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arXiv:2111.09622 · Physical Review Research

Error-mitigated deep-circuit quantum simulation of open systems: Steady state and relaxation rate problems

Anbang Wang, Jing-Ning Zhang, Ying Li

Published Nov 28, 2022• 6 citations
arXivHTMLPDF

Abstract

Deep-circuit quantum computation, like Shor's algorithm, is undermined by error accumulation, and near-future quantum techniques are far from adequate for full-fledged quantum error correction. Instead of resorting to shallow-circuit quantum algorithms, recent theoretical research suggests that digital quantum simulation (DQS) of closed quantum systems are robust against the accumulation of Trotter errors, as long as local observables are concerned. In this paper, we investigate the error-mitigation problem of open systems via DQS. First, we prove that the deviation in the steady state obtained from digital quantum simulation depends only on the error in a single Trotter step, which indicates that error accumulation may not be disastrous. By numerical simulation of the quantum circuits for the DQS of the dissipative XYZ model, we then show that the correct results can be recovered by quantum error mitigation as long as the error rate in the DQS is below a sharp threshold. We explain this threshold behavior by the existence of a dissipation-driven quantum phase transition. Finally, we propose a error-mitigation technique based on the scaling behavior in the vicinity of the critical point of a quantum phase transition. Our results expand the territory of near-future available quantum algorithms and stimulate further theoretical and experimental efforts in practical quantum applications.

DOI
10.1103/physrevresearch.4.043140

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Deep-circuit quantum computation, like Shor's algorithm, is undermined by error accumulation, and near-future quantum techniques are far from adequate for full-fledged quantum error correction. Instead of resorting to shallow-circuit quantum algorithms, recent theoretical research suggests that digital quantum simulation (DQS) of closed quantum systems are robust against the accumulation of Trotter errors, as long as local observables are concerned. In this paper, we investigate the error-mitigation problem of open systems via DQS. First, we prove that the deviation in the steady state obtained from digital quantum simulation depends only on the error in a single Trotter step, which indicates that error accumulation may not be disastrous. By numerical simulation of the quantum circuits for the DQS of the dissipative XYZ model, we then show that the correct results can be recovered by quantum error mitigation as long as the error rate in the DQS is below a sharp threshold. We explain this threshold behavior by the existence of a dissipation-driven quantum phase transition. Finally, we propose a error-mitigation technique based on the scaling behavior in the vicinity of the critical point of a quantum phase transition. Our results expand the territory of near-future available quantum algorithms and stimulate further theoretical and experimental efforts in practical quantum applications.
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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.
Digital quantum simulation of many-body dynamics faces a tension between algorithmic Trotter error and physical noise that accumulates with circuit depth. Typical higher-order product formulas mitigate the algorithmic error at the cost of deeper circuits. Our benchmark shows that, within the limited physical error budget, the feasible advantage is confined to absolute errors well below the 10−210^{-2}10−2 scale, which vanishes under noise levels of current quantum hardware. In sharp contrast to the previous Trotter error mitigation methods, we introduce a constant-depth multi-product formula (cd-MPF) that suppresses the Trotter error by combining multiple circuits at fixed circuit depth. We identify an auxiliary parameter ααα, which reshapes the Trotter error terms while leaving the target evolution invariant. The classical linear combination of the measured expectation values cancels the leading (Δt)2(Δt)^{2}(Δt)2 algorithmic contribution and steepens the Trotter-error scaling with the two-qubit circuit depth ddd from d−2d^{-2}d−2 to d−4d^{-4}d−4. Combined with physical-noise mitigation, our method can serve as a key ingredient for realizing long-time quantum dynamics simulation on near-term hardware.
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.
Understanding the dynamics of quantum systems is crucial in many areas of physics, but simulating many-body systems presents significant challenges due to the large Hilbert space to navigate and the exponential growth of computational overhead. Quantum computers offer a promising platform to overcome these challenges, particularly for simulating the time evolution with Hamiltonians. Trotterization is a widely used approach among available algorithms in this regard, and well suited for near-term quantum devices. However, it introduces algorithmic Trotter errors due to the non-commutativity of Hamiltonian components. Several techniques such as multi-product formulas have been developed to mitigate Trotter errors, but often require deep quantum circuits, which can introduce additional physical errors. In this work, we propose a resource-efficient scheme to reduce the algorithmic Trotter error with relatively shallow circuit depth. We develop a profiling method by introducing an auxiliary parameter to estimate the error effects in expectation values, enabling significant error suppression with a fixed number of Trotter steps. Our approach offers an efficient way of quantum simulation on near-term quantum processors with shallow circuits.
Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.