Consider $Q=H$. Let $|\lambda_n\rangle$ be the eigenbasis of $H$, i.e.
Now consider the $|x\rangle$ that minimises $\langle x|H|x\rangle$. Since the eigenbasis is an orthonormal basis, we can choose to write
where it is now ...
The idea is that you want to use the adiabatic theorem. This states (roughly) that if your Hamiltonian $H(s)$ has an energy gap $\Delta(s)$ between the ground and excited states, then provided
for some small $\epsilon>0$ (if you look up the formal statements, there's a few more conditions, but this will do for this ...
What is τ(s) and s in Schrödingers equation
Let me break it down for you:
We know s(t) is the "way" in time to change from state 1 to 0.
That of course happens for different cases faster or less fast. So you get different "long" ways to achieve that change of states.
That turns us into τ(s). So the above explanation is our s in τ(s).
The article of Devoret and Schoelkopf  and an update provided in Section 7.1 of Reagor  makes a comparison between Moore's law and an observed trend of exponentially improving $T_1$ and $T_2$ times for superconducting qubits. The trend they present shows a roughly exponential improvement from $10^0$ to $10^6$ nanoseconds for $T_2$ between various ...