Tensor-Networks-based Learning of Probabilistic Cellular Automata Dynamics

TL;DR

This work develops a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and applies this algorithm to probabilistic cellular automata processes, finding that the ability to learn these dynamics is a function of the bitwise difference between the rules.

Abstract

Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle problems in the classical domain. In this work, we focus on matrix product operators, a prominent numerical technique to study many-body quantum systems, especially in one dimension. It has been previously shown that such a tool can be used for classification, learning of deterministic sequence-to-sequence processes and of generic quantum processes. We further develop a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and apply this algorithm to probabilistic cellular automata. This new approach can accurately learn probabilistic cellular automata processes in different conditions, even when the process is a probabilistic mixture of different chaotic rules. In addition, we find that the ability to learn these dynamics is a function of the bit-wise difference between the rules and whether one is much more likely than the other.

Figures (7)

• Figure 1: Illustration of a single realization of a probabilistic cellular automata evolution between two different rules: rule 18 with probability $p_{18}$ and rule 51 with probability $p_{51}=1-p_{18}$. The left panel presents probability $p_{18} = 0.2$ while the right panel presents $p_{18} = 0.8$. At each time step, the sequence $\pmb{x}$ is updated to different $\pmb{x}^\prime$ following the two probabilistic cellular automata rules.
• Figure 2: Illustration of the probabilistic learning protocol. In the left portion, an input state and more than one output state are contracted with the MPO object which is in turn fed into the minimization protocol. In the right portion, we demonstrate the prediction, contracting an input state with the, now, trained, MPO, which is afterward subject to a sampling algorithm to recover a possible output with correct probability.
• Figure 3: MPO training for probabilistic cellular automata with rules 18 and 51, with $p_{18}^e = 0.7$ and $p_{51}^e = 0.3$ respectively. (a) Predicted probabilities $P^p_{18}$ and $p_{51}^p$ against the number of training sweeps. Different symbols present different MPO bond dimensions: $D_W = 2$ is shown in blue "+", $D_W = 4$ is shown in orange circles, and $D_W = 8$ is shown in green "x". The solid lines are for $p^p_{18}$ and the dashed lines are for $p^p_{51}$. The dotted horizontal line marks the exact probabilities $p^e_{18}$ and $p_{51}^e$. (b) Prediction error $\epsilon$ against the number of training sweeps. The different lines represent different $D_W$. (c) $\epsilon$ against different $D_W$. The different lines represent different training samples $N$. The parameters used are system size $L = 20$, number of samples $N=20000$, and number of sweeps $N_{\text{sweep}}=10$, unless otherwise specified in the subplot descriptions.
• Figure 4: MPO training for probabilistic cellular automata with different pairs of rules. For each rule, the probability of occurrence is $p^e_i = 0.5$. The different markers represent different MPO bond dimensions: blue "+" for $D_W = 2$, orange circles for $D_W = 4$, purple asterisks for $D_W = 6$, and green "x" for $D_W = 8$. The solid and dashed lines correspond to the predicted probabilities of different rules. The dotted lines correspond to the exact probabilities. (a) Predicted probabilities $p^p_{18}$ and $p_{51}^p$ against the number of training sweeps. (b) Predicted probabilities $p^p_{18}$ and $p_{30}^p$ against the number of training sweeps. The parameters used are system size $L = 20$ and number of samples $N=20000$.
• Figure 5: MPO training for probabilistic cellular automata with different pairs of rules. The probabilities of the rules are $p^e_{18} = 0.9$ and $p^e_{51/30} = 0.1$. Different markers correspond to different bond dimensions: blue "+" for $D_W = 2$, orange circles for $D_W = 4$, and green "x" for $D_W = 8$. The solid and dashed lines correspond to the predicted probabilities of different rules. The dotted lines correspond to the exact probabilities. (a) Predicted probabilities $p^p_{18}$ and $p_{51}^p$ against the number of training sweeps. (b) Predicted probabilities $p^p_{18}$ and $p_{30}^p$ against the number of training sweeps. The parameters used are system size $L = 20$ and number of samples $N=20000$.