Quantum Walk Dynamics Research Articles

Statistical moments of quantum-walk dynamics reveal topological quantum transitions.

Many phenomena in solid-state physics can be understood in terms of their topological properties. Recently, controlled protocols of quantum walk (QW) are proving to be effective simulators of such phenomena. Here we report the realization of a photonic QW showing both the trivial and the non-trivial topologies associated with chiral symmetry in one-dimensional (1D) periodic systems. We find that the probability distribution moments of the walker position after many steps can be used as direct indicators of the topological quantum transition: while varying a control parameter that defines the system phase, these moments exhibit a slope discontinuity at the transition point. Numerical simulations strongly support the conjecture that these features are general of 1D topological systems. Extending this approach to higher dimensions, different topological classes, and other typologies of quantum phases may offer general instruments for investigating and experimentally detecting quantum transitions in such complex systems.

Non-Markovian continuous-time quantum walks on lattices with dynamical noise

We address the dynamics of continuous-time quantum walks on one-dimensional disordered lattices inducing dynamical noise in the system. Noise is described as time-dependent fluctuations of the tunneling amplitudes between adjacent sites, and attention is focused on non-Gaussian telegraph noise, going beyond the usual assumption of fast Gaussian noise. We observe the emergence of two different dynamical behaviors for the walker, corresponding to two opposite noise regimes: slow noise (i.e. strong coupling with the environment) confines the walker into few lattice nodes, while fast noise (weak coupling) induces a transition between quantum and classical diffusion over the lattice. A phase transition between the two dynamical regimes may be observed by tuning the ratio between the autocorrelation time of the noise and the coupling between the walker and the external environment generating the noise. We also address the non-Markovianity of the quantum map by assessing its memory effects, as well as evaluating the information backflow to the system. Our results suggest that the non-Markovian character of the evolution is linked to the dynamical behavior in the slow noise regime, and that fast noise induces a Markovian dynamics for the walker.

Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice

Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin systems. For two-dimensional quantum walks using a four-state coin system, however, the only known limit distribution is for a walk using a parameterized Grover coin operation and analytical complexities have been a major obstacle for obtaining long-time limit distributions for other coins. In this work however, we present two new types of long-time limit distributions for walks using different forms of coin-flip operations in a four-state coin system. This opens the road towards understanding the dynamics and asymptotic behaviour for higher state coin system from a mathematical view point.

Limit distributions for different forms of four-state quantum walks on a two-dimensional lattice

Long-time limit distributions are key quantities for understanding the asymptotic dynamics of quantum walks, and they are known for most forms of one-dimensional quantum walks using two-state coin ...

Entanglement dynamics of two-particle quantum walks

This paper explores the entanglement dynamics generated by interacting two-particle quantum walks on degree-regular and degree-irregular graphs. We performed spectral analysis of the time-evolution of both the particle probability distribution and the entanglement between the two particles for various interaction strength. While the particle probability distributions are stable and not sensitive to perturbations in the interaction strength, the entanglement dynamics are found to be much more sensitive to system variations. This property may be utilised to probe small differences in the system parameters.

Quantum lattice Boltzmann is a quantum walk

Numerical methods for the 1-D Dirac equation based on operator splitting and on the quantum lattice Boltzmann (QLB) schemes are reviewed. It is shown that these discretizations fall within the class of quantum walks, i.e. discrete maps for complex fields, whose continuum limit delivers Dirac-like relativistic quantum wave equations. The correspondence between the quantum walk dynamics and these numerical schemes is given explicitly, allowing a connection between quantum computations, numerical analysis and lattice Boltzmann methods. The QLB method is then extended to the Dirac equation in curved spaces and it is demonstrated that the quantum walk structure is preserved. Finally, it is argued that the existence of this link between the discretized Dirac equation and quantum walks may be employed to simulate relativistic quantum dynamics on quantum computers.

Localized quantum walks as secured quantum memory

We show that a quantum walk process can be used to construct and secure quantum memory. More precisely, we show that a localized quantum walk with temporal disorder can be engineered to store the information of a single, unknown qubit on a compact position space and faithfully recover it on demand. Since the localization occurs with a finite spread in position space, the stored information of the qubit will be naturally secured from the simple eavesdropper. Our protocol can be adopted to any quantum system for which experimental control over quantum walk dynamics can be achieved.

