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arXiv:2005.02403 · Physical Review X

Quantum Advantage in Simulating Stochastic Processes

Kamil Korzekwa, Matteo Lostaglio

Published Apr 22, 2021• 33 citations
arXivHTMLPDF

Abstract

We investigate the problem of simulating classical stochastic processes through quantum dynamics and present three scenarios where memory or time quantum advantages arise.First, by introducing and analyzing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory.Second, by extending the notion of space-time cost of a stochastic process P to the quantum domain, we prove an advantage of the quantum cost of simulating P over the classical cost.Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.

DOI
10.1103/physrevx.11.021019

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We investigate the problem of simulating classical stochastic processes through quantum dynamics and present three scenarios where memory or time quantum advantages arise.First, by introducing and analyzing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory.Second, by extending the notion of space-time cost of a stochastic process P to the quantum domain, we prove an advantage of the quantum cost of simulating P over the classical cost.Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.
Stochastic processes underlie a vast range of natural and social phenomena.Some processes such as atomic decay feature intrinsic randomness, whereas other complex processes, e.g., traffic congestion, are effectively probabilistic because we cannot track all relevant variables.To simulate a stochastic system's future behavior, information about its past must be stored, and thus memory is a key resource.Quantum information processing promises a memory advantage for stochastic simulation.Here, we report the first experimental demonstration that a quantum stochastic simulator can encode the required information in fewer dimensions than any classical simulator, thereby achieving a quantum advantage in minimal memory requirements using an individual simulator.This advantage is in contrast to recent proof-of-concept experiments, where the memory saving would only become accessible in the limit of a large number of parallel simulations.In those examples, the minimal memory registers of individual quantum simulators had the same dimensionality as their classical counterparts.Our photonic experiment thus establishes the potential of new, practical resource savings in the simulation of complex systems.
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.
The classical embeddability problem asks whether a given stochastic matrix T , describing transition probabilities of a d -level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given T . More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.
Abstract Many inference scenarios rely on extracting relevant information from known data in order to make future predictions. When the underlying stochastic process satisfies certain assumptions, there is a direct mapping between its exact classical and quantum simulators, with the latter asymptotically using less memory. Here we focus on studying whether such quantum advantage persists when those assumptions are not satisfied, and the model is doomed to have imperfect accuracy. By studying the trade-off between accuracy and memory requirements, we show that quantum models can reach the same accuracy with less memory, or alternatively, better accuracy with the same memory. Finally, we discuss the implications of this result for learning tasks.
Quantum advantage is notoriously hard to find and even harder to prove. For example the class of functions computable with classical physics exactly coincides with the class computable quantum mechanically. It is strongly believed, but not proven, that quantum computing provides exponential speed-up for a range of problems, such as factoring. Here we address a computational scenario of randomness processing in which quantum theory provably yields, not only resource reduction over classical stochastic physics, but a strictly larger class of problems which can be solved. Beyond new foundational insights into the nature and malleability of randomness, and the distinction between quantum and classical information, these results also offer the potential of developing classically intractable simulations with currently accessible quantum technologies. Although quantum alternatives of various classical tasks are considered advantageous, this is typically extremely difficult to concretely prove. Here, the authors show that a quantum approach to randomness processing provides a reduction in resources and a larger class of solvable problems.