We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.

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We study the entanglement spectrum of highly excited eigenstates of two known models which exhibit a many-body localization transition, namely the one-dimensional random-field Heisenberg model and the quantum random energy model. Our results indicate that the entanglement spectrum shows a "two-component" structure: a universal part that is associated to Random Matrix Theory, and a non-universal part that is model dependent. The non-universal part manifests the deviation of the highly excited eigenstate from a true random state even in the thermalized phase where the Eigenstate Thermalization Hypothesis holds. The fraction of the spectrum containing the universal part decreases as one approaches the critical point and vanishes in the localized phase in the thermodynamic limit. We use the universal part fraction to construct an order parameter for measuring the degree of randomness of a generic highly excited state, which is also a promising candidate for studying the many-body localization transition. Two toy models based on RokhsarKivelson type wavefunctions are constructed and their entanglement spectra are shown to exhibit the same structure. Introduction-Quantum entanglement, a topic of much importance in quantum information theory, has also gained relevance in quantum many-body physics in the past few years [1,2]. In particular, the entanglement entropy provides a wealth of information about physical states, including novel ways to classify states of matter which do not have a local order parameter [3]. However, it has been realized only recently in various physical contexts that the entanglement entropy is not enough to fully characterize a generic quantum state. For example, the quantum complexity corresponding to the geometric structure of black holes cannot be fully encoded just by the entanglement entropy [4]. One natural step beyond the amount of entanglement is the specific pattern of entanglement, i.e., the entanglement spectrum. A recent result which motivates this direction is the relationship between irreversibility and entanglement spectrum statistics in quantum circuits [5,6]. It was shown that irreversible states display Wigner-Dyson statistics in the level spacing of entanglement eigenvalues, while reversible states show a deviation from Wigner-Dyson distributed entanglement levels and can be efficiently disentangled.

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Entanglement is usually quantified by von Neumann entropy, but its properties are much more complex than what can be expressed with a single number. We show that the three distinct dynamical phases known as thermalization, Anderson localization, and many-body localization are marked by different patterns of the spectrum of the reduced density matrix for a state evolved after a quantum quench. While the entanglement spectrum displays Poisson statistics for the case of Anderson localization, it displays universal Wigner-Dyson statistics for both the cases of many-body localization and thermalization, albeit the universal distribution is asymptotically reached within very different time scales in these two cases. We further show that the complexity of entanglement, revealed by the possibility of disentangling the state through a Metropolis-like algorithm, is signaled by whether the entanglement spectrum level spacing is Poisson or Wigner-Dyson distributed.Introduction.-Entanglement is usually quantified by a number, the entanglement entropy, defined as the von Neumann entropy of the reduced density matrix ρ A of a subsystem, and it is a key concept in many different physical settings, from novel phases of quantum matter [1-4] to cosmology [5,6]. However, there is a lot more information in the entanglement spectrum of ρ A , namely the full set of its eigenvalues (or its logarithms) [7]. Recently, a measurement protocol to access the entanglement spectrum of many-body states using cold atoms has been proposed [8]. The main goal of this letter is to explore the relationship between entanglement spectrum and dynamical behavior of a quantum many-body system.In Refs. [9,10] it was shown that the entanglement of a state generated by a quantum circuit can be simple or complex, in the sense that the state either can or cannot be disentangled by an entanglement cooling algorithm that resembles the Metropolis algorithm for finding the ground state of a Hamiltonian. The success or failure of the disentangling procedure is signaled by the so called entanglement spectrum statistics (ESS) [9,10], namely the distribution of the spacings between consecutive eigenvalues of ρ A . When such a distribution is Wigner-Dyson (WD), the cooling algorithm fails. This situation occurs when the gates in the circuit are sufficient for universal computing, either classical or quantum. On the other hand, for circuits that are not capable of universal computing, the states can be disentangled and they feature a (semi-)Poisson ESS.In this letter, we focus on systems whose dynamics is controlled by a time-independent quantum many-body Hamiltonian, as opposed to a random circuit. We study the entanglement complexity revealed by the ESS of the time-evolved state for Hamiltonians whose eigenstates yield one of three behaviors: 1) eigenstate thermalization (ETH) [11][12][13][14][15][16], 2) Anderson localization (AL), or 3) many-body localization (MBL) [17][18][19]. We find that the time-evolved states under Hamiltonians that feature AL follow a Poisson...

where k n = 2π L (n + 1/2) with n = 0, 1, . . . , L/2 − 1.

This paper is concerned with classical solutions to the interaction of two arbitrary planar rarefaction waves for the self-similar Euler equations in two space dimensions. We develop the direct approach, started in Chen and Zheng (in press) [3], to the problem to recover all the properties of the solutions obtained via the hodograph transformation of Li and Zheng (2009) [14]. The direct approach, as opposed to the hodograph transformation, is straightforward and avoids the common difficulties of the hodograph transformation associated with simple waves and boundaries. The approach is made up of various characteristic decompositions of the self-similar Euler equations for the speed of sound and inclination angles of characteristics.

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