# Improving the Solving of Optimization Problems:
## 4 Quantum Annealer
### Quantum Annealing  
Quantum annealing is the practical implementation of the theoretical and ideal AQC. It is based on AQC principles, but to make the physical implementation feasible, the possible final Hamiltonian is limited to the Ising form, which is reformulated as a quantum Hamiltonian formulation:  
\[H_{1}= \bigg{(}-\sum_{ij}J_{ij}\sigma_{i}^{x}\sigma_{j}^{z}-h\sum_{i} \sigma_{i}^{z}\bigg{)}, \tag{9}\]  
where \(\sigma_{i}^{z}\) is the Pauli matrix associated with the \(z\) component, allowing the optimization of only Ising/QUBO problems. On the other hand, the initial Hamiltonian is usually the transverse field \(H_{0}=-\sum_{i}\sigma_{i}^{x}\), where \(\sigma_{i}^{x}\) is the Pauli matrix associated with the \(x\) component. Moreover, in quantum annealing, the adiabaticity is no longer guaranteed since the estimation of the time \(T_{A}\) is harder than the optimization problem to solve [45], and it could be impractical to run the system evolution for such a long time. Therefore, the quantum annealing runs for a certain time and, even if it does not satisfy the adiabaticity, it is expected to be able to find a good approximation of the optimal solution. In fact, it is not strictly needed to remain in the ground state of \(H(t)\). Ultimately, upon measuring the state, it is sufficient for the amplitude of an optimal or adequately good solution in the final state to be large enough. This ensures that the probability of obtaining a useful result remains high.  
To better understand the quantum annealing evolution, a qualitative idea can be given considering a hydraulic model in Figure 7b of [46]:  
* In the beginning, the transverse field makes the bottom of the tank flat; thus, the water is uniformly distributed. This corresponds to having all the spins aligned along the \(x\)-axis (Figure 7a of [46]), i.e., state superposition.
* By increasing the longitudinal field and decreasing the strength of the transverse one, the bottom of the tank is gradually deformed and the water begins to flow towards the lowest points. This corresponds to having the spins that gradually modify their orientation to follow the instantaneous ground state.
* In the end, only the longitudinal field is present, which has deformed the bottom of the tank as a function of the problem, and the water is concentrated in the lowest point, i.e., that with minimum energy. Therefore, the final spin configuration corresponds to the optimal solution of the optimization problem expressed as the Ising Hamiltonian.  
Quantum annealing can be seen as an improvement of the simulated annealing [17], in which thermal fluctuations are replaced with quantum fluctuations [11]. The transverse Ising model form, proposed in [11], is the basis of the Hamiltonian used in quantum annealing:  
\[H(t)=-\sum_{ij}J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}-h\sum_{i}\sigma_{i}^{z}-\Gamma( t)\sum_{i}\sigma_{i}^{x}\, \tag{10}\]  
where \(\Gamma(t)\) is the transverse field strength (equivalent of B(t) of the previous section).  
The transverse field \(\Gamma(t)\) causes quantum tunneling between system eigenstates. The advantage of quantum annealing exploration is that the probability of overcoming an energy barrier with height \(\Delta\) is proportional to \(e^{-\frac{\sqrt{\Delta w}}{\Delta}}\) (where \(w\) is the width of the energy barrier and \(\Gamma\) the strength of the transverse field). On the other hand, in the simulated annealing case, it is equal to \(e^{-\frac{\Delta}{\Delta}}\) (where \(T\) is the temperature parameter and \(k_{B}\) is the Boltzmann constant). Consequently, quantum exploration is significantly more effective than classical one in case of problems with the energy landscape with a high amount of perturbation with many high and thin barriers (\(w\ll\Delta\)) [47], as shown in Figure 7c of [46].