Friction in quantum thermodynamics results from fast driving schemes that generate nonadiabatic excitations.

We discuss the energetic cost of superadiabatic models of quantum computation. Specifically, we investigate the energy-time complementarity in general transitionless controlled evolutions and in shortcuts to the adiabatic quantum search over an unstructured list. We show that the additional energy resources required by superadiabaticity for arbitrary controlled evolutions can be minimized by using probabilistic dynamics, so that the optimal success probability is fixed by the choice of the evolution time. In the case of analog quantum search, we show that the superadiabatic approach induces a non-oracular counter-diabatic Hamiltonian, with the same energy-time complexity as equivalent adiabatic implementations. 1 arXiv:1603.07778v2 [quant-ph] 30 Sep 2016 Coulamy et al.The superadiabatic speedup is intrinsically connected with an increase of the energy resources demanded by the quantum computer [14,15], which in turn implies a rather versatile computational cost that is controlled by the energetic capacity available to the physical apparatus. Here we show that this energytime complementarity can be exploited in quantum information processing. First, we consider controlled evolutions (CE) as a mechanism to implement superadiabatic universal QC [14] which generalizes the original adiabatic approach introduced in Ref. [16]. We then show that, within the superadiabatic scenario, the energetic cost can be minimized by replacing the deterministic realization of quantum gates for probabilistic implementations based on a probability distribution of a binary random variable described by an angle parameter. By doing so, the energy expense can be minimized by adjusting the probability distribution, provided the choice of the evolution time of the computational process. Second, we analyze the effects of the energy-time complementarity in analog quantum search [20], where the oracular approach designed by the local adiabatic Grover algorithm is known to be optimal [21,22]. In this case, we show that the superadiabatic approach naturally requires an unphysical non-oracular counter-diabatic Hamiltonian, with the energy-time complexity equivalent to non-oracular adiabatic implementations.The paper is organized as follows. In Section 2, we describe the adiabatic implementation of quantum gates via CE and several adiabatic quantum search approaches. We then provide their superadiabatic versions and introduce the metric for energetic cost used in our work. In Section 3, we investigate the energy complexity of the superadiabatic realizations of both quantum gates via CE and analog quantum search. In particular, we consider the properties of the probabilistic model of QC through CE and the consequences of the energy-time complementarity for the search problem. In Section 4, we present our conclusions and future perspectives.

We investigate the excitation dynamics at a first-order quantum phase transition (QPT). More specifically, we consider the quench-induced QPT in the quantum search algorithm, which aims at finding out a marked element in an unstructured list. We begin by deriving the exact dynamics of the model, which is shown to obey a Riccati differential equation. Then, we discuss the probabilities of success by adopting either global or local adiabaticity strategies. Moreover, we determine the disturbance of the quantum criticality as a function of the system size. In particular, we show that the critical point exponentially converges to its thermodynamic limit even in a fast evolution regime, which is characterized by both entanglement QPT estimators and the Schmidt gap. The excitation pattern is manifested in terms of quantum domain walls separated by kinks. The kink density is then shown to follow an exponential scaling as a function of the evolution speed, which can be interpreted as a Kibble-Zurek mechanism for first-order QPTs.

We investigate the local quantum uncertainty (LQU) between a block of L qubits and one single qubit in a composite system of n qubits driven through a quantum phase transition (QPT). A first-order QPT is analytically considered through a Hamiltonian implementation of the quantum search. In the case of second-order QPTs, we consider the transverse-field Ising chain via a numerical analysis through density matrix renormalization group. For both cases, we compute the LQU for finite-sizes as a function of L and of the coupling parameter, analyzing its pronounced behavior at the QPT.