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International Conference on Quantum Information

Fractional scaling of quantum walks on percolation lattices

Viv M. Kendon, Godfrey Leung, Joe Bailey, Paul Knott

Published Jan 1, 2011• 0 citations
Publisher

Abstract

Quantum walks have been used as simple models of quantum transport phenomena, applicable to systems as diverse as spin chains [1] and bio-molecules [2]. Here we investigate the properties of quantum walks on percolation lattices, disordered structures appropriate for modelling biological and experimentally realistic systems. Both bond (edge) and site percolation have similar definitions: with independent randomly chosen probability p the bond or site is present in the lattice. In two and higher dimensions, percolation lattices exhibit a phase transition from a set of small disconnected regions to a more highly connected structure (“one giant cluster”). On 2D Cartesian lattices, the critical probability pc = 0.5 (bond) and pc = 0.5927... (site). Below pc, the quantum walk will not be able to spread. Approaching pc from above, the spreading slows down completely, as the number of long-distance connected paths reduces to zero. For p = 1, the lattice is fully connected, and the standard quantum walk spreading applies (linear in t). In between, we find the quantum walks show fractional scaling of the spreading, i.e., proportional to tα for (0.5 < α < 1). The numerically determined scaling exponents α are shown in fig.

DOI
10.1364/icqi.2011.qmi29

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Quantum walks have been used as simple models of quantum transport phenomena, applicable to systems as diverse as spin chains [1] and bio-molecules [2]. Here we investigate the properties of quantum walks on percolation lattices, disordered structures appropriate for modelling biological and experimentally realistic systems. Both bond (edge) and site percolation have similar definitions: with independent randomly chosen probability p the bond or site is present in the lattice. In two and higher dimensions, percolation lattices exhibit a phase transition from a set of small disconnected regions to a more highly connected structure (“one giant cluster”). On 2D Cartesian lattices, the critical probability pc = 0.5 (bond) and pc = 0.5927... (site). Below pc, the quantum walk will not be able to spread. Approaching pc from above, the spreading slows down completely, as the number of long-distance connected paths reduces to zero. For p = 1, the lattice is fully connected, and the standard quantum walk spreading applies (linear in t). In between, we find the quantum walks show fractional scaling of the spreading, i.e., proportional to tα for (0.5 < α < 1). The numerically determined scaling exponents α are shown in fig.
Quantum walks can be used to model processes such as transport in spin chains and bio-molecules. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well-studied on regular structures and also shown to be sensitive to defects and imperfections. Using numerical simulation, we study the spreading properties of quantum walks on percolation lattices for both bond and site percolation. The randomly missing edges or sites provide a controlled amount of disorder in the regular Cartesian lattice. In one dimension (the line) we introduce a simple model of quantum tunneling to allow the walk to proceed past the missing edges or sites. This allows the quantum walk to spread faster than a classical random walk for short times, but at longer times the disorder localises the quantum walk. In two dimensions, we observe fractional scaling of the spreading with the number of steps of the walk. For percolation above the 85% level, we obtain faster spreading than classical random walks on the full lattice.
Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well studied on regular structures and also shown to be sensitive to defects and imperfections in the lattice. As a simple example of a disordered system, we consider percolation lattices, in which edges or sites are randomly missing, interrupting the progress of the quantum walk. We use numerical simulation to study the properties of coined quantum walks on these percolation lattices in one and two dimensions. In one dimension (the line), we introduce a simple notion of quantum tunnelling and determine how this affects the properties of the quantum walk as it spreads. On two-dimensional percolation lattices, we show how the spreading rate varies from linear in the number of steps down to zero as the percolation probability decreases towards the critical point. This provides an example of fractional scaling in quantum-walk dynamics.
Quantum percolation describes the problem of a quantum particle moving through a disordered system. While certain similarities to classical percolation exist, the quantum case has additional complexity due to the possibility of Anderson localisation. Here, we consider a directed discrete-time quantum walk as a model to study quantum percolation of a two-state particle on a two-dimensional lattice. Using numerical analysis we determine the fraction of connected edges required (transition point) in the lattice for the two-state particle to percolate with finite (non-zero) probability for three fundamental lattice geometries, finite square lattice, honeycomb lattice, and nanotube structure and show that it tends towards unity for increasing lattice sizes. To support the numerical results we also use a continuum approximation to analytically derive the expression for the percolation probability for the case of the square lattice and show that it agrees with the numerically obtained results for the discrete case. Beyond the fundamental interest to understand the dynamics of a two-state particle on a lattice (network) with disconnected vertices, our study has the potential to shed light on the transport dynamics in various quantum condensed matter systems and the construction of quantum information processing and communication protocols.
The study of the Ising model from a percolation perspective has played a significant role in the modern theory of critical phenomena. We consider the celebrated square-lattice Ising model and construct percolation clusters by placing bonds, with probability p, between any pair of parallel spins within an extended range beyond nearest neighbors. At the Ising criticality, we observe two percolation transitions as p increases: starting from a disordered phase with only small clusters, the percolation system enters into a stable critical phase that persists over a wide range p_{c_{1}}<p<p_{c_{2}}, and then develops a long-ranged percolation order with giant clusters for both up and down spins. At p_{c1} and for the stable critical phase, the critical behaviors agree well with those for the Fortuin-Kasteleyn random clusters and the spin domains of the Ising model, respectively. At p_{c2}, the fractal dimension of clusters and the scaling exponent along p direction are estimated as y_{h2}=1.9580(6) and y_{p2}=0.552(9), of which the exact values remain unknown. These findings reveal interesting geometric properties of the two-dimensional Ising model that has been studied for more than 100 years.
The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.
We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.