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arXiv:1803.05426 · New Journal of Physics

Memory-efficient tracking of complex temporal and symbolic dynamics with quantum simulators

Thomas J. Elliott, Andrew J. P. Garner, Mile Gu

Published Dec 12, 2018• 18 citations
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Abstract

Tracking the behaviour of stochastic systems is a crucial task in the statistical sciences. It has recently been shown that quantum models can faithfully simulate such processes whilst retaining less information about the past behaviour of the system than the optimal classical models. We extend these results to general temporal and symbolic dynamics. Our systematic protocol for quantum model construction relies only on an elementary description of the dynamics of the process. This circumvents restrictions on corresponding classical construction protocols, and allows for a broader range of processes to be modelled efficiently. We illustrate our method with an example exhibiting an apparent unbounded memory advantage of the quantum model compared to its optimal classical counterpart.

DOI
10.1088/1367-2630/aaf824

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Abstract Continuous-time stochastic processes pervade everyday experience, and the simulation of models of these processes is of great utility. Classical models of systems operating in continuous-time must typically track an unbounded amount of information about past behaviour, even for relatively simple models, enforcing limits on precision due to the finite memory of the machine. However, quantum machines can require less information about the past than even their optimal classical counterparts to simulate the future of discrete-time processes, and we demonstrate that this advantage extends to the continuous-time regime. Moreover, we show that this reduction in the memory requirement can be unboundedly large, allowing for arbitrary precision even with a finite quantum memory. We provide a systematic method for finding superior quantum constructions, and a protocol for analogue simulation of continuous-time renewal processes with a quantum machine.
A growing body of work has established the modeling of stochastic processes as a promising area of application for quantum technologies; it has been shown that quantum models are able to replicate the future statistics of a stochastic process while retaining less information about the past than any classical model must, even for a purely classical process. Such memory-efficient models open a potential future route to study complex systems in greater detail than ever before and suggest profound consequences for our notions of structure in their dynamics. Yet, to date methods for constructing these quantum models are based on having a prior knowledge of the optimal classical model. Here, we introduce a protocol for blind inference of the memory structure of quantum models---tailored to take advantage of quantum features---direct from time-series data, in the process highlighting the robustness of their structure to noise. This in turn provides a way to construct memory-efficient quantum models of stochastic processes while circumventing certain drawbacks that manifest solely as a result of classical information processing in classical inference protocols.
Isolating past information relevant for future prediction is central to quantitative science. Quantum models offer a promising approach, enabling statistically faithful modeling while using less past information than any classical counterpart. Here we introduce a class of phase-enhanced quantum models, representing the most general means of simulating a stochastic process unitarily in causal order. The resulting constructions surpass previous state-of-art methods---both in reducing the information they need to store about the past and in the minimal memory dimension they require to store this information. Moreover, these two features are generally competing factors in optimization---leading to an ambiguity in optimal modeling that is unique to the quantum regime. Our results simultaneously offer quantum advantages for stochastic simulation and illustrate further qualitative differences between classical and quantum notions of complexity.
Effective and efficient forecasting relies on identification of the relevant information contained in past observations -- the predictive features -- and isolating it from the rest. When the future of a process bears a strong dependence on its behaviour far into the past, there are many such features to store, necessitating complex models with extensive memories. Here, we highlight a family of stochastic processes whose minimal classical models must devote unboundedly many bits to tracking the past. For this family, we identify quantum models of equal accuracy that can store all relevant information within a single two-dimensional quantum system (qubit). This represents the ultimate limit of quantum compression and highlights an immense practical advantage of quantum technologies for the forecasting and simulation of complex systems.
Complex systems are embedded in our everyday experience. Stochastic modelling enables us to understand and predict the behaviour of such systems, cementing its utility across the quantitative sciences. Accurate models of highly non-Markovian processes - where the future behaviour depends on events that happened far in the past - must track copious amounts of information about past observations, requiring high-dimensional memories. Quantum technologies can ameliorate this cost, allowing models of the same processes with lower memory dimension than corresponding classical models. Here we implement such memory-efficient quantum models for a family of non-Markovian processes using a photonic setup. We show that with a single qubit of memory our implemented quantum models can attain higher precision than possible with any classical model of the same memory dimension. This heralds a key step towards applying quantum technologies in complex systems modelling.
Stochastic modeling of complex systems plays an essential, yet often computationally intensive, role across the quantitative sciences.Recent advances in quantum information processing have elucidated the potential for quantum simulators to exhibit memory advantages for such tasks.Heretofore, the focus has been on lossless memory compression, wherein the advantage is typically in terms of lessening the amount of information tracked by the model, while-arguably more practical-reductions in memory dimension are not always possible.Here, we address the case of lossy compression for quantum stochastic modeling of continuous-time processes, introducing a method for coarse graining in quantum state space that drastically reduces the requisite memory dimension for modeling temporal dynamics while retaining near-exact statistics.In contrast to classical coarse graining, this compression is not based on sacrificing temporal resolution and brings memory-efficient high-fidelity stochastic modeling within reach of present quantum technologies.