PyCTQW: A continuous-time quantum walk simulator on distributed memory computers

In the general field of quantum information and computation, quantum walks are playing an increasingly important role in constructing physical models and quantum algorithms. We have recently developed a distributed memory software package pyCTQW, with an object-oriented Python interface, that allows efficient simulation of large multi-particle CTQW (continuous-time quantum walk)-based systems. In this paper, we present an introduction to the Python and Fortran interfaces of pyCTQW, discuss various numerical methods of calculating the matrix exponential, and demonstrate the performance behavior of pyCTQW on a distributed memory cluster. In particular, the Chebyshev and Krylov-subspace methods for calculating the quantum walk propagation are provided, as well as methods for visualization and data analysis. Program summaryProgram title: pyCTQWCatalogue identifier: AEUN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: yesNo. of lines in distributed program, including test data, etc.: 130389No. of bytes in distributed program, including test data, etc.: 4971890Distribution format: tar.gzProgramming language: Fortran and Python.Computer: Workstation or cluster implementing MPI.Operating system: Any operating system with Fortran, python, and MPI installed.RAM: Depends on graph size and number of walkersClassification: 4.15, 14.External routines: PETSc [1–3], SLEPc [4–6], MPI, NumPy and SciPy [7–9], Matplotlib [10], NetworkX [11]Nature of problem:Simulates, visualizes and analyzes continuous-time quantum walks on arbitrary undirected graphs.Solution method:Distributed memory implementations of the matrix exponential, via a choice of Krylov-subspace and Chebyshev expansion techniques, are used to simulate the continuous-time quantum walkers. Visualization ability is provided via the supplied Python module and Matplotlib.Restrictions:The size of the quantum walking system is limited by the amount of available memory. The current package implements up to 3 simultaneous walkers with interactions, but it can be readily extended.Unusual features:In addition to utilizing a parallelized Krylov subspace method and Chebyshev approximation scheme to maximize efficiency, pyCTQW also provides functions for visualization of the quantum walk dynamics and calculation of multi-particle entanglement, and allows for arbitrary diagonal defects to be placed on graph nodes to explore transmission and resonance structures.Running time:Runtime varies depending on the size of the graph, number of processors used, and number of simultaneous walkers.

Trapping photons on the line: controllable dynamics of a quantum walk.

Optical interferometers comprising birefringent-crystal beam displacers, wave plates, and phase shifters serve as stable devices for simulating quantum information processes such as heralded coined quantum walks. Quantum walks are important for quantum algorithms, universal quantum computing circuits, quantum transport in complex systems, and demonstrating intriguing nonlinear dynamical quantum phenomena. We introduce fully controllable polarization-independent phase shifters in optical pathes in order to realize site-dependent phase defects. The effectiveness of our interferometer is demonstrated through realizing single-photon quantum-walk dynamics in one dimension. By applying site-dependent phase defects, the translational symmetry of an ideal standard quantum walk is broken resulting in localization effect in a quantum walk architecture. The walk is realized for different site-dependent phase defects and coin settings, indicating the strength of localization signature depends on the level of phase due to site-dependent phase defects and coin settings and opening the way for the implementation of a quantum-walk-based algorithm.

Noise-enhanced quantum transport on a closed loop using quantum walks

We study the effect of noise on the transport of a quantum state from a closed loop of $n-$sites with one of the sites as a sink. Using a discrete-time quantum walk dynamics, we demonstrate that the transport efficiency can be enhanced with noise when the number of sites in the loop is small and reduced when the number of sites in the loop grows. By using the concept of measurement induced disturbance we identify the regimes in which genuine quantum effects are responsible for the enhanced transport.

Enhancing the spreading of quantum walks on star graphs by additional bonds

We study the dynamics of continuous-time quantum walks (CTQW) on networks with highly degenerate eigenvalue spectra of the corresponding connectivity matrices. In particular, we consider the two cases of a star graph and of a complete graph, both having one highly degenerate eigenvalue, while displaying different topologies. While the CTQW spreading over the network--in terms of the average probability to return or to stay at an initially excited node--is in both cases very slow, also when compared to the corresponding classical continuous-time random walk (CTRW), we show how the spreading is enhanced by randomly adding bonds to the star graph or removing bonds from the complete graph. Then, the spreading of the excitations may become very fast, even outperforming the corresponding CTRW. Our numerical results suggest that the maximal spreading is reached halfway between the star graph and the complete graph. We further show how this disorder-enhanced spreading is related to the networks' eigenvalues.

Entanglement Generation in Spatially Separated Systems Using Quantum Walk

We present a scheme for generating entanglement between two spatially separated systems from the spatial entanglement generated by the interference effect during the evolution of a single-particle quantum walk. Any two systems which can interact with the spatial modes entangled during the walk evolution can be entangled using this scheme. A notable feature is the ability to control the quantum walk dynamics and its localization at desired pair lattice sites irrespective of separation distance resulting in a substantial control and improvement in the entanglement output. Implementation schemes to entangle spatially separated atoms using quantum walk on a single atom is also presented.

