The Adiabatic Theorem - How to derive Schrödinger equation in the "s" variable

I'm studying Adiabatic Quantum computing from the book
"Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice" by Catherine C. McGeoch at D-Wave.
The section THE ADIABATIC THEOREM in the book is based on the paper
"How Powerful is Adiabatic Quantum Computation? (Wim van Dam et al., 2001)" - section 2.

In both sources above, Schrödinger equation in variable $$t$$ is introduced first: $$i \hbar \frac{d }{dt}|\Phi_t\rangle = \mathcal{H}(t) |\Phi_t\rangle.$$

Then, in the book the authors said this before arriving at Schrödinger equation in $$s$$. $$\frac{d }{ds}|\Phi_s\rangle = -i \tau(s) \widetilde{\mathcal{H}}(s) |\Phi_s\rangle.$$

I have difficulty understanding this derivation. Honestly, I don't even know the formula of $$\tau(s)$$. The only clue about it is the sentence

Let $$\tau(s)$$ determine the rate at which the Hamiltonian changes as a function of $$s$$,

which I'm not sure what it means. I've also looked in the paper (section 2) and just found that this $$\tau$$ got a name.

a delay factor $$\tau(s)$$

But no extra information can be found there.

Can someone please help explain what $$\tau$$ is and/or, if possible, explain to me about the whole derivation of the second Schrödinger equation (in variable $$s$$) ?

• +1. Welcome to the site and we hope to see much more of you in the future! Thank you for contributing your question here ;) Jul 16 '21 at 20:12
• @user1271772 Thank you for your warm welcome. Jul 17 '21 at 15:44

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 $$\frac{d\Delta}{ds}\ll\epsilon,$$ for some small $$\epsilon>0$$ (if you look up the formal statements, there's a few more conditions, but this will do for this explanation), the error in the final evolution can be bounded by $$\delta(\epsilon)$$. Now, let's say that you change your Hamiltonian according to the function $$\tau(s)$$. Then $$\frac{d\Delta}{ds}=\frac{d\Delta}{d\tau}\frac{d\tau}{ds}.$$ So, the idea is that you can try and bound $$\frac{d\Delta}{d\tau}$$ by some properties you know about the Hamiltonian. This means that you can suggest a functional form for $$\frac{d\tau}{ds}$$, and integrate to find an $$\tau(s)$$ that guarantees you have sufficient accuracy.

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 other variable left is τ, that τ in combination of the function s will result as our known τ(s).

Putting everything together let us know that τ(s) is the rate how a Hamiltonian changes depending on it's way [in time] to reach the other state (1 -> 0). Here the mentioned "rate" arises between way and time s(t) consolidate in τ to τ(s).

I hope I could answer your question and give you a better understanding of that missing piece of a larg and growing puzzle. Cheers!

• Why do you say there is a quantum phase transition here? Jul 17 '21 at 20:45
• You are right this is misleading. Searched a name for that. I changed it. Thanks for your hint! Jul 17 '21 at 20:47
• Thanks for your answer. I try to read your explanation in English but still do not understand it. Is it possible that you give some example of s(t) and τ(s), or write some math formula to make things precise? Jul 18 '21 at 16:52
• note that you can include math directly in the posts. See e.g. quantumcomputing.meta.stackexchange.com/q/49/55
– glS
Jul 18 '21 at 18:16
• Oh wow, thanks :) Take makes a lot easier! Jul 18 '21 at 18:30