Can quantum cellular automata simulate quantum continuous processes?

[Suekdccia]

Can quantum cellular automata/quantum game of life simulate quantum continuous processes in the continuous limit?

At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document

it is said that: "For example, several works simulate quantum field theoretical equations in the continuous limit of a QCA dynamics."

But a user on reddit told me that:

"the continuous limit is not the same as being continuous. The "continuous limit" is a trick of physics that goes back to Boltzmann's work on thermodynamics in the 19th century, and the people who work with QCA in the continuous limit use the same trick he came up with. Discrete dynamic models often present simpler math for some systems, than do the more accurate continuum dynamics models. So you find a way to get an answer to a question or prediction for an event using discrete dynamics, but have the quantized units' quanta appear in the solution. At this point you might think, "ok, just make the quanta 0 to represent a continuum, and problem solved." However, typically, if those values are zero, you get some kind of undefined answer like 0/0, 1infinity, or something else like that. So instead you do something familiar to anyone who's taken calculus - you take the limit of the solution as the quantized units approach zero. Boom, done. This is not actually a continuum simulation. It let's you predict what a continuum simulation would yield, if you could build one. It gives you the mathematical power to calculate the state of a system at any moment in continuous time. But you are not actually calculating the state of the system at every moment in time. Depending on the system, you might get an algebraic solution for the state of the system with respect to continuous time, in which case you do actually have the state of the system at every moment in time, but this is not guaranteed to occur - it may be a computational solution instead. And it's still not the same as having literally simulated the continuum."

I sent an email to the author and I have contacted with other physicists/mathematicians, and they told me that what this user said were sloppy words and nothing tangible/meaningful.

Then, was this user wrong? Is continuous limit the same as continuum? Can the cellular automata mentioned in the article really simulate (quantum) continuous processes/behaviour (with perfect accuracy, not only just to an arbitrary accuracy)?

 

[eljose79]

I cannot definitively say whether or not the user on Reddit was wrong. However, based on my understanding of the topic, it is important to distinguish between the continuous limit and the continuum.

The continuous limit is a mathematical technique used to approximate a continuous system with a discrete one. In this case, the discrete system is the quantum cellular automaton (QCA). The continuous limit allows for the simulation of continuous processes by taking the limit of the discrete system as the time or space intervals approach zero. This can provide useful insights and predictions for the behavior of a continuous system, but it is not the same as actually simulating the continuous system.

On the other hand, the continuum refers to a system that is truly continuous, without any discrete elements. It is a theoretical concept that cannot be fully realized in physical systems, but it can be approximated through the use of mathematical models and simulations.

Therefore, while the QCA in the continuous limit may provide valuable information about the behavior of a continuous system, it is not the same as simulating the continuum. It is important to be precise with language when discussing these concepts in order to avoid confusion and misinterpretation.