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arXiv:2203.03164 · Physical Review Research

Speeding up quantum adiabatic processes with a dynamical quantum geometric tensor

Jin-Fu Chen

Published Jun 28, 2022• 8 citations
arXivHTMLPDF

Abstract

For adiabatic controls of quantum systems, the nonadiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For the system initiated from the $n\mathrm{th}$ eigenstate, the transition probability in the optimal protocol is bounded by ${P}_{n}(t)\ensuremath{\le}4{\mathcal{L}}_{n}^{2}/{\ensuremath{\tau}}^{2}$ with the operation time $\ensuremath{\tau}$ and the quantum adiabatic length ${\mathcal{L}}_{n}$ induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising model.

DOI
10.1103/physrevresearch.4.023252

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For adiabatic controls of quantum systems, the nonadiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For the system initiated from the nthn\mathrm{th}nth eigenstate, the transition probability in the optimal protocol is bounded by Pn(t)\ensuremath≤4Ln2/\ensuremathτ2{P}_{n}(t)\ensuremath{\le}4{\mathcal{L}}_{n}^{2}/{\ensuremath{\tau}}^{2}Pn​(t)\ensuremath≤4Ln2​/\ensuremathτ2 with the operation time \ensuremathτ\ensuremath{\tau}\ensuremathτ and the quantum adiabatic length Ln{\mathcal{L}}_{n}Ln​ induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising model.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary sufficiently slow protocols using adiabatic perturbation theory, and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those which drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses, or target individual excited states in quantum chaotic systems, the geometric bound for the QSL can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in non-integrable spin chains using optimal control.
We propose a method to obtain optimal protocols for adiabatic ground-state preparation near the adiabatic limit, extending earlier ideas from [D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. 108, 190602 (2012)] to quantum non-dissipative systems. The space of controllable parameters of isolated quantum many-body systems is endowed with a Riemannian quantum metric structure, which can be exploited when such systems are driven adiabatically. Here, we use this metric structure to construct optimal protocols in order to accomplish the task of adiabatic ground-state preparation in a fixed amount of time. Such optimal protocols are shown to be geodesics on the parameter manifold, maximizing the local fidelity. Physically, such protocols minimize the average energy fluctuations along the path. Our findings are illustrated on the Landau-Zener model and the anisotropic XY spin chain. In both cases we show that geodesic protocols drastically improve the final fidelity. Moreover, this happens even if one crosses a critical point, where the adiabatic perturbation theory fails.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians.We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor.The conjecture implies a geometric lower bound for the quantum speed limit (QSL).We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory.Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution.When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound.However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length.We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.
Efficient adiabatic control schemes, where one steers a quantum system along an adiabatic path ensuring minimal excitations while achieving a desired final state, that enable fast, high-fidelity operations are essential for any practical quantum computation. However, current optimization protocols are not universally tractable due to stringent requirements imposed by the microscopic systems encoding the qubit, including complex energy level structures and unwanted transitions, and generally require a trade-off between speed and fidelity of the operation. Here, we address these challenges by developing a general framework for optimal control based on the quantum metric tensor. This framework allows for fast and high-fidelity adiabatic pulses, even for a dense energy spectrum, based solely on the Hamiltonian of the system instead of the time evolution propagator and independent of the size of the underlying Hilbert space. Furthermore, our framework suppresses diabatic transitions and state-dependent crosstalk effects without the need for additional control fields. As an example, we study the adiabatic charge transfer in a double quantum dot to find optimal control pulses with improved performance. We show that for the geometric protocol, the transfer fidelities are lower bounded F>99%\mathcal{F}>99\%F>99% for ultrafast $20~\mbox{ns}$ pulses, regardless of the size of the anti-crossing, while being robust against miscalibration errors and quasistatic noise.
We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.
Stimulated Raman adiabatic passage is a prominent and widely applied technique that transfers population between quantum states in three-level quantum systems. In this paper, we propose an optimized version of stimulated Raman adiabatic passage, which can be implemented faster than the conventional version. In order to achieve quantum adiabaticity in shorter times, we use the geometric optimization method based on the dynamical quantum geometric tensor to regularize nonadiabatic transitions and to eliminate nonadiabatic effects during the evolution process. Optimal stimulated Raman adiabatic passage can shorten the operation time required for adiabatic population transfer appreciably, thereby avoiding decoherence resulting from long-time evolution. Moreover, optimal stimulated Raman adiabatic passage is robust against systematic and random errors. Our results provide an optimal route for the realization of fast and accurate population transfer in three-level quantum systems.