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Quantum localization bounds Trotter errors in digital quantum simulation

Heyl, M., Hauke, P., Zoller, P.

Published Apr 12, 2019• 97 citations
OpenAlex

Abstract

A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.

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Quantum localization bounds Trotter errors in digital quantum simulation

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A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.

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This work aims at giving Trotter errors in digital quantum simulation (DQS) of collective spin systems an interpretation in terms of quantum chaos of the kicked top. In particular, for DQS of such systems, regular dynamics of the kicked top ensures convergence of the Trotterized time evolution, while chaos in the top, which sets in above a sharp threshold value of the Trotter step size, corresponds to the proliferation of Trotter errors. We show the possibility to analyze this phenomenology in a wide variety of experimental realizations of the kicked top, ranging from single atomic spins to trapped-ion quantum simulators which implement DQS of all-to-all interacting spin-1/2 systems. These platforms thus enable in-depth studies of Trotter errors and their relation to signatures of quantum chaos, including the growth of out-of-time-ordered correlators.

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Abstract This work aims at giving Trotter errors in digital quantum simulation (DQS) of collective spin systems an interpretation in terms of quantum chaos of the kicked top. In particular, for DQS of such systems, regular dynamics of the kicked top ensures convergence of the Trotterized time evolution, while chaos in the top, which sets in above a sharp threshold value of the Trotter step size, corresponds to the proliferation of Trotter errors. We show the possibility to analyze this phenomenology in a wide variety of experimental realizations of the kicked top, ranging from single atomic spins to trapped-ion quantum simulators which implement DQS of all-to-all interacting spin-1/2 systems. These platforms thus enable in-depth studies of Trotter errors and their relation to signatures of quantum chaos, including the growth of out-of-time-ordered correlators.

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Digital quantum simulation (DQS) is one of the most promising paths for achieving first useful real-world applications for quantum processors. Yet even assuming rapid progress in device engineering and development of fault-tolerant quantum processors, algorithmic resource optimisation will long remain crucial to exploit their full power. Currently, Trotterisation provides state-of-the-art resource scaling. And recent theoretical studies of Trotterised Ising models suggest that even better performance than expected may be possible up to a distinct breakdown threshold in empirical performance. Here, we study multiple paradigmatic DQS models with experimentally realisable Trotterisations, and evidence the universality of a range of Trotterisation performance behaviours, including not only the threshold, but also new features in the pre-threshold regime that is most important for practical applications. In each model, we observe a distinct Trotterisation threshold shared across widely varying performance signatures; we further show that an onset of quantum chaotic dynamics causes the performance breakdown and is directly induced by digitisation errors. In the important pre-threshold regime, we are able to identify new distinct regimes displaying qualitatively different quasiperiodic performance behaviours, and show analytic behaviour for properly defined operational Trotter errors. Our results rely crucially on diverse new analytical tools, and provide a previously missing unified picture of Trotterisation behaviour across local observables, the global quantum state, and the full Trotterised unitary. This work provides new insights and tools for addressing important questions about the algorithm performance and underlying theoretical principles of sufficiently complex Trotterisation-based DQS, that will help in extracting maximum simulation power from future quantum processors.

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Robustness of digital quantum simulators against Trotter errors

Philipp Hauke, Markus Heyl, P. Zoller

In digital quantum simulation, the Trotter error of unitary time-evolution operators increases polynomially with simulated system size and time. In contrast, we argue that local observables can enter a regime where the Trotter error becomes constant.

