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A: Subtree reconstruction algorithm
Here, inference of the classical statistical complexity is achieved through the subtree reconstruction algorithm [7]. It works by explicitly building an machine of a stochastic process, from which may readily be deduced. The steps are detailed below.
1. Constructing a tree structure. The subtree construction begins by drawing a blank node to signify the start of the process with outputs . A moving window of size is chosen to parse through the process. Starting from the blank node, successive nodes are created with a directed link for every in each moving window . For any sequence starting from within whose path can be traced with existing directed links and nodes, no new links and nodes are added. New nodes with directed links are added only when the does not have an existing path. This is illustrated in Fig. S1
For example, suppose , giving rise to six nodes that branch outwards in serial from the initial blank node. If , the first five nodes gain no new branches, while the sixth node gains a new branch connecting to a new node with a directed link. Each different element of has its individual set of directed links and nodes, allowing a maximum of branches that originate from the blank node.
2. Assigning probabilities. The probability for each branch from the first node to occur can be determined by the ratio of the number of occurences the associated strings to the total number of strings. Correspondingly, this allows each link to be denoted with an output with its respective transition probability .
3. Subtree comparison. Next, starting from the initial node, the tree structure of outputs is compared against all other nodes. Working through all reachable nodes from the initial node, any nodes with identical and branch structure of size are given the same label. Because of finite data and finite , a test is used to account for statistical artefacts. The test will merge nodes that have similarenough tree structures. This step essentially enforces the causal equivalence relation on the nodes.
4. Constructing the machine.
It is now possible to analyse each individuallylabelled node with their single output and transition probability to the next node. An edgeemitting hidden Markov model of the process can then be drawn up. This edgeemitting hidden Markov model represents the (inferred)
machine of the process.5. Computing the statistical complexity. The hidden Markov model associated with the machine has a transition matrix giving the probability of the next output being given we are in causal state , and
being the causal state of the updated past. The steadystate of this (i.e., the eigenvector
satisfying ) gives the steadystate probabilities of the causal states. Taking , we then have the Shannon entropy of this distribution gives the statistical complexity:(S1) 
B: Quantum Models
Quantum models are based on having a set of nonorthogonal memory states in onetoone correspondence with the causal states . These quantum memory states are constructed to satisfy
(S2) 
for some suitable unitary operator [4, 30]. Here, is the probability of output given the past is in causal state , and is a deterministic update function that updates the memory state to that corresponding to the causal state of the updated past. Sequential application of then replicates the desired statistics (see Fig. S2).
Then, is the steadystate of the quantum model’s memory, and the quantum statistical memory is then given by the the von Neumann entropy of this state:
(S3) 
C: Quantum Inference Protocol
A quantum model can be systematically constructed from the machine of a process, and so a quantum model can be inferred from data by first inferring the machine. However, the quantum model will then inherit errors associated with the classical inference method, such as erroneous pairing/separation of pasts into causal states (due to e.g., the test in subtree reconstruction). For this reason, a quantumspecific inference protocol was recently developed [23] that bypasses the need to first construct a machine, thus circumventing some of these errors. Moreover, it offers a means to infer the quantum statistical memory of a quantum model without explicitly constructing said model.
It functions by scanning the through the stochastic process in moving windows of size , in order to estimate the probabilities , from which the marginal and conditional distributions and can be determined. From these, we construct a set of inferred quantum memory states , satisfying
(S4) 
for some suitable unitary operator . When is greater than or equal to the Markov order of the process, and the probabilities used are exact, this recovers the same quantum memory states to the exact quantum model Eq. (S2), where the quantum memory states associated to two different pasts are identical iff the pasts belong to the same causal state. Otherwise, if is sufficiently long to provide a ‘good enough’ proxy for the Markov order, and the data stream is long enough for accurate estimation of the length sequence probabilities, then the quantum model will still be a strong approximation with a similar memory cost. From the steadystate of these inferred quantum memory states, the quantum statistical memory can be inferred [23].
However, the explicit quantum model need not be constructed as part of the inference of the quantum statistical memory. The spectrum of the quantum model steadystate is identical to that of its Gram matrix [24]. For the inferred quantum model, this Gram matrix is given by
(S5) 
The associated conditional probabilities can either be estimated from compiling the using as a proxy for the Markov order, or directly by frequency counting of strings of length of in the data stream. Then, the quantum inference protocol yields an estimated quantum statistical memory :
(S6) 
D: Methodology (Extended)
In this work, we study finitewidth analogues of ECA. To avoid boundary effects from the edges of the ECA, to obtain a ECA state of width for up to timesteps we generate an extended ECA of width with periodic boundary conditions and keep only the centremost cells; this is equivalent to generating a width ECA with open boundaries (see Fig. S3). Note however that the choice of boundary condition showed little quantitative effect upon our results.
The state of an ECA can be interpreted as a stochastic pattern. That is, given an ECA at time in state , we can interpret this as a finite string of outcomes from a stochastic process with the same alphabet. We can then apply the tools of computational mechanics to this finite string, inferring the classical statistical complexity through the subtree reconstruction algorithm, and the quantum statistical memory from the quantum inference protocol. For both inference methods we use , as little qualitative difference was found using larger (see Fig. S6). For the subtree reconstruction we set the tolerance of the test to 0.05.
We apply the inference methods to ECA states of width . For each ECA rule we generate an initial state for where each cell is randomly assigned or with equal probability, and then evolve for steps (in Fig. S6 we analyse select rules for up to , finding that the qualitative features of interest are already captured at the shorter number of timesteps). Note that this is many, many orders of magnitude smaller than the time for which a typical finitewidth ECA is guaranteed to cycle through alreadyvisited states () [17]. We then apply the inference methods to the states at
; evaluating at every timestep shows little qualitative difference beyond highlighting the short periodicity of some Class II rules. We repeat five times for each rule, and plot the mean and standard deviation of
and (see Fig. S4 and Fig. S5).E: Longer , larger
Here [Fig. S6] we present plots supporting that our choice of , appear to be sufficiently large to capture the qualitative features of interest, showing little difference when they are extended.
The exception to this is Rule 110, which appears to plateau at longer times. We believe this to be attributable to the finite width of the ECA studied – as there are a finite number of gliders generated by the initial configuration, over time as the gliders annihilate there will be fewer of them to interact and propagate further correlations. This is illustrated in Fig. S7.
I F: Rule 18 and kinks
Within the dynamics of Rule 18 (Wolfram Class III), a phenomenon referred to as ‘kinks’ has been discovered [16]. These kinks are identified with the presence of two adjacent black cells in the ECA state, and have been shown capable of undergoing random walks [13], and annihilate when they meet. These kinks can be seen by applying a filter to the dynamics of Rule 18, replacing all cells by white, with the exception of adjacent pairs of black cells (see Fig. S8). The movement and interaction of kinks is reminiscent to that of a glider system like that of Rules 54 and 11; though, while information may be encoded into these kinks, they are noisy due to their seemingly random motion.
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