Speeding up critical system dynamics through optimized evolution

Abstract

The number of defects which are generated on crossing a quantum phase transition can be minimized by choosing properly designed time-dependent pulses. In this work we determine what are the ultimate limits of this optimization. We discuss under which conditions the production of defects across the phase transition is vanishing small. Furthermore we show that the minimum time required to enter this regime is T ∼ π/∆, where ∆ is the minimum spectral gap, unveiling an intimate connection between an optimized unitary dynamics and the intrinsic measure of the Hilbert space for pure states. Surprisingly, the dynamics is non-adiabatic, this result can be understood by assuming a simple two-level dynamics for the many-body system. Finally we classify the possible dynamical regimes in terms of the action s = T ∆. PACS numbers:The rapid progress in the experimental realization and manipulation of quantum systems [1] is opening the rich and intriguing perspective of the exploitation of quantum physics to realize quantum technologies like quantum simulators [2] and quantum computers [3,4]. These achievements pave the way to the simulation of condensed matter systems giving the possibility of studying different states of matter in controlled experiments [5]. Despite the impressive results obtained so far, this is a formidable technological and theoretical challenge due to the complexity of the systems in analysis and the experimental requirements. Indeed, the level of control needed on the quantum system is unprecedented: one should be able to prepare a system in a desired initial state, perform the desired evolution and finally measure the state in a very precise way. Moreover, the whole experiment should be performed faster than the system decoherence time that eventually will destroy any quantum information capability. Quantum optimal control (OC) theory, the study of optimization strategies to improve the outcome of a quantum process, can be an extremely powerful tool to cope with these issues [6-10]. It allows not only to optimize the desired experiment outcome but also to speed up the process itself. Traditionally employed in atomic and molecular physics [11,12], OC has been recently applied with success to the optimization of the dynamics of many-body systems [13][14][15], allowing to achieve the ultimate bound imposed by quantum mechanics, the so called quantum speed limit (QSL) [16]. Indeed as intuitively suggested by the time-energy uncertainty principle, the time required by a state to reach another distinguishable state has to be longer than the inverse of its energy fluctuations [17]. This implies that a quantum system cannot evolve at an arbitrary speed in its Hilbert space, but a minimum time is required to perform a transformation between orthogonal states [18][19][20][21][22]. For time-independent Hamiltonians this bound has been exactly determined [16]; the QSL has been formally generalized also to time-dependent Hamiltonians, but so far has been computed only in a few simple cases [13,[23][24][25][26]. A s...

Exploring the quantum speed limit with computer games

“…This has previously been associated with a universal-type sin 2 ( π 2 T /T QSL ) behavior [8]. Here we do not observe this behavior (see Extended Data Fig.…”

“…To better understand this, we have made a low-dimensional rendering of the optimization landscape. These studies show why traditional optimization methods fail near the quantum speed limit [7,8,9], and they bring promise that combined analyses of optimization land- Quantum physics holds the potential of unprecedented technological advances in the realms of computing [10] and simulations [11]. To ensure functionality, all quantum operations must be executed near perfection, requiring highly optimized operations with fidelities above F ≥ 0.999 [12].…”

“…With limited duration, the quantum optimization problem loses the favourable properties stated earlier and local optimizers are not almost guaranteed to converge. In this case the prevalent and hitherto successful paradigm is using multistart of such local algorithms [15,16,8]. For a time-dependent Hamiltonians the QSL is conventionally computed by assuming that it coincides with the process duration for which the multistarting of local optimization fails [16,8].…”

“…[20,25]. We believe our gamification strategy is extendable to these problems and other physical interactions including, but not limited to, many-body dynamics in optical lattices [26] and Bose-Einstein condensates [25].Recall that the standard approach for solving a problem such as BHW is to use a multistart of tailored local optimization algorithms, such as the Krotov algorithm (KA) [14,16,8]. The choice of an initial seed is a central challenge in such complex optimizations.…”

“…Recall that the standard approach for solving a problem such as BHW is to use a multistart of tailored local optimization algorithms, such as the Krotov algorithm (KA) [14,16,8]. The choice of an initial seed is a central challenge in such complex optimizations.…”