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# Directing quantum motion: the art of time-reversal symmetry breaking

Hello Zoltan, Jacob and hello to everyone

Ref:

Directing Quantum Motion: the Art of Time-Reversal Symmetry Breaking

I coded the first part of this blog/paper:

Catwalk: Topology and Laplacian

First of all someone needs to add a definition for the weighted Dd, only weighted Ad was defined, so I used this paper's:

page 2-2, Dg second equation from top

I already use that def in the code and it seems working.

Used John's other blog:

Quantum Network Theory (Part 1)

I showed that there is no convergence for the infinite steps by an actual example, 20 million steps compared to 40 million and clearly both matrices and their element's norm did not match at all.

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Please if someone could check e^dt taylor series expansions, EQ 2 to 6.

What I am doing is that all the derivations are actually code that could be run by the readers.

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John

I really really really like this stuff! specially if mixed with Petri nets.

I was wondering, and if it is not imposing on your busy schedule, sometime in future to give me some realistic and useful systems to code e.g. some particle theory thingy or any other such physical system.

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I will try to add the BACKGROUND MATERIAL , specially the bra-ket computations.

Dara

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For mathematicians and physicists the presentation of the computations might be redundant i.e. why same lines of code was repeated all over the file.

There is a wisdom: for someone out of the field it is impossible to have a grasp of the concepts, the oft-repeated code allows the reader to investigate to tear it apart and run separate lines and look at the interim computations. If I turn everything into neat function calls the abstraction defeats the purpose of the whole endeavour.

This approach allows for strong intuition building by an outsider who is new to the ideas, so much as possibly guessing some theorems by looking at repeated properties via the code. This is more effective to do with running code than explanations in English.

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John how should I use the "Symmetric normalized Laplacian" that you have in your other blog's diagram for Zoltan's paper.

Dara

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edited September 2014

Hello Zoltan and Jacob, hello to everyone

Zoltan I tried to duplicate your computations in paper "Quantum Transport Enhancement by Time-Reversal Symmetry Breaking" i.e.EQ 1-3:

Catwalk Mechana

I am very close but cannot get the EQ 3 of your paper, somehow my computations at your EQ 2 cannot proceed to EQ 3 in your paper. I suspect the culprit is the |><| code.

Let me walk you thru the computations:

The programming is setup with Notation Package in Mathematica such that the algebraic expressions form your paper are almost the same in the code see EQ 4 for HCQW.

The underlying algebras are not assuming any specific algebra, so the mechanics (Mechana) of the algebraic evaluations are abstracted. See how I changed 'sites' array between different possible definitions of spin vectors.

CirclePlus or the Tensor summation are abstract i.e. have no computational semantic and they are non-commutative. I used them, best I understand from your paper and references, to indicate multiple directions of movement, possible for each site, which I stored in the array 'sites'.

Note the actual sum sigma symbol does summation of the underlying algebraic constructs. That could run through a very large sites array and I know how to parallelize on GPU and CPUs to evaluate large HCQW or use the Solvers. That being said, you do not need to change your notations, you just have to suffer the throes of explaining them to Dara :)

Dara

PS I am not emotionally impaired so you could say that is wrong, fix that do it this way, the idea is to learn from your research

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edited September 2014

Dara wrote:

John how should I use the “Symmetric normalized Laplacian” that you have in your other blog’s diagram for Zoltan’s paper.

I think it's really Jacob and Zoltan who might know good examples of what to do with this symmetric normalized Laplacian... I'll ask Jacob.

Here's a paper that mentions normalized Laplacians:

Here's a strange and interesting result. Take a connected graph $G$ that is not a complete graph (i.e., not every vertex is connected to every other by an edge). Let $\lambda_2$ be the second smallest eigenvalue of the symmetric normalized Laplacian. (Be careful about the definitions here: some people's "second smallest eigenvalue" would be some other people's "second largest eigenvalue", due to different sign conventions. Let $f$ be the corresponding eigenfunction. Let $S_\alpha$ be the set of vertices $v$ with

$$f(v) \le \alpha v$$ for some constant $\alpha \ge 0$. Then subgraph of $G$ having $S_\alpha$ as its set of vertices and all edge between these is connected!!!

