the distance between the distributions of the unitary operators generated by the ansatz and the maximally expressive uniform distribution of operators (Haar measure) [64]. * Trainability: It is the ability to find the best parameter values to optimize the ansatz according to the Hamiltonian expectation value within a feasible time frame. An ansatz is not considered trainable when the cost function gradients vanish exponentially as a function of the optimization parameters, which is referred to as the barren plateau problem. The trainability depends on the number of parameters, their mutual dependence, and the optimization landscape. This concept is related to expressibility, as a good ansatz should accurately approximate the quantum state of the best expectation value but an overly accurate one could make the training of its parameters intractable. * Circuit depth: The depth of a quantum circuit can influence the final estimation due to noise and decoherence problems; hence, it should be carefully managed in NISQ hardware applications. A further classification of ansatz can be based on whether their structure changes during optimization:
* Fixed structures remain unchanged during optimization, except for their parameters. * Adaptive structures evolve during the optimization process by incorporating additional operators. Some gradually grow at each optimization step, others embed part of the ansatz into the optimization alongside the Hamiltonian, while some modify the existing structure to learn an optimal configuration. For both the ansatz categories, ref. [63] provides various examples highlighting their structures and discussing pros and cons. The classical optimization of the quantum circuit parameters must be sufficiently fast to maintain the effectiveness of quantum optimization, while still providing enough accuracy to converge to the Hamiltonian ground state. The training of the ansatz parameters is NP-hard [65]. However, several heuristic methods, especially those based on gradient calculation, are employed for this purpose. An overview of classical optimization approaches suitable for VQE is presented in [63]. The adaptive nature of VQE, whose parameters are optimized within the different iterations of the algorithm, allows balancing the systematic errors caused by the imperfection of control systems [66]. This makes VQE particularly well suited for NISQ hardware, as it can mitigate the noise problems commonly affecting these systems, whereas algorithms relying on error correction may provide more accurate results in future FTQCs.