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arXiv:1609.04408 · New Journal of Physics

Provably unbounded memory advantage in stochastic simulation using quantum mechanics

Andrew J. P. Garner, Qing Liu, Jayne Thompson, Vlatko Vedral, Mile Gu

Published Oct 12, 2017• 28 citations
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Abstract

Simulating the stochastic evolution of real quantities on a digital computer requires a trade-off between the precision to which these quantities are approximated, and the memory required to store them. The statistical accuracy of the simulation is thus generally limited by the internal memory available to the simulator. Here, using tools from computational mechanics, we show that quantum processors with a fixed finite memory can simulate stochastic processes of real variables to arbitrarily high precision. This demonstrates a provable, unbounded memory advantage that a quantum simulator can exhibit over its best possible classical counterpart.

DOI
10.1088/1367-2630/aa82df

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Simulating the stochastic evolution of real quantities on a digital computer requires a trade-off between the precision to which these quantities are approximated, and the memory required to store them. The statistical accuracy of the simulation is thus generally limited by the internal memory available to the simulator. Here, using tools from computational mechanics, we show that quantum processors with a fixed finite memory can simulate stochastic processes of real variables to arbitrarily high precision. This demonstrates a provable, unbounded memory advantage that a quantum simulator can exhibit over its best possible classical counterpart.
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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.
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.
Stochastic models are highly relevant tools in science, engineering, and society. Recent work suggests emerging quantum computing technologies can substantially decrease the memory requirements for simulating stochastic models. Here we show that some of these recent quantum memory-enhanced algorithms can be either implemented or approximated classically. In other words, we show that it is possible to develop quantum-inspired classical algorithms that require much less memory than the best classical algorithms known to date. Being classical, such algorithms could be implemented in state-of-the-art high-performance computers, which could potentially enhance the study of large-scale complex systems. Furthermore, since memory is the main bottleneck limiting the performance of classical supercomputers in one of the most promising avenues to demonstrate quantum 'supremacy', we expect adaptations of these ideas may potentially further raise the bar for near-term quantum computers to reach such a milestone.
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.
Classical stochastic processes can be generated by quantum simulators instead of the more standard classical ones, such as hidden Markov models. One reason for using quantum simulators has recently come to the fore: they generally require less memory than their classical counterparts. Here, we examine this quantum advantage for strongly coupled spin systems-in particular, the Dyson one-dimensional Ising spin chain with variable interaction length. We find that the advantage scales with both interaction range and temperature, growing without bound as interaction range increases. In particular, simulating Dyson's original spin chain with the most memory-efficient classical algorithm known requires infinite memory, while a quantum simulator requires only finite memory. Thus, quantum systems can very efficiently simulate strongly coupled one-dimensional classical spin systems.