Transitionless quantum driving is a useful approach to mimic the adiabatic evolution of a quantum system so that the system can rapidly evolve from its initial state to the target state without transitions between instantaneous eigenstates of the reference Hamiltonian. To quantify the energetic cost in the transitionless quantum driving process, the instantaneous cost is defined as the Frobenius norm of the driving Hamiltonian and the integral of instantaneous cost is used to describe the total cost in the entire evolution. In this paper, we find that the minimal integral of instantaneous cost has a geometric property being equal to the length of the evolution path on the Riemannian manifold spanned by the control parameters with a positive definite metric, but independent of the evolution details such as the changing rate of the parameters. Based on this property, we further show that the optimal transitionless quantum driving with the minimum total cost corresponds to the geodesic path on the Riemannian manifold, which provides a method for optimizing transitionless quantum driving.
Transitionless quantum driving is a useful approach to mimic the adiabatic evolution of a quantum system so that the system can rapidly evolve from its initial state to the target state without transitions between instantaneous eigenstates of the reference Hamiltonian. To quantify the energetic cost in the transitionless quantum driving process, the instantaneous cost is defined as the Frobenius norm of the driving Hamiltonian and the integral of instantaneous cost is used to describe the total cost in the entire evolution. In this paper, we find that the minimal integral of instantaneous cost has a geometric property being equal to the length of the evolution path on the Riemannian manifold spanned by the control parameters with a positive definite metric, but independent of the evolution details such as the changing rate of the parameters. Based on this property, we further show that the optimal transitionless quantum driving with the minimum total cost corresponds to the geodesic path on the Riemannian manifold, which provides a method for optimizing transitionless quantum driving.
We investigate the optimization of quantum control from a differential geometric perspective. In our approach, optimal control minimizes the cost associated with evolving a quantum state, with the cost quantified by the length of the trajectory on a relevant Riemannian manifold. We demonstrate the optimization protocol in systems with SU(2) and SU(1,1) dynamical symmetries, which encompass a broad range of physical systems. In these systems, the time evolution can be represented by trajectories on a three-dimensional manifold. Given the initial and final states, the minimum-cost quantum control corresponds to a geodesic on the manifold. When the trajectory between the initial and final states is specified, the minimum-cost control corresponds to a geodesic within a submanifold embedded in the three-dimensional space. This framework provides a geometric method for optimizing shortcuts to adiabatic driving.
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.
The quantum adiabatic theorem conventionally posits that a quantum system initialized in an instantaneous eigenstate of a time-dependent Hamiltonian will remain in that corresponding instantaneous eigenstate, provided the parameter changes are sufficiently slow and no level crossings occur. Under this idealized framework, standard quantum mechanics neglects any intrinsic energetic cost associated merely with traversing a trajectory through the projective Hilbert space or the underlying parameter manifold, assuming that adiabatic evolution is inherently frictionless and dissipation-free in closed systems. In this work, we challenge this traditional paradigm by introducing the concept of Quantum-State Path Friction (QSPF). We propose that even under strictly ideal adiabatic conditions, an irreducible energetic penalty exists, which scales quadratically with the velocity of the quantum state along its geometric path. We formally define the path work as W_path = \int \eta_Q (ds_Q / dt)^2 dt, where ds_Q^2 = g_{ij} d\lambda_i d\lambda_j denotes the quantum geometric line element governed by the quantum metric tensor g_{ij}, and \eta_Q represents the quantum friction coefficient. This formulation establishes a foundational framework for a new class of dissipation laws in quantum dynamics, bridging quantum information geometry, non-equilibrium thermodynamics, and optimal quantum control.
Transitionless quantum driving, also known as counterdiabatic driving, is a unique shortcut technique to adiabaticity, enabling a fast-forward evolution to the same target quantum states as those in the adiabatic case. However, as nothing is free, the fast evolution is obtained at the cost of stronger driving fields. Here, given the system initially gets prepared in equilibrium states, we construct relations between the dynamical evolution speed and the cost rate of transitionless quantum driving in two scenarios: one that preserves the transitionless evolution for a single energy eigenstate (individual driving), and the other that maintains all energy eigenstates evolving transitionlessly (collective driving). Remarkably, we find that individual driving may cost as much as collective driving, in contrast to the common belief that individual driving is more economical than collective driving in multilevel systems. We then present a potentially practical proposal to demonstrate the above phenomena in a three-level Landau-Zener model using the electronic spin system of a single nitrogen-vacancy center in diamond.
Many years have passed since the conception of the quintessential method of shortcut to adiabaticity known as counterdiabatic driving (or transitionless quantum driving). Yet, this method appears to be energetically cost-free and thus continually challenges the task of quantifying the amount of energy it demands to be accomplished. This paper proposes that the energy cost of controlling a closed quantum system using the counterdiabatic method can also be assessed using the instantaneous excess work during the process and related quantities, as the time-averaged excess work. Starting from the Mandelstam-Tamm bound for driven dynamics, we have shown that the speed-up of counterdiabatic driving is linked with the spreading of energy between the eigenstates of the total Hamiltonian, which is necessarily accompanied by transitions between these eigenstates. Nonetheless, although excess work can be used to quantify energetically these transitions, it is well known that the excess work is zero throughout the entire process under counterdiabatic driving. To recover the excess work as an energetic cost quantifier for counterdiabatic driving, we will propose a different interpretation of the parameters of the counterdiabatic Hamiltonian, leading to an excess work different from zero. We have illustrated our findings with the Landau-Zener model.
We present an information geometric characterization of quantum driving schemes specified by su(2;C) time-dependent Hamiltonians in terms of both complexity and efficiency concepts. Specifically, starting from pure output quantum states describing the evolution of a spin-1/2 particle in an external time-dependent magnetic field, we consider the probability paths emerging from the parametrized squared probability amplitudes of quantum origin. The information manifold of such paths is equipped with a Riemannian metrization specified by the Fisher information evaluated along the parametrized squared probability amplitudes. By employing a minimum action principle, the optimum path connecting initial and final states on the manifold in finite time is the geodesic path between the two states. In particular, the total entropy production that occurs during the transfer is minimized along these optimum paths. For each optimum path that emerges from the given quantum driving scheme, we evaluate the so-called information geometric complexity (IGC) and our newly proposed measure of entropic efficiency constructed in terms of the constant entropy production rates that specify the entropy minimizing paths being compared. From our analytical estimates of complexity and efficiency, we provide a relative ranking among the driving schemes being investigated. Moreover, we determine that the efficiency and the temporal rate of change of the IGC are monotonic decreasing and increasing functions, respectively, of the constant entropic speed along these optimum paths. Then, after discussing the connection between thermodynamic length and IGC in the physical scenarios being analyzed, we briefly examine the link between IGC and entropy production rate. Finally, we conclude by commenting on the fact that an higher entropic speed in quantum transfer processes seems to necessarily go along with a lower entropic efficiency together with a higher information geometric complexity.