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arXiv:1810.09668 · Physical Review A

Optimal stochastic modeling with unitary quantum dynamics

Qing Liu, Thomas J. Elliott, Felix C. Binder, Carlo Di Franco, Mile Gu

Published Jun 18, 2019• 39 citations
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Abstract

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.

DOI
10.1103/physreva.99.062110

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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.
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.
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 behavior 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.
Stochastic processes are as ubiquitous throughout the quantitative sciences as they are notorious for being difficult to simulate and predict. In this Letter, we propose a unitary quantum simulator for discrete-time stochastic processes which requires less internal memory than any classical analogue throughout the simulation. The simulator's internal memory requirements equal those of the best previous quantum models. However, in contrast to previous models, it only requires a (small) finite-dimensional Hilbert space. Moreover, since the simulator operates unitarily throughout, it avoids any unnecessary information loss. We provide a stepwise construction for simulators for a large class of stochastic processes hence directly opening the possibility for experimental implementations with current platforms for quantum computation. The results are illustrated for an example process.
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.
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.
Causal asymmetry is one of the great surprises in predictive modeling: The memory required to predict the future differs from the memory required to retrodict the past. There is a privileged temporal direction for modeling a stochastic process where memory costs are minimal. Models operating in the other direction incur an unavoidable memory overhead. Here, we show that this overhead can vanish when quantum models are allowed. Quantum models forced to run in the less-natural temporal direction not only surpass their optimal classical counterparts but also any classical model running in reverse time. This holds even when the memory overhead is unbounded, resulting in quantum models with unbounded memory advantage.