Quantum Information and Measurement (QIM)•Jan 1, 2017•0 citations
A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.
A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.
This work aims at giving Trotter errors in digital quantum simulation (DQS) of collective spin systems an interpretation in terms of quantum chaos of the kicked top. In particular, for DQS of such systems, regular dynamics of the kicked top ensures convergence of the Trotterized time evolution, while chaos in the top, which sets in above a sharp threshold value of the Trotter step size, corresponds to the proliferation of Trotter errors. We show the possibility to analyze this phenomenology in a wide variety of experimental realizations of the kicked top, ranging from single atomic spins to trapped-ion quantum simulators which implement DQS of all-to-all interacting spin-1/2 systems. These platforms thus enable in-depth studies of Trotter errors and their relation to signatures of quantum chaos, including the growth of out-of-time-ordered correlators.
Digital quantum simulation (DQS) is one of the most promising applications of quantum computing. Trotterised DQS models quantum dynamics by decomposing the evolution into a sequence of gate-based Trotter steps. Using larger Trotter steps reduces the required gate count, but it is now understood that the Trotterisation step size cannot be increased beyond a certain threshold value, beyond which there is a sudden breakdown in simulation accuracy, due to the onset of quantum chaos. To date, this behaviour has only been studied for closed systems not subjected to the environmental noise and decoherence to which real-world quantum computers are highly sensitive. In this work, we study a range of experimentally relevant open-system quantum simulator models and show that Trotterisation thresholds persist even when open-system effects are included, even in the long-time, steady-state limit, once all coherence has been lost. We show that open DQS can accurately reproduce the dynamics of open target systems prior to the threshold, but still exhibits a sudden performance breakdown beyond it. We introduce new, statistical techniques to rigorously characterise the onset of quantum chaos beyond the Trotterisation threshold for both approximately unitary and genuine dissipative quantum chaos, based on random matrix theory (RMT), building on techniques developed for closed systems. Finally, we demonstrate that dissipative quantum chaos emerges when the dissipation is in balance with the presence of underlying unitary quantum chaos; the dissipation must be strong enough to induce genuinely dissipative behaviour, yet not so strong as to suppress the underlying chaotic structure.
Abstract This work aims at giving Trotter errors in digital quantum simulation (DQS) of collective spin systems an interpretation in terms of quantum chaos of the kicked top. In particular, for DQS of such systems, regular dynamics of the kicked top ensures convergence of the Trotterized time evolution, while chaos in the top, which sets in above a sharp threshold value of the Trotter step size, corresponds to the proliferation of Trotter errors. We show the possibility to analyze this phenomenology in a wide variety of experimental realizations of the kicked top, ranging from single atomic spins to trapped-ion quantum simulators which implement DQS of all-to-all interacting spin-1/2 systems. These platforms thus enable in-depth studies of Trotter errors and their relation to signatures of quantum chaos, including the growth of out-of-time-ordered correlators.
Digital quantum simulation (DQS) is one of the most promising paths for achieving first useful real-world applications for quantum processors. Yet even assuming rapid progress in device engineering and development of fault-tolerant quantum processors, algorithmic resource optimisation will long remain crucial to exploit their full power. Currently, Trotterisation provides state-of-the-art resource scaling. And recent theoretical studies of Trotterised Ising models suggest that even better performance than expected may be possible up to a distinct breakdown threshold in empirical performance. Here, we study multiple paradigmatic DQS models with experimentally realisable Trotterisations, and evidence the universality of a range of Trotterisation performance behaviours, including not only the threshold, but also new features in the pre-threshold regime that is most important for practical applications. In each model, we observe a distinct Trotterisation threshold shared across widely varying performance signatures; we further show that an onset of quantum chaotic dynamics causes the performance breakdown and is directly induced by digitisation errors. In the important pre-threshold regime, we are able to identify new distinct regimes displaying qualitatively different quasiperiodic performance behaviours, and show analytic behaviour for properly defined operational Trotter errors. Our results rely crucially on diverse new analytical tools, and provide a previously missing unified picture of Trotterisation behaviour across local observables, the global quantum state, and the full Trotterised unitary. This work provides new insights and tools for addressing important questions about the algorithm performance and underlying theoretical principles of sufficiently complex Trotterisation-based DQS, that will help in extracting maximum simulation power from future quantum processors.