No abstract

In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.The quest for simple representations and models of the physical world, often phrased as Occam's famous razor, underlies most scientific pursuits. In this spirit, computational mechanics seeks the most memory-efficient predictive models for stochastic processes-models which track relevant past information about a process, in order to generate statistically faithful future predictions [1-5]. The classically minimal models, ε-machines, have been used in diverse contexts from neuroscience to nonequilibrium contextuality [6][7][8][9][10][11][12][13][14][15]. Recently, it was shown that quantum extensions of ε-machines can further reduce their memory [16], leading recent studies to find memoryefficient quantum means of predictive modeling [17][18][19][20][21][22][23][24][25][26][27].In condensed matter, on the other hand, simplicity is sought after for the description of quantum many-body systems. Tensor networks, such as matrix product states (MPS), for instance, provide an efficient and useful description of one-dimensional quantum systems-i.e., spin chains [28][29][30]. This has led to reliable and powerful numerical methods for probing and simulating properties of multi-partite systems, whose study would be otherwise intractable [31][32][33][34].In this Letter, we develop a connection between εmachines and the MPS representation, shown in Fig. 1. We associate each stochastic process with a suitable quantum state of a spin chain called q-sample, measurement of which generates the corresponding stochastic process. We then show that the classical ε-machine of the process leads to a systematic MPS representation of the q-sample. Conversely, applying MPS methods to this state allows us to construct a q-simulator for the associated stochastic process -the best known quantum model [17,21,22]. Lastly, we show that the entanglement across the past-future bipartition of the q-sample coincides exactly with the quantum memory requirements * Yangchengran92@gmail.com † quantum@felix-binder.net ‡ mgu@quantumcomplexity.org Fig. 1. This Letter connects the complexity of stochastic processes and the representational complexity of spin chains. These (i) lin...

In modelling complex processes, the potential past data that influence future expectations are immense. Models that track all this data are not only computationally wasteful but also shed little light on what past data most influence the future. There is thus enormous interest in dimensional reduction -finding automated means to reduce the memory dimension of our models while minimizing its impact on its predictive accuracy. Here we construct dimensionally reduced quantum models by machine learning methods that can achieve greater accuracy than provably optimal classical counterparts. We demonstrate this advantage on present-day quantum computing hardware. Our algorithm works directly off classical time-series data and can thus be deployed in real-world settings. These techniques illustrate the immediate relevance of quantum technologies to time-series analysis and offer a rare instance where the resulting quantum advantage can be provably established.

Quantifying how distinguishable two stochastic processes are is at the heart of many fields, such as machine learning and quantitative finance. While several measures have been proposed for this task, none have universal applicability and ease of use. In this article, we suggest a set of requirements for a well-behaved measure of process distinguishability. Moreover, we propose a family of measures, called divergence rates, that satisfy all of these requirements. Focusing on a particular member of this family-the coemission divergence rate-we show that it can be computed efficiently, behaves qualitatively similar to other commonly used measures in their regimes of applicability, and remains well behaved in scenarios where other measures break down.

In the design of stochastic models, there is a constant trade-off between model complexity and accuracy.Here we prove that quantum models enable a more favorable trade-off. We present a technique for identifying fundamental upper bounds on the predictive accuracy of dimensionality-constrained classical models. We identify quantum models that surpass this bound by creating an algorithm that learns quantum models given time-series data. We demonstrate that this quantum accuracy advantage is attainable in a present-day noisy quantum device. These results illustrate the immediate relevance of quantum technologies to time-series analysis and offer an instance where their resulting accuracy advantage can be provably established.