arXiv subjects

Stefan Boettcher

Publications and source records attributed to Stefan Boettcher.

At least 37 records · Page 2Linked to original sources

arXiv2018

Complexity Bounds on Quantum Search Algorithms in finite-dimensional Networks

We establish a lower bound concerning the computational complexity of Grover's algorithms on fractal networks. This bound provides general predictions for the quantum advantage gained for searching unstructured lists. It yields a fundamental criterion, derived from quantum transport properties, for the improvement a quantum search algorithm achieves over the corresponding classical search in a network based solely on its spectral dimension, $d_{s}$. Our analysis employs recent advances in the interpretation of the venerable real-space renormalization group (RG) as applied to quantum walks. It clarifies the competition between Grover's abstract algorithm, i.e., a rotation in Hilbert space, and quantum transport in an actual geometry. The latter is characterized in terms of the quantum walk dimension $d_{w}^{Q}$ and the spatial (fractal) dimension $d_{f}$ that is summarized simply by the spectral dimension of the network. The analysis simultaneously determines the optimal time for a quantum measurement and the probability for successfully pin-pointing a marked element in the network. The RG further encompasses an optimization scheme devised by Tulsi that allows to tune this probability to certainty, leaving quantum transport as the only limiting process. It considers entire families of problems to be studied, thereby establishing large universality classes for quantum search, which we verify with extensive simulations. The methods we develop could point the way towards systematic studies of universality classes in computational complexity to enable modification and control of search behavior.

cond-mat.stat-mech↗

arXiv2018

Staggered Quantum Walk on Hexagonal Lattices

A discrete-time staggered quantum walk was recently introduced as a generalization that allows to unify other versions, such as the coined and Szegedy's walk. However, it also produces new forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.

quant-ph↗

arXiv2018

Mesoscopic real space structures in aging spin-glasses: the Edwards-Anderson model

Isothermal simulational data for the 3D Edwards-Anderson spin glass are collected at several temperatures below $T_{\rm c}$ and, in analogy with a recent model of dense colloidal suspensions,interpreted in terms of clusters of contiguous spins overturned by quakes, non-equilibrium events linked to record sized energy fluctuations. We show numerically that, to a good approximation, these quakes are statistically independent and constitute a Poisson process whose average grows logarithmically in time. The overturned clusters are local projections on one of the two ground states of the model, and grow likewise logarithmically in time. Data collected at different temperatures $T$ can be collapsed by scaling them with $T^{1.75}$, a hitherto unnoticed feature of the E-A model, which we relate on the one hand to the geometry of configuration space and on the other to experimental memory and rejuvenation effects. The rate at which a cluster flips is shown to decrease exponentially with the size of the cluster, as recently assumed in a coarse grained model of dense colloidal dynamics. The evolving structure of clusters in real space is finally sssociated to the decay of the thermo-remanent magnetization. Our analysis provides an unconventional coarse-grained description of spin glass aging as statistically subordinated to a Poisson quaking process and highlights record dynamics as a viable common theoretical framework for aging in different systems.

cond-mat.stat-mech↗

arXiv2018

Two-Time Correlations for Probing the Aging Dynamics of Jammed Colloids

We present results for the aging dynamics of a jammed 2D colloidal system obtained with molecular dynamics simulations. We performed extensive simulations to gather detailed statistics about rare rearrangement events. With a simple criterion for identifying irreversible events based on Voronoi tessellations, we find that the rate of those events decelerates hyperbolically. We track the probability density function for particle displacements, the van-Hove function, with sufficient statistics as to reveal its two-time dependence that is indicative of aging. Those displacements, measured from a waiting time $t_{w}$ after the quench up to times $t=t_{w}+Δt$, exhibit a data collapse as a function of $Δt/t_{w}$. These findings can be explained comprehensively as manifestations of "record dynamics", i.e., a relaxation dynamic driven by record-breaking fluctuations. We show that an on-lattice model of a colloid that was built on record dynamics indeed reproduces the experimental results in great detail.

cond-mat.soft↗

arXiv2017

Renormalization of Discrete-Time Quantum Walks with non-Grover Coins

We present an in-depth analytic study of discrete-time quantum walks driven by a non-reflective coin. Specifically, we compare the properties of the widely-used Grover coin ${\cal C}_{G}$ that is unitary and reflective (${\cal C}_{G}^{2}=\mathbb{I}$) with those of a $3\times3$ "rotational" coin ${\cal C}_{60}$ that is unitary but non-reflective (${\cal C}_{60}^{2}\not=\mathbb{I}$) and satisfies instead ${\cal C}_{60}^{6}=\mathbb{I}$, which corresponds to a rotation by $60^{\circ}$. While such a modification apparently changes the real-space renormalization group (RG) treatment, we show that nonetheless this non-reflective quantum walk remains in the same universality class as the Grover walk. We first demonstrate the procedure with ${\cal C}_{60}$ for a 3-state quantum walk on a one-dimensional (\emph{1d}) line, where we can solve the RG-recursions in closed form, in the process providing exact solutions for some difficult non-linear recursions. Then, we apply the procedure to a quantum walk on a dual Sierpinski gasket (DSG), for which we reproduce ultimately the same results found for ${\cal C}_{G}$, further demonstrating the robustness of the universality class.

