Physical consequences of P≠\neqNP and the DMRG-annealing conjecture

Computational complexity theory contains a corpus of theorems and conjectures regarding the time a Turing machine will need to solve certain types of problems as a function of the input size. Nature {\em need not} be a Turing machine and, thus, these theorems do not apply directly to it. But {\em classical simulations} of physical processes are programs running on Turing machines and, as such, are subject to them. In this work, computational complexity theory is applied to classical simulations of systems performing an adiabatic quantum computation (AQC), based on an annealed extension of the density matrix renormalization group (DMRG). We conjecture that the computational time required for those classical simulations is controlled solely by the {\em maximal entanglement} found during the process. Thus, lower bounds on the growth of entanglement with the system size can be provided. In some cases, quantum phase transitions can be predicted to take place in certain inhomogeneous systems. Concretely, physical conclusions are drawn from the assumption that the complexity classes {\bf P} and {\bf NP} differ. As a by-product, an alternative measure of entanglement is proposed which, via Chebyshev's inequality, allows to establish strict bounds on the required computational time.Comment: Accepted for publication in JSTA

Power accretion in social systems

We consider a model of power distribution in a social system where a set of agents plays a simple game on a graph: The probability of winning each round is proportional to the agent’s current power, and the winner gets more power as a result. We show that when the agents are distributed on simple one-dimensional and two-dimensional networks, inequality grows naturally up to a certain stationary value characterized by a clear division between a higher and a lower class of agents. High class agents are separated by one or several lower class agents which serve as a geometrical barrier preventing further flow of power between them. Moreover, we consider the effect of redistributive mechanisms, such as proportional (nonprogressive) taxation. Sufficient taxation will induce a sharp transition towards a more equal society, and we argue that the critical taxation level is uniquely determined by the system geometry. Interestingly, we find that the roughness and Shannon entropy of the power distributions are a very useful complement to the standard measures of inequality, such as the Gini index and the Lorenz curveWe acknowledge financial support from the Spanish Government through Grants No. FIS2015-69167-C2-1-P, No. FIS2015-66020-C2- 1-P, and No. PGC2018-094763-B-I0

Building an adiabatic quantum computer simulation in the classroom

We present a didactic introduction to adiabatic quantum computation (AQC) via the explicit construction of a classical simulator of quantum computers. This constitutes a suitable route to introduce several important concepts for advanced undergraduates in physics: quantum many-body systems, quantum phase transitions, disordered systems, spin-glasses, and computational complexity theory. (C) 2018 American Association of Physics Teachers.The authors want to acknowledge the faculty and students of the Facultad de Informática of UCM (Madrid) for their kind invitation to deliver this crash course, particularly to I. Rodríguez-Laguna and N. Martí. The authors would also like to thank G. Sierra for very useful comments on the manuscript. This work was funded by the Spanish government through Grant Nos. FIS2015-69167-C2-1-P and FIS2015-66020-C2-1-

Circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer

We consider the Kardar-Parisi-Zhang equation for a circular interface in two dimensions, unconstrained by the standard small-slope and no-overhang approximations. Numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short-time regime with the interface fluctuations of a self-avoiding ring or two-dimensional vesicle, and a long-time regime governed by the Tracy-Widom distribution expected for this geometry. For small-noise amplitudes, scaling behavior is only of the latter type. Large noise is also seen to renormalize the bare physical parameters of the ring, akin to analogous parameter renormalization for equilibrium three-dimensional membranes. Our results bear particular importance on the relation between relevant universality classes of scale-invariant systems in two dimensions.We thank M. Castro, A. Celi, M. Nicoli, and T. Lagatta for very useful discussions. This work has been partially supported through Grant No. FIS2012-38866-C05-01 (MINECO, Spain)

Reconstruction of the second layer of Ag on Pt(111)

The reconstruction of an Ag monolayer on Ag/Pt(111) is analyzed theoretically, employing a vertically extended Frenkel-Kontorova model whose parameters are derived from density functional theory. Energy minimization is carried out using simulated quantum annealing techniques. Our results are compatible with the STM experiments, where a striped pattern is initially found which transforms into a triangular reconstruction upon annealing. In our model we recognize the first structure as a metastable state, while the second one is the true energy minimum

