quantum system initialized in the ground state of a time-dependent Hamiltonian remains in the instantaneous ground state, as long as it is driven sufficiently slowly. Since the ground state of $H_p$ encodes the solution to the optimization problem, the idea behind QA is to adiabatically prepare this ground state by initializing the quantum system in the easy-to-prepare ground state of $H_b$, which corresponds to a superposition of all $2^N$ states of the computational basis. The system is driven slowly to the problem Hamiltonian, $H(\tau=1) \approx H_p$.

![[]{data-label="fig:chimera"}](./Fig1_working_qubits_v2.pdf){width="45.00000%"}

Determining the optimum value of $t_a$ is an important and non-trivial problem in itself. In principle, the adiabatic theorem states that over sufficient adiabatic time $t_a$, the state ${\mathinner{|{\psi(t_a)}\rangle}}$ will converge to the solution of the problem ${\mathinner{|{\psi_{g} (t_a)}\rangle}}$. Notice that the parameter $t_a$ determines the rate at which $H(t)$ varies. Following the notation from Farhi [*et al*]{} [@farhi2000], consider $H(t)=
\tilde{H} (t/ t_a) = \tilde{H}(\tau)$, with instantaneous values of $\tilde{H}(\tau)$ defined by $$\tilde{H}(\tau){\mathinner{|{l;\tau}\rangle}}=E_{l}(\tau){\mathinner{|{l;\tau}\rangle}}$$ with $$E_{0}(\tau) \le E_{1}(\tau) \le \cdots \le E_{M-1}(\tau)$$ where $M$ is the dimension of the Hilbert space, e.g., for the case of $N$ qubits $M=2^N$. According to the adiabatic theorem, if the gap between the two lowest levels, $E_{1}(\tau) 