quant-ph↗

arXiv2017

Analysis of Coined Quantum Walks with Renormalization

We introduce a new framework to analyze quantum algorithms with the renormalization group (RG). To this end, we present a detailed analysis of the real-space RG for discrete-time quantum walks on fractal networks and show how deep insights into the analytic structure as well as generic results about the long-time behavior can be extracted. The RG-flow for such a walk on a dual Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained explicitly from elementary algebraic manipulations, after transforming the unitary evolution equation into Laplace space. Unlike for classical random walks, we find that the long-time asymptotics for the quantum walk requires consideration of a diverging number of Laplace-poles, which we demonstrate exactly for the closed form solution available for the walk on a 1d-loop. In particular, we calculate the probability of the walk to overlap with its starting position, which oscillates with a period that scales as $N^{d_{w}^{Q}/d_{f}}$ with system size $N$. While the largest Jacobian eigenvalue $λ_{1}$ of the RG-flow merely reproduces the fractal dimension, $d_{f}=\log_{2}λ_{1}$, the asymptotic analysis shows that the second Jacobian eigenvalue $λ_{2}$ becomes essential to determine the dimension of the quantum walk via $d_{w}^{Q}=\log_{2}\sqrt{λ_{1}λ_{2}}$. We trace this fact to delicate cancellations caused by unitarity. We obtain identical relations for other networks, although the details of the RG-analysis may exhibit surprisingly distinct features. Thus, our conclusions -- which trivially reproduce those for regular lattices with translational invariance with $d_{f}=d$ and $d_{w}^{Q}=1$ -- appear to be quite general and likely apply to networks beyond those studied here.

quant-ph↗

arXiv2017

Evaluation of Spectral Zeta-Functions with the Renormalization Group

We evaluate spectral zeta-functions of certain network Laplacians that can be treated exactly with the renormalization group. As specific examples we consider a class of Hanoi networks and those hierarchical networks obtained by the Migdal-Kadanoff bond moving scheme from regular lattices. As possible applications of these results we mention quantum search algorithms as well as synchronization, which we discuss in more detail.

math-ph↗

arXiv2017

Renormalization of the Unitary Evolution Equation for Coined Quantum Walks

We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep insights into the relation between classical and quantum walks, its analytic basis has remained obscure. Our discussion here is laying the groundwork for a rigorous implementation of the RG for this important class of transport and algorithmic problems, although some instances remain unresolved. Specifically, we find that the RG fixed-point analysis of the classical walk, which typically focuses on the dominant Jacobian eigenvalue $λ_{1}$, with walk dimension $d_{w}^{RW}=\log_{2}λ_{1}$, needs to be extended to include the subdominant eigenvalue $λ_{2}$, such that the dimension of the quantum walk obtains $d_{w}^{QW}=\log_{2}\sqrt{λ_{1}λ_{2}}$. With that extension, we obtain analytically previously conjectured results for $d_{w}^{QW}$ of Grover walks on all but one of the fractal networks that have been considered.

quant-ph↗

arXiv2016

Record Dynamics: Direct Experimental Evidence from Jammed Colloids

In a broad class of complex materials a quench leads to a multi-scaled relaxation process known as aging. To explain its commonality and the astounding insensitivity to most microscopic details, record dynamics (RD) posits that a small set of increasingly rare and irreversible events, so called quakes, controls the dynamics. While key predictions of RD are known to concur with a number of experimental and simulational results, its basic assumption on the nature of quake statistics has proven extremely difficult to verify experimentally. The careful distinction of rare ("record") cage-breaking events from in-cage rattle accomplished in previous experiments on jammed colloids, enables us to extract the first direct experimental evidence for the fundamental hypothesis of RD that the rate of quakes decelerates with the inverse of the system age. The resulting description shows the predicted growth of the particle mean square displacement and of a mesoscopic lengthscale with the logarithm of time.

cond-mat.soft↗

arXiv2016

Continuous Time Quantum Walks in finite Dimensions

We consider the quantum search problem with a continuous time quantum walk for networks of finite spectral dimension d_{s} of the network Laplacian. For general networks of fractal (integer or non-integer) dimension d_{f}, for which in general d_{f}\not=d_{s}, it suggests that d_{s} is the scaling exponent that determines the computational complexity of the search. Our results are consistent with those of Childs and Goldstone [Phys. Rev. A 70 (2004), 022314] for lattices of integer dimension, where d=d_{f}=d_{s}. For general fractals, we find that the Grover limit of quantum search can be obtained whenever d_{s}>4. This complements the recent discussion of mean-field (i.e., d_{s}\to\infty) networks by Chakraborty et al. [Phys. Rev. Lett. 116 (2016), 100501] showing that for all those networks spatial search by quantum walk is optimal.