Entanglement in correlated random spin chains, RNA folding and kinetic roughening

Average block entanglement in the 1D XX-model with uncorrelated random couplings is known to grow as the logarithm of the block size, in similarity to conformal systems. In this work we study random spin chains whose couplings present long range correlations, generated as gaussian fields with a power-law spectral function. Ground states are always planar valence bond states, and their statistical ensembles are characterized in terms of their block entropy and their bond-length distribution, which follow power-laws. We conjecture the existence of a critical value for the spectral exponent, below which the system behavior is identical to the case of uncorrelated couplings. Above that critical value, the entanglement entropy violates the area law and grows as a power law of the block size, with an exponent which increases from zero to one. Interestingly, we show that XXZ models with positive anisotropy present the opposite behavior, and strong correlations in the couplings lead to lower entropies. Similar planar bond structures are also found in statistical models of RNA folding and kinetic roughening, and we trace an analogy between them and quantum valence bond states. Using an inverse renormalization procedure we determine the optimal spin-chain couplings which give rise to a given planar bond structure, and study the statistical properties of the couplings whose bond structures mimic those found in RNA folding.We would like to thank J Cuesta for insights into the statistical mechanics of RNAfolding, and F Iglói and Z Zimborás for useful remarks. This work was funded by grants FIS-2012-33642 and FIS-2012-38866-C05-1,from the Spanish government, QUITEMAD+S2013/ICE-2801 from the Madrid regional government and SEV-2012-0249 of the ‘Centro de Excelencia Severo Ochoa’ Programme

Topology and the Kardar-Parisi-Zhang universality class

We study the role of the topology of the background space on the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class. To do so, we study the growth of balls on disordered 2D manifolds with random Riemannian metrics, generated by introducing random perturbations to a base manifold. As base manifolds we consider cones of different aperture angles theta, including the limiting cases of a cylinder (theta = 0, which corresponds to an interface with periodic boundary conditions) and a plane (theta = pi/2, which corresponds to an interface with circular geometry). We obtain that in the former case the radial fluctuations of the ball boundaries approach the Tracy-Widom (TW) distribution of the largest eigenvalue of random matrices in the Gaussian orthogonal ensemble (TW-GOE), while on cones with any aperture angle theta not equal 0 fluctuations correspond to the TW- GUE distribution related with the Gaussian unitary ensemble. We provide a topological argument to justify the relevance of TW-GUE statistics for cones, and state a conjecture which relates the KPZ universality subclass with the background topology.The work of S.N.S., J.R.-L., and R.C. was funded by MINECO (Spain) Grants Nos. FIS2012-33642, FIS2012-38866-C05-01, and FIS2015-66020-C2-1-P. A.C. acknowledges financial support from the EU grants EQuaM (FP7/2007-2013 Grant No. 323714), OSYRIS (ERC-2013-AdG Grant No. 339106), SIQS (FP7-ICT-2011-9 No. 600645), QUIC (H2020-FETPROACT-2014 No. 641122), Spanish MINECO grants (Severo Ochoa SEV-2015-0522 and FOQUS FIS2013-46768-P), Generalitat de Catalunya (2014 SGR 874), and Fundació Cellex

Random geometry and the Kardar-Parisi-Zhang universality class

We consider a model of a quenched disordered geometry in which a random metric is defined on R-2, which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as R-x, with a fluctuation exponent x similar or equal to 1/3, while the lateral spread of the minimizing geodesic between two points at a distance L grows as L-zeta, with wandering exponent value zeta similar or equal to 2/3. Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang universality class of surface kinetic roughening, with. and. relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy-Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW-GUE statistics with good accuracy in arrival times.We want to acknowledge very useful discussions with K Takeuchi and S Ferreira. This work has been supported by the Spanish government (MINECO) through grant FIS2012-38866-C05-01. JR-L also acknowledges MINECO grants FIS2012-33642, TOQATA and ERC grant QUAGATUA. TLʼs research and travel was supported in part by NSF PIRE grant OISE-07-30136