- E_{0}(\tau)$, is greater than zero for all $0 \le \tau \le 1$, and taking $$t_a \gg \frac{\varepsilon}{g_{min}^{2}}$$ with the minimum gap, $g_{min}^{2}$, defined by $$g_{min}= \min_{0 \le \tau \le 1} (E_{1}(\tau) - E_{0}(\tau)),$$ and $\varepsilon$ given by $$\varepsilon = \max_{0 \le \tau \le 1} \vert {\langle l=1;\tau|\frac{d\tilde{H}}{d\tau} \vert l=0;\tau\rangle} \mid,$$ then we can make $$\vert {\langle l=0;\tau=1|\psi(t_a)\rangle} \vert$$ arbitrarily close to 1. In other words, the existence of a nonzero gap guarantees that ${\mathinner{|{\psi (t)}\rangle}}$ remains very close to the ground state of $H(t)$ for all $0 \le t \le t_a$, if $t_a$ is sufficiently large.

The adiabatic condition presented above applies to zero-temperature noise-free (i.e., 100% quantum coherent) conditions, where the dynamics are well-described by $H(t)$ in Eq. \[h-AQO\] and the scaling of $t_a$ as a function of the problem size determines the complexity (and usefulness) of the quantum algorithm. In a realistic experimental implementation, the quantum processor will operate at a finite temperature, and in addition to thermal fluctuations, other types of noise are unavoidable, leading to dissipation processes not captured in $H(t)$; deviations from adiabaticity affecting the performance of the quantum algorithm seem to be a delicate balance between the quantum coherence effects and the interaction 