quant-ph↗

arXiv2016

Record Dynamics in the Parking Lot Model

We present an analytical and numerical study of the parking lot model (PLM) of granular relaxation and make a connection to the aging dynamics of dense colloids. As we argue, the PLM is a Kinetically Constrained Model which features astronomically large equilibration times and displays a characteristic aging behavior on all observable time scales. The density of parked cars displays quasi-equilibrium Gaussian fluctuations interspersed by increasingly rare intermittent events, quakes, which can lead to an increase of the density to new record values. Defining active clusters as the shortest domains of parked cars which must be re-arranged to allow further insertions, we find that their typical length grows logarithmically with time for low enough temperatures and show how the number of active clusters on average gradually decreases as the system approaches equilibrium. We further characterize the aging process in terms of the statistics of the record sized fluctuations in the interstitial free volume which lead to quakes and show that quakes are uncorrelated and that they can be approximately described as a Poisson process in logarithmic time.

cond-mat.soft↗

arXiv2016

Continuous-Time Quantum Search on Balanced Trees

We examine the effect of network heterogeneity on the performance of quantum search algorithms. To this end, we study quantum search on a tree for the oracle Hamiltonian formulation employed by continuous-time quantum walks. We use analytical and numerical arguments to show that the exponent of the asymptotic running time $\sim N^β$ changes uniformly from $β=0.5$ to $β=1$ as the searched-for site is moved from the root of the tree towards the leaves. These results imply that the time complexity of the quantum search algorithm on a balanced tree is closely correlated with certain path-based centrality measures of the searched-for site.

quant-ph↗

arXiv2015

Jamming in Hierarchical Networks

We study the Biroli-Mezard model for lattice glasses on a number of hierarchical networks. These networks combine certain lattice-like features with a recursive structure that makes them suitable for exact renormalization group studies and provide an alternative to the mean-field approach. In our numerical simulations here, we first explore their equilibrium properties with the Wang-Landau algorithm. Then, we investigate their dynamical behavior using a grand-canonical annealing algorithm. We find that the dynamics readily falls out of equilibrium and jams in many of our networks with certain constraints on the neighborhood occupation imposed by the Biroli-Mezard model, even in cases where exact results indicate that no ideal glass transition exists. But while we find that time-scales for the jams diverge, our simulations cannot ascertain such a divergence for a packing fraction distinctly above random close packing. In cases where we allow hopping in our dynamical simulations, the jams on these networks generally disappear.

cond-mat.dis-nn↗

arXiv2015

Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks

We derive the determinant of the Laplacian for the Hanoi networks and use it to determine their number of spanning trees (or graph complexity) asymptotically. While spanning trees generally proliferate with increasing average degree, the results show that modifications within the basic patterns of design of these hierarchical networks can lead to significant variations in their complexity. To this end, we develop renormalization group methods to obtain recursion equations from which many spectral properties can be obtained. This provides the basis for future applications to explore the physics of several dynamic processes.

cond-mat.stat-mech↗

arXiv2015

Relation between Random Walks and Quantum Walks

Based on studies on four specific networks, we conjecture a general relation between the walk dimensions $d_{w}$ of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that $d_{w}$ of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggests that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $ρ(x,t)$ in space and time of this quantum walk equation exhibits a scaling collapse for a variable $x^{d_{w}}/t$ in the weak limit, which defines $d_{w}$ and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for $ρ(x,t)$ in each case with extensive numerical simulation. The exact values for $d_{w}$ in themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.

cond-mat.stat-mech↗

arXiv2015

Classification of critical phenomena in hierarchical small-world networks

A classification of critical behavior is provided in systems for which the renormalization group equations are control-parameter dependent. It describes phase transitions in networks with a recursive, hierarchical structure but appears to apply also to a wider class of systems, such as conformal field theories. Although these transitions generally do not exhibit universality, three distinct regimes of characteristic critical behavior can be discerned that combine an unusual mixture of finite- and infinite-order transitions. In the spirit of Landau's description of a phase transition, the problem can be reduced to the local analysis of a cubic recursion equation, here, for the renormalization group flow of some generalized coupling. Among other insights, this theory explains the often-noted prevalence of the so-called inverted Berezinskii-Kosterlitz-Thouless transitions in complex networks. As a demonstration, a one-parameter family of Ising models on hierarchical networks is considered.

cond-mat.stat-mech↗

arXiv2015

Moments of Coinless Quantum Walks on Lattices

The properties of the coinless quantum walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than random-walk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on $d$-dimensional lattices. The mean square displacement for large times is explicitly calculated for the one- and two-dimensional lattices and, using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.

quant-ph↗

arXiv2014

From explosive to infinite-order transitions on a hyperbolic network

We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous ("explosive") transition in ordinary percolation. To this end, we solve the $q$-state Potts model in the analytic continuation for non-integer $q$ with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in $q$ shows that the discontinuous transition is generic in the regime $q<2$ that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, $q=2$. For $q>2$ the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless (BKT) type.

cond-mat.stat-mech↗