Minimum memory for generating rare events

Not All Fluctuations Are Created Equal: Spontaneous Variations in Thermodynamic Function

We identify macroscopic functioning arising during a thermodynamic system's typical and atypical behaviors, thereby describing system operations over the entire set of fluctuations. We show how to use the information processing second law to determine functionality for atypical realizations and how to calculate the probability of distinct modalities occurring via the large-deviation rate function, extended to include highly correlated, memoryful environments and systems. Altogether, the results complete a theory of functional fluctuations for complex thermodynamic nanoscale systems operating over finite periods. In addition to constructing the distribution of functional modalities, one immediate consequence is a cautionary lesson: ascribing a single, unique functional modality to a thermodynamic system, especially one on the nanoscale, can be misleading, likely masking an array of simultaneous, parallel thermodynamic transformations that together may also be functional. In this way, functional fluctuation theory alters how we conceive of the operation of biological cellular processes, the goals of engineering design, and the robustness of evolutionary adaptation.

Prediction and generation of binary Markov processes: Can a finite-state fox catch a Markov mouse?

Understanding the generative mechanism of a natural system is a vital component of the scientific method. Here, we investigate one of the fundamental steps toward this goal by presenting the minimal generator of an arbitrary binary Markov process. This is a class of processes whose predictive model is well known. Surprisingly, the generative model requires three distinct topologies for different regions of parameter space. We show that a previously proposed generator for a particular set of binary Markov processes is, in fact, not minimal. Our results shed the first quantitative light on the relative (minimal) costs of prediction and generation. We find, for instance, that the difference between prediction and generation is maximized when the process is approximately independently, identically distributed.

Extreme Quantum Advantage when Simulating Classical Systems with Long-Range Interaction

Classical stochastic processes can be generated by quantum simulators instead of the more standard classical ones, such as hidden Markov models. One reason for using quantum simulators has recently come to the fore: they generally require less memory than their classical counterparts. Here, we examine this quantum advantage for strongly coupled spin systems-in particular, the Dyson one-dimensional Ising spin chain with variable interaction length. We find that the advantage scales with both interaction range and temperature, growing without bound as interaction range increases. In particular, simulating Dyson's original spin chain with the most memory-efficient classical algorithm known requires infinite memory, while a quantum simulator requires only finite memory. Thus, quantum systems can very efficiently simulate strongly coupled one-dimensional classical spin systems.