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To preface my question, I'm coming to adiabatic quantum computing from a background in classical computer science with little knowledge of quantum physics, so simple, step-by-step explanations or references to helpful literature on this subject would be greatly appreciated.

A folk theorem of adiabatic quantum computation states that the minimum time $T$ required to track the ground state of a time-dependent Hamiltonian $H(t)$ evolving from $t=0$ to $t=1$ is on the order of $\frac{\mathcal{E}}{\gamma^2}$, where $\mathcal{E}$ is usually the magnitude of an eigenvalue of $H(t)$ and $\gamma$ is the minimum spectral gap (i.e., the difference between the two smallest eigenvalues) of $H(t)$ for $t \in [0,1]$. My question is, what are the units of time for the quantity $T$? Is $T$ measured in seconds, is it a measurement of "computation steps" in the classical complexity theory sense, or is it somehow dimensionless? And if $T$ has no dimension, then how is it supposed to be understood as a quantity, and more practically, is there some way to convert it to more natural units of time?

I've checked the Physics Stack Exchange for similar questions, and the closest thing I could find was What are the units of time when planck's constant is equal to 1? I'm sorry to say that I couldn't quite follow the answer to that question, and in any event, I'm not sure if it addresses precisely this subject.

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  • $\begingroup$ As you formula shows, T has units of inverse energy! $\endgroup$ Aug 31, 2020 at 8:45

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Use dimensional analysis to restore the missing power of $\hbar$, remembering that it has the dimensions of energy times time. In normal units, $T$ is on the order of $\hbar\mathcal{E}/\gamma^2$. For example, you could measure the energies $\mathcal{E}$ and $\gamma$ in joules and the time in seconds if you like SI units.

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