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Quantum random walk in Wolfram Model

I was thinking, how a random walk in a 1D chain could be realized in a wolfram model (on a quantum level). I started making something, but it doesn't seem to work as expected.

Suppose, we have a string substitution system with 3 rules: ABA->AAB, ABA->ABA, ABA->BAA (on each step B can move left, right, or not move at all). The starting string is a sequence with one B in a row of As: AAAABAAAA (could be an infinite sequence as well). This model represents a random walk, where B can move to the left, to the right, or remain still.

From here I am starting to struggle with the basic formalism. According to Wolfram, the transition between two states is supposed to be expressed by a path integral. First question:

Is state AAAABAAAA at the time t=0 is the same state as AAAABAAAA at time t=10? In other words, does the "transition amplitude" depend on the time we consider the states at? (in quantum mechanics there is universal time, so all transition amplitudes are functions of time)

I initially thought that an answer to the questions must be "states at different update iterations are different states for the purpose of transition amplitude calculations" (after all, we can enforce each time step to be different by modifying the rules a little bit), but I am not sure anymore, because if it is true, the complex phases of all paths will be the same. The last statement follows from the length of all path leading from one point to another being the same, and Lagrangian density at each node of the causal graph also being the same (so all paths, leading from one point to the other have the same action). If phases for all paths are the same, the random walk will be classical, and not quantum.

Can someone help me to clarify the situation? Was my example wrong? What would be the right way to do a quantum random walk?
POSTED BY: Pavlo Bulanchuk
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Just to point out that Jonathan's bulletin A Short Note on the Double-Slit Experiment and Other Quantum Interference Effects in the Wolfram Model is highly speculative and not everybody agrees with his approach. Here is another approach to quantum interference in the framework of the Wolfram Model.
POSTED BY: José Manuel Rodríguez Caballero

Thank you for the second reference. It was helpful, but unfortunately it left me disappointed. The first two examples in the reference are just regular constructions of the double slit experiment in Lagrangian and Hamiltonian formalisms and have nothing to do with Wolfram model. The third example is a Wolfram model, but it is more "construction of two negatively interfering paths", rather than "a double slit experiment". Each transition in the model was constructed by hand, making it not scalable (and I am still not sure how exactly we assign phases after the Knuth–Bendix completion).
POSTED BY: Pavlo Bulanchuk

I am very curious about different path phases as well. The way it was presented is extremely confusing. Clearly, Gorard's model in the first part did not use any destructive interference to calculate weights of final states. Moreover, as you mention, if we apply Gorard's instructions for the phase to my example, phase difference between any two paths is π, which is logically impossible.
POSTED BY: Pavlo Bulanchuk

I think you could find some insight in this bulletin by Gorard. There, he analyzes a system quite similar to yours, namely the rule {X0->0X, 0X->X0} starting from a string like 00000X00000, which is essentially your random walk, but forced to move right or left each step. In the bulletin, to each state it is assigned a real number, that is proportional to the number of different paths that can lead to the state. In this way Gorard is able to show the effect of diffraction, so it doesn't seem that this kind of system can exhibit interference patterns (I think that when you refer to a quantum random walk, you mean that you would like to see an interference pattern emerge).

So how do we get interference? Gorard's answer is to start from a different initial condition: if you start with something like 000000X000X00000, with two Xs, you get a peculiar interference pattern.

In the second part of the paper Gorard explains how all this is connected with his paper on quantum mechanics. I couldn't understand many parts of it, but I'd like to discuss it with you.

In particular, I would like to quote the following, which I think could be helpful in understanding how to calculate the phase of the action along a path:

So how does a completion procedure achieve the destructive interference effects that are so crucial for reproducing the results of the double-slit experiment just shown? As I described in my quantum mechanics paper, the basic idea is to think of the phase difference between two paths in the multiway system as corresponding to the ratio of branchlike- to spacelike-separated events along those paths (or, strictly speaking, to twice the ArcTan of that ratio).
POSTED BY: Ruggero Valli

I realize that this quote may raise more doubts than it solves. In particular, how does one calculate the ratio of branchlike- to spacelike-separated events along two paths?

Do I have to compare every event in one path with every event in the other path, or do I first define a foliation of the multiway causal graph and then compare only events on the same slice? What should I do if two events are both spacelike and branchlike separated? This last question worries me particularly because Gorard writes:

spacelike-separation is, in general, a special case of branchlike-separation, since the application of a pair of purely spacelike-separated updating events will usually yield a branch pair in the multiway graph

Does this mean that every spacelike-separated event is also branchlike-separated?

More importantly, I don't see how to apply this reasoning to the system in your question. The rule ABA->AAB, ABA->ABA, ABA->BAA with initial condition AAAABAAAA can only generate pure branchlike separations. With this I mean that there is no way to do two updates "simultaneously", because all the rules apply only to the part with the B, the right hand side of the rules always "overlap". If no event is spacelike-separated, the ratio becomes infinite, and the phase difference between any two given paths is always $\pi$. This conclusion, apart from giving the wrong physical result, is also logically impossible.

I am sorry, I know I should answer your question, not pose other questions. My hope is that someone will eventually come and clarify all these doubts that we are exposing in these days.
POSTED BY: Ruggero Valli
POSTED BY: Pavlo Bulanchuk
POSTED BY: José Manuel Rodríguez Caballero
POSTED BY: Pavlo Bulanchuk

I agree with José, Wolfram model is 100% deterministic (I don't get why you say that it behaves like a probabilistic cellular automaton). A deterministic theory could still satisfy Bell's inequalities in many ways for example by being nonlocal or superdeterministic. The Wolfram model is definitely superdeterministic, I am not sure if it is also nonlocal. Therefore it is in principle able to satisfy the Bell's inequalities. The real question is whether it actually does, because I have not yet seen a convincing proof of it, but I think that before a proof of Bell's inequalities, one would need a better understanding of the way quantum mechanics could be embedded in the model.
POSTED BY: Ruggero Valli
POSTED BY: Pavlo Bulanchuk
POSTED BY: Ruggero Valli
POSTED BY: Pavlo Bulanchuk
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