The quantum speed limit (QSL), or the energy-time uncertainty relation, describes the fundamental maximum rate for quantum time evolution and has been regarded as being unique in quantum mechanics. In this study, we obtain a classical speed limit corresponding to the QSL using the Hilbert space for the classical Liouville equation. Thus, classical mechanics has a fundamental speed limit, and QSL is not a purely quantum phenomenon but a universal dynamical property of the Hilbert space. Furthermore, we obtain similar speed limits for the imaginary-time Schrödinger equations such as the master equation.Introduction.-Noncommutativity is one of the most important components of quantum mechanics. The Heisenberg uncertainty principle [1] stems from the canonical commutation relations [2]. Because this consequence cannot appear in classical systems, Heisenbergs uncertainty principle is a purely quantum phenomenon. The product of energy and time has the same dimensions as the product of position and momentum, which naively implies the existence of a similar relation between energy and time. However, the time operator, which satisfies the canonical commutation relations for the Hamiltonian, does not exist in a realistic model [3], and thus, there is no energy-time uncertainty principle that strictly corresponds to Heisenberg's uncertainty principle. Properly formulating the uncertainty relation for energy and time is a delicate issue that is still being discussed [4,5].The first rigorous derivation of an analogous uncertainty principle for energy and time was given by Mandelstam and Tamm [6] in which they determined that the product of the energy variance and time required for a state to be orthogonal to its initial state was greater than Planck's constant. This result implied that quantum mechanics has a fundamental speed limit characterized by Planck's constant, and thus, this inequality is called the energy-time uncertainty relation or quantum speed limit (QSL). The quantum speed limit can be also regarded as a trade-off between energy and time in the variance of a state. Investigating the restrictions on the time evolution of quantum dynamics is an interesting and important problem, and there are many related works: an alternative quantum speed limit [7], the shortest time for quantum computation [8], cases on mixed states [9], time-dependent systems [10][11][12], and open systems [13][14][15], geometric derivations of the QSL [16][17][18][19][20], and various applications [21][22][23][24][25][26][27][28].Note that the QSL is a strictly different concept than Heisenberg's uncertainty principle. Nevertheless, since QSL appears in a similar context to Heisenberg's uncertainty principle, QSL has been considered a purely quantum phenomenon with no corresponding concept in classical mechanics. Recent studies [12,[29][30][31] have im-

Using shortcuts to adiabaticity, we solve the time-dependent Schrödinger equation that is reduced to a classical nonlinear integrable equation. For a given time-dependent Hamiltonian, the counterdiabatic term is introduced to prevent nonadiabatic transitions. Using the fact that the equation for the dynamical invariant is equivalent to the Lax equation in nonlinear integrable systems, we obtain the counterdiabatic term exactly. The counterdiabatic term is available when the corresponding Lax pair exists and the solvable systems are classified in a unified and systematic way. Multisoliton potentials obtained from the Korteweg-de Vries equation and isotropic XY spin chains from the Toda equations are studied in detail. Introduction.-Ideal control of quantum systems has attracted interest recently from both theoretical and practical viewpoints. Rapid technological advances make it possible to manipulate quantum systems precisely, and designing the optimal Hamiltonian is a realistic important problem. The meaning of optimality is not obvious, and various methods have been proposed theoretically in different contexts. In the methods using shortcuts to adiabaticity, the Hamiltonian is designed so that the state follows an adiabatic passage of a reference Hamiltonian [1][2][3][4][5]. This technique was realized in several experiments [6][7][8][9] and is expected to be applied to the adiabatic quantum computation called quantum annealing [10].

We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.

We formulate the theory of shortcuts to adiabaticity in classical mechanics. For a reference Hamiltonian, the counterdiabatic term is constructed from the dispersionless Korteweg-de Vries (KdV) hierarchy. Then the adiabatic theorem holds exactly for an arbitrary choice of time-dependent parameters. We use the Hamilton-Jacobi theory to define the generalized action. The action is independent of the history of the parameters and is directly related to the adiabatic invariant. The dispersionless KdV hierarchy is obtained from the classical limit of the KdV hierarchy for the quantum shortcuts to adiabaticity. This correspondence suggests some relation between the quantum and classical adiabatic theorems.Shortcuts to adiabaticity (STA) is a method controlling dynamical systems. The implementation of the method results in dynamics that are free from nonadiabatic transitions for an arbitrary choice of timedependent parameters in a reference Hamiltonian. It was developed in quantum systems [1][2][3][4] and its applications have been studied in various fields of physics and engineering [5]. It is important to notice that this method, decomposing the Hamiltonian into the reference term and the counterdiabatic term, is applied to any dynamical systems and offers a novel insight into the systems.It is an interesting problem to find the corresponding method in classical mechanics from several points of view as we discuss in the following. Jarzynski studied STA for the classical system by using the adiabatic invariant [6]. He found the form of the counterdiabatic term in a certain system based on a generator of the adiabatic transport. Although several applications have been discussed [7][8][9], the complete formulation of the classical STA is still under investigation.In the quantum system, the adiabatic theorem is described by the adiabatic state constructed from the instantaneous eigenstate of a reference Hamiltonian H 0 . When the time-dependence of the parameters in the Hamiltonian is weak, the solution of the Schrödinger equation can be approximated by the adiabatic state. On the other hand, the adiabatic theorem in the classical system is described by the phase volume defined in periodic systems. The closed trajectory in phase space for a fixed parameter gives the adiabatic invariantwhere E 0 denotes the instantaneous energy. J is defined instantaneously and the adiabatic theorem states that J is approximately conserved when the parameter change is slow. Thus the quantum and classical adiabatic theorems look very different and the relation between them is not obvious. The quantum STA is introduced so that the quantum adiabatic theorem holds exactly and we expect that the same holds for the classical case. These two formulations will allow us to make a link between two theorems. In this letter, we develop the theory of the classical STA. First, we formulate the classical STA in a general way so that the counterdiabatic term can be calculated, in principle, from the derived formula. Second, we show that the adiabatic the...

We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue ‘Quantum annealing and computation: challenges and perspectives’.