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arXiv:2610.03026 · Physical Review Letters

Spatiotemporal quantum advantages for rare-event sampling

Akshat Gupta, Jianjun Chen, Chengran Yang, Mile Gu, Caterina Doglioni, Jayne Thompson, Thomas J. Elliott

quant-phcond-mat.stat-mechSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

In a world fraught with uncertainty, it behooves us to consider the most disastrous possible outcomes, in order that we can best mitigate against them. Often these are extreme risk events -- of high consequence, but low probability. Yet, the rarity of such events limits our ability to study them; there is a paucity of past occurrences to draw upon, and sampling such rare events from models is typically costly in time. Here, we introduce an approach for rare-event sampling based on quantum stochastic models. This integrates the memory advantages of quantum stochastic modelling -- and its associated efficient means of preparing quantum sample states -- with the `speed-ups' of quantum amplitude amplification, to yield simultaneous reductions in spatial and temporal computational complexity compared to classical Monte Carlo-based approaches.

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8+3 pages, 3+5 figures

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In a world fraught with uncertainty, it behooves us to consider the most disastrous possible outcomes, in order that we can best mitigate against them. Often these are extreme risk events -- of high consequence, but low probability. Yet, the rarity of such events limits our ability to study them; there is a paucity of past occurrences to draw upon, and sampling such rare events from models is typically costly in time. Here, we introduce an approach for rare-event sampling based on quantum stochastic models. This integrates the memory advantages of quantum stochastic modelling -- and its associated efficient means of preparing quantum sample states -- with the `speed-ups' of quantum amplitude amplification, to yield simultaneous reductions in spatial and temporal computational complexity compared to classical Monte Carlo-based approaches.
Financial crashes, cascading failures in infrastructure, and critical errors in AI systems are frequently triggered by events that occur with extremely small probability. Efficiently discovering and sampling events with probability below a threshold is therefore of critical interest. Yet this task is highly non-trivial using existing classical or quantum methods. Being rare, such events require an immense sampling overhead to collect sufficient data samples. Moreover, because the rare events are not known in advance, they cannot be flagged for amplification using standard techniques. Here, we introduce a quantum algorithm for rare-event discovery and sampling without first learning which events are rare. The algorithm achieves the optimal quantum scaling with the rarity threshold. We further demonstrate that this can achieve a quadratic speedup for heavy-tailed systems whose tail has nonvanishing total mass, and translates into a robust polynomial speedup for stationary stochastic processes, with the exponent determined by its entropy-rate structure.
Many potential future events have devastating consequences, from the onset of earthquakes to financial black swan events. Accurate predictions of their probability of occurrence are vital. Yet as the stochastic systems underlying such events become increasingly complex, these probabilities can depend on ever-growing amounts of past data, which require ever more memory to track. The accuracy of our estimates is ultimately limited by both computational and memory constraints; the former to reduce sampling error, the latter to avoid sampling bias. Traditional quantum algorithms -- such as amplitude estimation -- address only the former, and become untenable especially in near-future quantum devices where quantum memory is a premium. Here, we offer quantum-algorithmic means to reduce systematic bias while maintaining a quadratic speed-up. Our results thus advance the timeline for the practical deployment of quantum technologies, and illustrate their additional benefits when simulating processes of growing complexity.
Abstract Quantum technologies offer a promising route to the efficient sampling and analysis of stochastic processes, with potential applications across the sciences. Such quantum advantages rely on the preparation of a quantum sample state of the stochastic process, which requires a memory system to propagate correlations between the past and future of the process. Here, we introduce a method of lossy quantum dimension reduction that allows this memory to be compressed, not just beyond classical limits, but also beyond current state-of-the-art quantum stochastic sampling approaches. We investigate the trade-off between the saving in memory resources from this compression, and the distortion it introduces. We show that our approach can be highly effective in low distortion compression of both Markovian and strongly non-Markovian processes alike. We further discuss the application of our results to quantum stochastic modelling more broadly.
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.
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.