Paper Writing

[LucaYoy]

@as2457 Hi Adam, im trying to prove the claim of the polar optimisation that the overlap always increases whenever we add a new gate however im stuck and im not sure how to start. Do you have a hint of what I can try until our next meeting?

I have tried to write down what it means to contract the indices when we add the old gate and new gate back in and then try to compare the two expression in hope that I see that the new one is bigger; however im kind of stuck.

[AdamSmith-physics]

@LucaYoy OK, let me give you a few hints and if you can't get it then we can go through it together.

There are a few steps.

• First we want to show that minimising the (Frobenius) distance between U and E is equivalent to maximizing the Frobenius inner product between U and E.
• Next you want to plug in the value for U using SVD from E and show this equals Tr(S), where S is the singular values.
• Then you want to work out what the trace is for a general unitary U. Now you can use the Cauchy Schwarz inequality and the properties of unitary matrices to show this is less than or equal to Tr(S)
• Actually, from Cauchy Schwarz you will additionally be able to show that Tr(U^dag E) = Tr(S) if and only if U is of the form we said. And so this unitary is optimal and unique!

[AdamSmith-physics]

Actually its really only the triangle inequality that you need. Which is a consequence of the Cauchy-Schwarz

[LucaYoy]

@as2457, @NicholasSynesi Here are my correlation function plots for reference:

Entropy: entropyPlots14_110.pdf entropyPlots14_111.pdf entropyPlots14_11.50.pdf

I: matrixCmPlots14_11.50.pdf matrixCmPlots14_110.pdf matrixCmPlots14_111.pdf

J: JPlots14_11.50log=True.pdf JPlots14_110log=True.pdf JPlots14_111log=True.pdf

Energy: energyPlot14_11.50.pdf energyPlot14_110.pdf energyPlot14_111.pdf

[AdamSmith-physics]

This looks much better! Make sure you update the figures

[NicholasSynesi]

@as2457 Hi, so as it turns out, most of the plots we went through on Thursday were incorrect! For 3 and 5 layers we were seeing some wild behaviour and this was just an issue on my end and all plots are now fixed and are about what we would expect. Just one question about the physics now, how do we explain this behaviour as seen on our von Neumann entropy plots? Where only for model 1, we see the jumpy behaviour for 3 layers but not on the other two models. The other Renyi entropy plots have these questions also.

[LucaYoy]

@as2457 I have written the proof in the appendix, It is not exactly as the one you wrote but uses the same ideas just written in a way to match the discussions in our paper. Could you give it a quick read and tell me if it makes sense? Thank you

[AdamSmith-physics]

@NicholasSynesi I would attribute the zig-zag pattern the alternating pattern of the brickwall. For few layer the brickwall is essentially trying to increase the entropy up to the required value. With more layers it is then able to refine the spectrum and the shape in these plots. This matches why the models with the most entanglement have more of this zig-zagging. For model 3, I think this is maybe an interference pattern due to the limited size of the system and the added interactions.

[AdamSmith-physics]

@LucaYoy I had a very very quick look and it looks good.

[AdamSmith-physics]

@NicholasSynesi @LucaYoy Please make sure you add the latest results, use chapters, and address my comments before submission. Also make sure you update all of the discussion that related to the previous incorrect data!

Often you also seem to be missing the point. The goal is not only to "successfully" approximate the states. It is also to understand what information we are missing for shallower circuits, and what information it is still able to capture well.

[LucaYoy]

@as2457 Hi Adam, could you please clarify the point about implementation on a quantum computer of the optimisation.

is it that instead of ED we do VQE on the QC and then we use our same algorithm but now the target state is the VQE state

or is it that we alter our algorithm and try to make it suitable for the QC maybe by changing it to a more VQE orientation so that it can run on the actual machines but still use ED as the method of "comparing against"?