Among the predictive hidden Markov models that describe a given stochastic process, the ϵ-machine is strongly minimal in that it minimizes every Rényi-based memory measure. Quantum models can be smaller still. In contrast with the ϵ-machine ’s unique role in the classical setting, however, among the class of processes described by pure-state hidden quantum Markov models, there are those for which there does not exist any strongly minimal model. Quantum memory optimization then depends on which memory measure best matches a given problem’s circumstance.

Quantum coherence allows for reduced-memory simulators of classical processes. Using recent results in single-shot quantum thermodynamics, we derive a minimal work cost rate for quantum simulators that is quasistatically attainable in the limit of asymptotically infinite parallel simulation. Comparing this cost with the classical regime reveals that quantizing classical simulators not only results in memory compression but also in reduced dissipation. We explore this advantage across a suite of representative examples.

Modularity dissipation identifies how locally implemented computation entails costs beyond those required by Landauer's bound on thermodynamic computing. We establish a general theorem for efficient local computation, giving the necessary and sufficient conditions for a local operation to have zero modularity cost. Applied to thermodynamically-generating stochastic processes it confirms a conjecture that classical generators are efficient if and only if they satisfy retrodiction, which places minimum-memory requirements on the generator. This extends immediately to quantum computation: Any quantum simulator that employs quantum memory compression cannot be thermodynamically efficient.

Predictive states for stochastic processes are a nonparametric and interpretable construct with relevance across a multitude of modeling paradigms. Recent progress on the self-supervised reconstruction of predictive states from time-series data focused on the use of reproducing kernel Hilbert spaces. Here, we examine how Wasserstein distances may be used to detect predictive equivalences in symbolic data. We compute Wasserstein distances between distributions over sequences ("predictions") using a finite-dimensional embedding of sequences based on the Cantor set for the underlying geometry. We show that exploratory data analysis using the resulting geometry via hierarchical clustering and dimension reduction provides insight into the temporal structure of processes ranging from the relatively simple (e.g., generated by finite-state hidden Markov models) to the very complex (e.g., generated by infinite-state indexed grammars).

Predictive equivalence in discrete stochastic processes has been applied with great success to identify randomness and structure in statistical physics and chaotic dynamical systems and to inferring hidden Markov models. We examine the conditions under which predictive states can be reliably reconstructed from time-series data, showing that convergence of predictive states can be achieved from empirical samples in the weak topology of measures. Moreover, predictive states may be represented in Hilbert spaces that replicate the weak topology. We mathematically explain how these representations are particularly beneficial when reconstructing high-memory processes and connect them to reproducing kernel Hilbert spaces.

Multiregional input-output (MRIO) tables, in conjunction with Leontief analysis, are widely-used to assess the geographical distribution of carbon emissions and the economic activities that cause them. Majorization, a tool originating in economics that has found utility in statistical mechanics, can provide insight into how Leontief analysis links disparities in emissions with global income inequality. We examine Leontief analysis as a model, drawing out similarities with modern nonequilibrium statistical mechanics. Paralleling the physical concept of thermo-majorization, we define the concept of eco-majorization and show it is a sufficient condition to determine the directionality of embodied emission flows. Surprisingly, relatively small trade deficits and a geographically heterogeneous emissions-per-dollar ratio greatly increases the appearance of eco-majorization, regardless of any further content in the MRIO tables used. Our results are bolstered by a statistical analysis of null models of MRIO tables, based on data provided by the Global Trade Aggregation Project9

We introduce a quantum algorithm for memory-efficient biased sampling of rare events generated by classical memoryful stochastic processes. Two efficiency metrics are used to compare quantum and classical resources for rare-event sampling. For a fixed stochastic process, the first is the classical-to-quantum ratio of required memory. We show for two example processes that there exists an infinite number of rare-event classes for which the memory ratio for sampling is larger than r, for any large real number r. Then, for a sequence of processes each labeled by an integer size N, we compare how the classical and quantum required memories scale with N. In this setting, since both memories can diverge as N→, the efficiency metric tracks how fast they diverge. An extreme quantum memory advantage exists when the classical memory diverges in the limit N→, but the quantum memory has a finite bound. We then show that finite-state Markov processes and spin chains exhibit memory advantage for sampling of almost all of their rare-event classes.

Given a classical channel—a stochastic map from inputs to outputs—the input can often be transformed into an intermediate variable that is informationally smaller than the input. The new channel accurately simulates the original but at a smaller transmission rate. Here, we examine this procedure when the intermediate variable is a quantum state. We determine when and how well quantum simulations of classical channels may improve upon the minimal rates of classical simulation. This inverts Holevo’s original question of quantifying the capacity of quantum channels with classical resources: We determine the lowest-capacity quantum channel required to simulate a classical channel. We also show that this problem is equivalent to another, involving the local generation of a distribution from common entanglement.