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arXiv:1712.02368 · Physical Review X

Causal Asymmetry in a Quantum World

Jayne Thompson, Andrew J. P. Garner, John R. Mahoney, James P. Crutchfield, Vlatko Vedral, Mile Gu

Published Jul 18, 2018• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevx.8.031013

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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.
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.
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.
Why is it that a ticking clock typically becomes less accurate when subject to outside noise but rarely the reverse? Here, we formalise this phenomenon by introducing process causal asymmetry—a fundamental difference in the amount of past information an autonomous agent must track to transform one stochastic process to another over an agent that transforms in the opposite direction. We then illustrate that this asymmetry can surprisingly reverse when agents possess a quantum memory. Thus, the spontaneous direction in which processes get ‘simpler’ may be different, depending on whether quantum information processing is allowed or not.
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.
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.
What makes one quantum process more complex than another? We adopt an operational approach inspired by classical complexity science, which characterises structure through the memory required for causal generation. We develop a fully-quantum extension of computational mechanics for quantum space series: Correlated sequences of quantum outputs that do not return to interact with their generator. For stationary, memoryful quantum space series admitting finite-dimensional, primitive causal models, we show that any memory-minimal causal generator can be chosen to produce outputs isometrically, without a discarded environment. The resulting canonical matrix product state representation uniquely determines the irreducible memory required for generation. In particular, the minimum memory entropy is exactly the entanglement entropy across the past-future bipartition, giving entanglement a direct operational interpretation as irreducible memory for causal generation. The same structure determines the minimum memory dimension through the corresponding Schmidt rank. Within this class, we further show that the fundamental entropy rate vanishes, that causal generation is symmetric under time reversal, and that all memoryful processes possess infinite quantum Markov order with respect to arbitrary sharp measurements. These results establish an operational framework for quantifying intrinsic quantum structure, revealing deep connections between causal generation, complexity, memory, entropy, and entanglement.