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Quantum random walk in Wolfram Model

Posted 10 months ago
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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?

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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).

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.

The reference is actually great. It clarifies my questions about the way one assigns weights and treats different nodes (the question was from the other thread, but I didn't get a clear answer there). From what I can see, the evolution of the system ("oX"->"Xo", "oX"->"Xo") is essentially a probabilistic cellular automaton, where each possible substitution is assigned the same probability (or weight). This immediately rises a number of concerns about Gorard's results and Wolfram model in general:

  1. The first concern is very specific. The substitution system ("oX"->"Xo", "oX"->"Xo") resembles classical random walk very much. The only difference from a classical random walk is that a state "oooXXooo" can bounce only into the state "ooXooXoo", while in the classical random walk it can also bounce into "oooXXooo". This is a minor difference, and it cannot create "waves" in the probability distribution of the position of the letter X. The wavy interference pattern, that Gorard gets, is the result of the string sorting algorithm, which sets the order in which the strings are presented. This point was discussed in detail by Matt Kelly in the comments to the bulletin. To be honest, I am puzzled by Gorard being seemingly unaware about the interpretation of the rules as a modified random walk, then not being surprised after obtaining interference in such system, and then not addressing the issue of the ambiguity of the sorting algorithm, which is the source of the interference pattern.

  2. My second concern is about Wolfram model in general. If my understanding is correct, and a Wolfram model can be interpreted as a probabilistic cellular automaton, it makes it very difficult to make anything resembling quantum fields. The reason is the following: probabilistic cellular automatons are observationally equivalent to deterministic automatons with hidden variables. It is problematic to reproduce quantum mechanics by a local theory of hidden variables due to Bell's theorem (there are some loopholes in the theorem, but no one managed to actually use them, so... may be the loopholes are just an illusion). This result is very concerning to the prospects of a Wolfram model being able to model the real world.

I will try to answer this concern:

My second concern is about Wolfram model in general. If my understanding is correct, and a Wolfram model can be interpreted as a probabilistic cellular automaton, it makes it very difficult to make anything resembling quantum fields. The reason is the following: probabilistic cellular automatons are observationally equivalent to deterministic automatons with hidden variables. It is problematic to reproduce quantum mechanics by a local theory of hidden variables due to Bell's theorem (there are some loopholes in the theorem, but no one managed actually to use them, so... may be the loopholes are just an illusion). This result is very concerning to the prospects of a Wolfram model being able to model the real world.

The Wolfram Model is 100% deterministic. According to the Wolfram Model, the randomness observed in nature, including the outcomes of quantum measurements, is not true randomness, but pseudorandomness. Concerning the "compatibility" of the Wolfram model with Bell theorem, here is the answer of Jonathan concerning this question. In my case, I prefer to explore the option of superdeterminism as it was developed by Gerard 't Hooft, e.g., in this preprint. I agree that the use of superdeterminism is not mainstream physics yet, but it is not excluded in principle.

Thank you for clarification. I am not sure it resolves my concerns though.

The Wolfram Model is 100% deterministic.

Every probabilistic model is 100% deterministic with respect to the probability distribution function (including probabilistic cellular automatons). The question one should be asking is "can evolution of the weights of the states in a Wolfram model be represented by the evolution of probabilities in a probabilistic cellular automaton?". And if the answer "Yes", then WM will have big problems modeling any interacting quantum fields.

Now concerning Gorard's response

By allowing for the existence of causal connections not only between updating events on the same branch of evolutionary history, but also between updating events on distinct branches of evolution history

Is he talking about Knuth–Bendix completion algorithm here, or the regular update rules? Because the update rules in a Wolfram model are fairly local, by which I mean that there exists a space in which they are local.

Concerning superdeterminism:

In my case, I prefer to explore the option of superdeterminism as it was developed by Gerard 't Hooft

Superdeterminism is interesting, but not particularly useful for developing working theories. In specific cases, the results of Bell's theorem can be reformulated without the requirement of the free will, but with an additional equality imposed onto the three-variable correlation function (see chapter 3.6.1. The mouse dropping function in Hooft's book, Eq. 3.26). The specific form of the correlation function is a necessary and sufficient condition for a local deterministic classical theory to satisfy Bell's theorem (it follows from the way Eq. 3.26 was derived). Unfortunately, The requirement is by itself is extremely constraining, because any noise in the model usually disrupts correlations between variables over time (assuming the variables are associated with distant non-communicating structures). I am not saying it is impossible to create a theory which would satisfy a particular form of the three variable correlation equation, but it is extremely hard (no one succeeded in making one so far, and I doubt that anyone ever will).

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.

I say the Wolfram model behaves like a probabilistic cellular automaton, because the evolution of the distribution of weights over different states in the WM behaved exactly like the distribution of probabilities over different states in the equivalent probabilistic cellular automaton, namely, a 1D automaton, where there are two rules ("oX"->"Xo" and "Xo"->"oX"), with each having 50% probability of being applied at each time step. In other words, each possible evolution path in WM is assigned the same weight, and then the weights are added together. This is precisely one of the ways how you could do evolution of probabilities in the corresponding probabilistic cellular automaton, or any other randomly evolving system.

The important difference of quantum mechanical systems from the diffusion systems is that different paths in quantum mechanics add up with different complex phases, allowing interference effects. The problem with Gorard's example was that all paths had the same phase, which negated all possible quantum effects

Thank you for the explanation, I now understand why you were talking about a probabilistic cellular automaton. You are absolutely right when you say that if all the paths are summed with the same phase the only thing you can obtain is diffusion and not quantum mechanics. However, I suspect that the random walk we were discussing is only a special case, where by incident or construction every path has the same phase.

The quote where Gorard mentions the ArcTan seems to imply that in general every path will have a different phase, therefore allowing quantum interference. It would be nice if someone could explain how to calculate in practice these phase difference with a real example.

I found this lecture by Gorard: https://youtu.be/NupBxcmAwYs?t=4211

At 1:10:10, he states that he counts the number of paths to calculate the amplitude: "And then the natural measure that we define on evolution graphs in terms of the path weights, just purely in terms of paths counting, then gives you a way of defining the magnitude for the amplitude, which is the elementary weighting of the elements in the superposition".

And then at 1:12:05 he reconfirms that he sums up the number of paths leading from one vertex to another.

There doesn't seem to be any indication that there may be cases when one would do anything different, like sum paths weighed by different complex phases.

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.

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.

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).

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