A method for exact simulation of quantum dynamics

We propose a method to simulate exactly quantum dynamics for finite-dimensional quantum systems and analyze its complexity. We apply this method to the exact simulation of any unitary using the dynamics of a time-varying discrete quantum walk on a graph. For this example, in the special case where the graph is a cycle, we show that the complexity grows like Nlog N, N being the number of vertices of the graph.

QwViz: Visualisation of quantum walks on graphs

qwViz is a software package for interactive visualisation of the time-evolution of quantum walks on arbitrarily complex graphs. The package is written in C and uses OpenGL to generate graphics in real-time. The qwViz package can be used to directly simulate discrete-time quantum walks on undirected graphs when provided with the adjacency matrix of the graph. For more detailed studies, qwViz can also be used to visualise externally generated quantum walk data written in an XML-based file format (QWML). Various aspects of the visualisation can be customised and manipulated in real-time, allowing quantum walk dynamics to be probed at various length and time scales. Program summaryProgram title:qwVizCatalogue identifier: AEJN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJN_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: GNU General Public Licence version 3No. of lines in distributed program, including test data, etc.: 42 881No. of bytes in distributed program, including test data, etc.: 97 152Distribution format: tar.gzProgramming language: CComputer: 32-bit and 64-bit workstationOperating system: Linux, Mac OS X 10.5 (and later)RAM: Depends on size of graph, typically less than 50 MBClassification: 4.15, 14External routines: OpenGL, GLUT, Graphviz [1]Nature of problem: Simulation and visualisation of quantum walks on arbitrarily complex undirected graphs.Solution method: The program uses OpenGL to produce 3D visualisations of time-dependent probability distributions arising from quantum walks on graphs. Graph layouts are automatically generated using Graphviz libraries.Restrictions: Graph layouts are two-dimensional, with the third spatial dimension being used to represent the probability of finding the quantum walker at a certain location.Unusual features: The software can be used in active or dual-stereo modes for 3D visualisation of quantum walks. Images and image sequences for movies can be exported in TIFF and TGA formats.Additional comments: Examples of various input files and an XML schema are provided. Source codes written in C and Fortran are also supplied for generating QWML files from external quantum walk simulations.Running time: Computing quantum walk data and graph layout for a 500-step quantum walk on a fifth-generation Sierpinski gasket (366 vertices) took less than 2 seconds on a 2.53 GHz Intel Core 2 Duo processor with 4 GB of RAM and 3 MB L2 shared cache under Mac OS X 10.6.6. The same simulation for a hyper-branched fractal with 1331 vertices took less than 25 seconds. GNU C compiler with optimisation option -O2 was used for these tests. Once data has been computed, the interactive visualisation can be manipulated in real-time.

Coined quantum walks on percolation graphs

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.

Tailoring discrete quantum walk dynamics via extended initial conditions

We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line. By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape. In particular, for a class of initial conditions, the evolution is dictated by the Schrödinger equation of a free particle. As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.

Connection between continuous and discrete time quantum walks. From D-dimensional lattices to general graphs

I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice with periodic boundary conditions as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [15] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [4]. By highlighting the main features of the limiting procedure, I then extend it to general regular graphs. For a given discrete time quantum walk on a graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics.

Limits of quantum speedup in photosynthetic light harvesting

It has been suggested that excitation transport in photosynthetic light-harvesting complexes features speedups analogous to those found in quantum algorithms. Here we compare the dynamics in these light-harvesting systems to the dynamics of quantum walks, in order to elucidate the limits of such quantum speedups. For the Fenna–Matthews–Olson complex of green sulfur bacteria, we show that while there is indeed speedup at short times, this is short lived (70 fs) despite longer-lived (ps) quantum coherence. Remarkably, this timescale is independent of the details of the decoherence model. More generally, we show that the distinguishing features of light-harvesting complexes not only limit the extent of quantum speedup but also reduce the rates of diffusive transport. These results suggest that quantum coherent effects in biological systems are optimized for efficiency or robustness rather than the more elusive goal of quantum speedup.

Dynamics of continuous-time quantum walks in restricted geometries

We study quantum transport on finite discrete structures and we model the process by means of continuous-time quantum walks. A direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time. Indeed, a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms. By means of analytical and numerical investigations, we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum-walk performance. We further highlight the interplay between the quantum-walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long-time averages.

Continuous-time quantum random walks require discrete space

Quantum random walks are shown to have non-intuitive dynamics which makes them an attractive area of study for devising quantum algorithms for long-standing open problems as well as those arising in the field of quantum computing. In the case of continuous-time quantum random walks, such peculiar dynamics can arise from simple evolution operators closely resembling the quantum free-wave propagator. We investigate the divergence of quantum walk dynamics from the free-wave evolution and show that, in order for continuous-time quantum walks to display their characteristic propagation, the state space must be discrete. This behavior rules out many continuous quantum systems as possible candidates for implementing continuous-time quantum random walks.