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arXiv:2404.15922 · Physical Review Letters

Single-Atom Verification of the Optimal Trade-Off between Speed and Cost in Shortcuts to Adiabaticity

J. -W. Zhang, J. -T. Bu, J. C. Li, Weiquan Meng, W. -Q. Ding, B. Wang, W. -F. Yuan, H.-J. Du, G.-Y. Ding, W.-J. Chen, L. Chen, Fan Zhou, Zhenyu Xu, M. Feng

Published May 24, 2024• 12 citations
arXivHTMLPDF

Abstract

The approach of shortcuts to adiabaticity enables the effective execution of adiabatic dynamics in quantum information processing with enhanced speed. Owing to the inherent trade-off between dynamical speed and the cost associated with the transitionless driving field, executing arbitrarily fast operations becomes impractical. To understand the accurate interplay between speed and energetic cost in this process, we propose theoretically and verify experimentally a new trade-off, which is characterized by a tightly optimized bound within s-parametrized phase spaces. Our experiment is carried out in a single ultracold ^{40}Ca^{+} ion trapped in a harmonic potential. By exactly operating the quantum states of the ion, we execute the Landau-Zener model as an example, where the quantum speed limit as well as the cost are governed by the spectral gap. We witness that our proposed trade-off is indeed tight in scenarios involving both initially eigenstates and initially thermal equilibrium states. Our work helps understanding the fundamental constraints in shortcuts to adiabaticity and illuminates the potential of underutilized phase spaces that have been traditionally overlooked.

DOI
10.1103/physrevlett.132.213602

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The approach of shortcuts to adiabaticity enables the effective execution of adiabatic dynamics in quantum information processing with enhanced speed. Owing to the inherent trade-off between dynamical speed and the cost associated with the transitionless driving field, executing arbitrarily fast operations becomes impractical. To understand the accurate interplay between speed and energetic cost in this process, we propose theoretically and verify experimentally a new trade-off, which is characterized by a tightly optimized bound within s-parametrized phase spaces. Our experiment is carried out in a single ultracold ^{40}Ca^{+} ion trapped in a harmonic potential. By exactly operating the quantum states of the ion, we execute the Landau-Zener model as an example, where the quantum speed limit as well as the cost are governed by the spectral gap. We witness that our proposed trade-off is indeed tight in scenarios involving both initially eigenstates and initially thermal equilibrium states. Our work helps understanding the fundamental constraints in shortcuts to adiabaticity and illuminates the potential of underutilized phase spaces that have been traditionally overlooked.
Achieving effectively adiabatic dynamics is a ubiquitous goal in almost all areas of quantum physics. Here, we study the speed with which a quantum system can be driven when employing transitionless quantum driving. As a main result, we establish a rigorous link between this speed, the quantum speed limit, and the (energetic) cost of implementing such a shortcut to adiabaticity. Interestingly, this link elucidates a trade-off between speed and cost, namely, that instantaneous manipulation is impossible as it requires an infinite cost. These findings are illustrated for two experimentally relevant systems-the parametric oscillator and the Landau-Zener model-which reveal that the spectral gap governs the quantum speed limit as well as the cost for realizing the shortcut.
The implementation of information processing on a quantum device is a fundamental challenge for many technologies. As a matter of fact, the faster one wants to implement a quantum operation, the higher is the thermodynamic cost of realizing the quantum process. Here, we theoretically propose and experimentally verify a trade-off between quantum speed and energy cost using Schatten 2-norm. Our findings demonstrate that this trade-off remains tight for any instant of time, whether dealing with initial eigenstates or initial thermal equilibrium states, as illustrated by the Landau-Zehner model. This observation underscores the significance of coherence of the evolved states. By extending our method to open system, we find that quantum speed can be significantly affected by environmental decoherence effect. These results illuminate the fundamental limits of quantum state dynamics and hold promise for potential applications in quantum sensing and quantum computing.
It is known that high intensity fields are usually required to implement shortcuts to adiabaticity via transitionless quantum driving (TQD). Here, we show that this requirement can be relaxed by exploiting the gauge freedom of generalized TQD, which is expressed in terms of an arbitrary phase when mimicking the adiabatic evolution. We experimentally investigate the performance of generalized TQD in comparison to both traditional TQD and adiabatic dynamics. By using a Yb+171 trapped ion hyperfine qubit, we implement a Landau–Zener adiabatic Hamiltonian and its (traditional and generalized) TQD counterparts. We show that the generalized theory provides energy-optimal Hamiltonians for TQD, with no additional fields required. In addition, the optimal TQD Hamiltonian for the Landau–Zener model is investigated under dephasing. Even using less intense fields, optimal TQD exhibits fidelities that are more robust against a decohering environment, with performance superior to that provided by the adiabatic dynamics.
Abstract Shortcuts to adiabaticity provide a general approach to mimic adiabatic quantum processes via arbitrarily fast evolutions in Hilbert space. For these counter-diabatic evolutions, higher speed comes at higher energy cost. Here, the counter-diabatic theory is employed as a minimal energy demanding scheme for speeding up adiabatic tasks. As a by-product, we show that this approach can be used to obtain infinite classes of transitionless models, including time-independent Hamiltonians under certain conditions over the eigenstates of the original Hamiltonian. We apply these results to investigate shortcuts to adiabaticity in decohering environments by introducing the requirement of a fixed energy resource. In this scenario, we show that generalized transitionless evolutions can be more robust against decoherence than their adiabatic counterparts. We illustrate this enhanced robustness both for the Landau–Zener model and for quantum gate Hamiltonians.
Adiabatic time-optimal quantum controls are extensively used in quantum technologies to break the constraints imposed by short coherence times. However, practically it is crucial to consider the trade-off between the quantum evolution speed and instantaneous energy cost of process because of the constraints in the available control Hamiltonian. Here, we experimentally show that using a transmon qubit that, even in the presence of vanishing energy gaps, it is possible to reach a highly time-optimal adiabatic quantum driving at low energy cost in the whole evolution process. This validates the recently derived general solution of the quantum Zermelo navigation problem, paving the way for energy-efficient quantum control which is usually overlooked in conventional speed-up schemes, including the well-known counter-diabatic driving. By designing the control Hamiltonian based on the quantum speed limit bound quantified by the changing rate of phase in the interaction picture, we reveal the relationship between the quantum speed limit and instantaneous energy cost. Consequently, we demonstrate fast and high-fidelity quantum adiabatic processes by employing energy-efficient driving strengths, indicating a promising strategy for expanding the applications of time-optimal quantum controls in superconducting quantum circuits.
Achieving precise control and efficient manipulation of different quantum states is one of the most important goals of quantum technology. We study the quantum control and quantum speed limits of single-well systems by taking a harmonic oscillator whose frequency is a power function of time as an example when the transitionless quantum driving is imposed. For different eigenstates, we reveal the effects of control time and power exponent on Bures angle, fidelity, time-averaged cost, and quantum-speed-limit time. We find that the shortest control time for any eigenstate is proportional to the control parameter (i.e., power exponent). For different eigenstates meeting the same high fidelity requirements, it is shown that an optimal control parameter can always be obtained. Our findings provide theoretical support for the rapid and efficient manipulation of quantum states in single-well systems by using shortcut-to-adiabaticity technique experimentally.