In other words, the vertices where $f(v) \le \alpha v$ don't lie in two separate "chunks".

This is Lemma 2.1.

This is a somewhat surprising result, which might be fun to test numerically. However, it would be even better to find some practical use for studying the symmetric normalized Laplacian (or other graph Laplacians). For this, I think Daniel Estrada's book Network Theory might help - it's full of real-world examples. Unfortunately I don't have that book here with me in Singapore.

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Hi Dara. The thing I like about your approach is that you're using Mathematica. We've done all of our simulations in Matlab and python, neither of which really offer symbolic manipulations. Let me try to get a copy of mathematica so I can play around with your simulations.

The first thing to notice however, is that the spin-operators are not needed for this. So let's forget them for now. They just offer a means to embed the jump operators. So I'd skip that part for now, and just simulate a 3-site ring with the simplistic jumping operators. In this case, the Hamiltonian will just be a 3x3 matrix.

• could we start a new thread for this? How about we call it, 'experiments in quantum versus stochastic walks'?

And there I'd like to work with you to simulate the 3x3 Hamiltonian, and produce this 2D plot

from the paper, chiral quantum walks.

If you wanted to consider something called, 'charge conjugation symmetry', then you would need to consider spin. We didn't try that one yet since it's use is not as clear. We can try it later on perhaps. I'm also going to have a look at the paper John has posted and think about it.

By the way, now I'm much more interested in some of these questions as they relate to stochastic Petri nets. So I want to also talk to you about some of those ideas, after we get a few of these quantum walk simulations to work. I think we have a few deep questions in that area that touch on some of the problems people in the area of 'complex networks' have faced when considering reaction networks. Their approach does not work, and I think they need to rethink the entire thing with a 'quantum approach' to make it work. I can back up what I'm saying by providing references showing how their stuff does not really work, and I have a few ideas I'm working on to try to fix it.

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Thanx John , look at Estrada's book and the paper you had mentioned, try to work out a few examples, very curious.

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Hello Jacob

could we start a new thread for this?

As you like.

We’ve done all of our simulations in Matlab and python, neither of which really offer symbolic manipulations

That is why I used Mathematica, it is a horrible programming language but more effective for our uses here. On the symbolics side version 10 has a huge new step forward which makes it more functional than before. Also remember that these symbolics could be parallelized + mixed with numerics and graphics.

And there I’d like to work with you to simulate the 3x3 Hamiltonian, and produce this 2D plot

Allrighty I try to study that part of the CHRIAL QUANTUM WALKS paper.

If you wanted to consider something called, ’charge conjugation symmetry’, then you would need to consider spin.

Actually I am quite agnostic on these topics, I was mislead by my mind thinking you were using spin, but what the heck, as long as I could master the symbolics we could anything! However I have no preferences on the research side, focused on the coding + what is most educational worthy.

By the way, now I’m much more interested in some of these questions as they relate to stochastic Petri nets.

You have my attention!

Dara

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Hello Jacob

Was my Ket-Bra outerproduct correct? It seems that is what is used in the Chiral paper too

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what is h.c in (3) in Chiral Quantum Walks

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Hi Dara,

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Jacob in chiral paper EQ (3)

H= ...|3><1| + h.c.

What does "+ h.c." mean? To apply hermitian conjugate to the RHS?

D

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|3><1| + h.c. means |3><1| + |1><3|

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Ok I will redo the computations.

tij = -tji ? in other words one needs to flip the sign for time reversal?

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I would produce that 2D plot, first. It's just for a particle on a 3 node ring. If I had a copy of Mathematica I could get it going with you. But I don't, so the best I can do is try to help if you get stuck.

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Dear Jacob

Chiral Catwalk

Please review this, you see the interim computation in full + the plots.

I am still off, something is not the same as your paper, I suspect 'sites' are the wrong form and shape.

Dara

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Hello Jacob

If you are serious about Mathematica for programming your work with John, if you download a trial version of Mathematica 10 and see if you like it and can work with it, I might have an extra license lying around that you could use at the end of 30 days

Dara