# What are the units of time in the Quantum Adiabatic Theorem?

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.

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

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.