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arXiv:2008.06076 · Physical Review A

Achieving fast high-fidelity optimal control of many-body quantum dynamics

Jesper Hasseriis Mohr Jensen, Frederik Møller, Jens Jakob Sørensen, Jacob Friis Sherson

Published Nov 16, 2021• 15 citations
arXivHTMLPDF

Abstract

We demonstrate the efficiency of a recent exact-gradient optimal control methodology by applying it to a challenging many-body problem, crossing the superfluid to Mott-insulator phase transition in the Bose-Hubbard model. The system size necessitates a matrix product state representation and this seamlessly integrates with the requirements of the algorithm. We observe fidelities in the range 0.99--0.9999 with associated minimal process duration estimates displaying an exponential fidelity-duration trade-off across several orders of magnitude. The corresponding optimal solutions are characterized in terms of a predominantly linear sweep across the critical point followed by bang-bang-like structure. This is quite different from the smooth and monotonic solutions identified by earlier gradient-free optimizations which are hampered in locating the higher complexity protocols in the regime of high fidelities at low process durations. Overall, the comparison suggests significant methodological improvements also for many-body systems in the ideal open-loop setting. Acknowledging that idealized open-loop control may deteriorate in actual experiments, we discuss the merits of using such an approach in combination with closed-loop control---in particular, high-fidelity physical insights extracted with the former can be used to formulate practical, low-dimensional search spaces for the latter.

DOI
10.1103/physreva.104.052210

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We demonstrate the efficiency of a recent exact-gradient optimal control methodology by applying it to a challenging many-body problem, crossing the superfluid to Mott-insulator phase transition in the Bose-Hubbard model. The system size necessitates a matrix product state representation and this seamlessly integrates with the requirements of the algorithm. We observe fidelities in the range 0.99--0.9999 with associated minimal process duration estimates displaying an exponential fidelity-duration trade-off across several orders of magnitude. The corresponding optimal solutions are characterized in terms of a predominantly linear sweep across the critical point followed by bang-bang-like structure. This is quite different from the smooth and monotonic solutions identified by earlier gradient-free optimizations which are hampered in locating the higher complexity protocols in the regime of high fidelities at low process durations. Overall, the comparison suggests significant methodological improvements also for many-body systems in the ideal open-loop setting. Acknowledging that idealized open-loop control may deteriorate in actual experiments, we discuss the merits of using such an approach in combination with closed-loop control---in particular, high-fidelity physical insights extracted with the former can be used to formulate practical, low-dimensional search spaces for the latter.
We apply recent state-of-the-art optimal control techniques to a challenging many-body problem: driving the superfluid-Mott insulator transition in an optical lattice. At system sizes well beyond the reach of exact diagonalization approaches, and thus requiring a matrix product state ansatz, we obtain fidelities in the range 0.99-0.9999 and beyond with associated quantum speed limit estimates. Whereas previous efforts yielded lower fidelity solutions with smooth, monotonic controls, we efficiently identify a rich hierarchy of bang-bang-like solutions. These facilitate the non-adiabatic quantum interference pathways using sequential tunneling and phase-imprinting dynamics necessary for high fidelity. Mapping out the optimal solutions at various process durations, we observe a characteristic, exponential dependence for the fidelity across several orders of magnitude. Overall, we achieve these results by utilizing the counter-intuitive fact that appropriate dynamical approximations lead to a more precise and significantly cheaper implementation of optimal control than a full dynamical solution. In discussing the technique's generality, this demonstration may pave the way for fulfilling the comprehensive demands for efficient high-fidelity control in very high-dimensional systems which has hitherto not been feasible.
We apply recent state-of-the-art optimal control techniques to a challenging many-body problem: driving the superfluid-Mott insulator transition in an optical lattice. At system sizes well beyond the reach of exact diagonalization approaches, and thus requiring a matrix product state ansatz, we obtain fidelities in the range 0.99-0.9999 and beyond with associated quantum speed limit estimates. Whereas previous efforts yielded lower fidelity solutions with smooth, monotonic controls, we efficiently identify a rich hierarchy of bang-bang-like solutions. These facilitate the non-adiabatic quantum interference pathways using sequential tunneling and phase-imprinting dynamics necessary for high fidelity. Mapping out the optimal solutions at various process durations, we observe a characteristic, exponential dependence for the fidelity across several orders of magnitude. Overall, we achieve these results by utilizing the counter-intuitive fact that appropriate dynamical approximations lead to a more precise and significantly cheaper implementation of optimal control than a full dynamical solution. In discussing the technique's generality, this demonstration may pave the way for fulfilling the comprehensive demands for efficient high-fidelity control in very high-dimensional systems which has hitherto not been feasible.
We determine the optimal quantum manipulation protocols for implementing high-fidelity, fast single-qubit gates. We demonstrate that the time-optimal pulse sequence is a ``bang-bang'' sequence: discrete pulses of either positive or negative maximum amplitude or zero. The non-adiabatic bang-bang pulse sequence minimizes gate duration providing a speedup for low-frequency architectures. We first derive the protocol for a transversally driven two-level system, extracting exact analytic expressions for minimum gate times. We then extend this framework to general multilevel architectures, identifying conditions that enable the coherent suppression of leakage errors. Using the fluxonium circuit as a representative case study, we optimize X/2X/2X/2 and Y/2Y/2Y/2 gates through a combination of discrete bang sequences and continuous waveform smoothing. This approach preserves near-optimal execution speeds while mitigating transitions outside the computational subspace. Open-system simulations demonstrate that these sequences outperform commensurate and resonant pulse schemes across different fluxonium regimes, achieving low-error manipulation significantly faster than standard resonant control, even in the presence of 1/f1/f1/f flux noise and dissipation.
We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glasslike transition controlled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm t-distributed stochastic neighbor embedding, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.
Accurate derivatives are important for locally traversing and converging efficiently in quantum optimization landscapes. By deriving analytically exact control derivatives (gradient and Hessian) for unitary control tasks, we show here that the computational feasibility of meeting this accuracy requirement depends on the choice of propagation scheme and problem representation. Even when exact propagation is sufficiently cheap it is, perhaps surprisingly, much more efficient to optimize the (appropriately) approximate propagators: Approximations in the dynamics are traded off for significant complexity reductions in the exact derivative calculations. Importantly, past the initial analytical considerations, only standard numerical techniques are explicitly required with straightforward application to realistic systems. These results are numerically verified for two concrete problems of increasing Hilbert space dimensionality. The best schemes obtain unit fidelity to machine precision, whereas the results for other schemes are separated consistently by orders of magnitude in computation time and in worst-case ten orders of magnitude in achievable fidelity. Since these gaps continually increase with system size and complexity, this methodology allows numerically efficient optimization of very high-dimensional dynamics, e.g., in many-body contexts, operating in the high-fidelity regime.
Quantum many-body systems are emerging as key elements in the quest for quantum-based technologies and in the study of fundamental physics. In this study, we address the challenge of achieving fast and high-fidelity evolutions across quantum phase transitions, a crucial requirement for practical applications. We introduce a control technique based on dynamical invariants tailored to ensure adiabatic-like evolution within the lowest-energy subspace of the many-body systems described by the transverse-field Ising and long-range Kitaev models. By tuning the controllable parameter according to analytical control results, we achieve high-fidelity evolutions operating close to the speed limit. Remarkably, our approach leads to the breakdown of Kibble-Zurek scaling laws, offering tunable and significantly improved time scaling behavior. We provide detailed numerical simulations to illustrate our findings, demonstrating scalability with the system size and robustness against noisy controls and disorder, as well as its applicability to a non-integrable system.