Continuous-time Quantum Walks Research Articles

Quantum transport on honeycomb networks

We study the transport properties on honeycomb networks motivated by graphene structures by using the continuous-time quantum walk (CTQW) model. For various relevant topologies we consider the average return probability and its long-time average as measures for the transport efficiency. These quantities are fully determined by the eigenvalues and the eigenvectors of the connectivity matrix of the network. For all networks derived from graphene structures we notice a nontrivial interplay between good spreading and localization effects. Flat graphene with similar number of hexagons along both directions shows a decrease in transport efficiency compared to more one-dimensional structures. This loss can be overcome by increasing the number of layers, thus creating a graphite network, but it gets less efficient when rolling up the sheets so that a nanotube structure is considered. We found peculiar results for honeycomb networks constructed from square graphene, i.e. the same number of hexagons along both directions of the graphene sheet. For these kind of networks we encounter significant differences between networks with an even or odd number of hexagons along one of the axes.

Continuous-time quantum walk based centrality testing on weighted graphs

Centrality measure is an essential tool in network analysis and widely used in the domain of computer science, biology and sociology. Taking advantage of the speedup offered by quantum computation, various quantum centrality measures have been proposed. However, few work of quantum centrality involves weighted graphs, while the weight of edges should be considered in certain real-world networks. In this work, we extend the centrality measure based on continuous-time quantum walk to weighted graphs. We testify the feasibility and reliability of this quantum centrality using an ensemble of 41,675 graphs with various topologies and comparing with the eigenvector centrality measure. The average Vigna’s correlation index of all the tested graphs with all edge weights in [1, 10] is as high as 0.967, indicating the pretty good consistency of rankings by the continuous-time quantum walk centrality and the eigenvector centrality. The intuitive consistency of the top-ranked vertices given by this quantum centrality measure and classical centrality measures is also demonstrated on large-scale weighted graphs. Moreover, the range of the continuous-time quantum walk centrality values is much bigger than that of classical centralities, which exhibits better distinguishing ability to pick the important vertices from the ones with less importance. All these results show that the centrality measure based on continuous-time quantum walk still works well on weighted graphs.

An information propagation model for social networks based on continuous-time quantum walk

An information propagation model for social networks based on continuous-time quantum walk

Quantum-classical distance as a tool to design optimal chiral quantum walks

Continuous-time quantum walks (CTQWs) provide a valuable model for quantum transport, universal quantum computation, and quantum spatial search, among others. Recently, the empowering role of new degrees of freedom in the Hamiltonian generator of CTQWs, which are the complex phases along the loops of the underlying graph, was acknowledged for its interest in optimizing or suppressing transport on specific topologies. We argue that the quantum-classical distance, a figure of merit which was introduced to capture the difference in dynamics between a CTQW and its classical, stochastic counterpart, guides the optimization of parameters of the Hamiltonian to achieve better quantum transport on cycle graphs and spatial search for the quantum speed limit without an oracle on complete graphs, the latter also implying fast uniform mixing. We compare the variations of this quantity with the 1-norm of coherence and the inverse participation ratio, showing that the quantum-classical distance is linked to both, but in a topology-dependent relation, which is key to spot the most interesting quantum evolution in each case.

Quantum walk on a comb with infinite teeth

We study continuous time quantum walk on a comb with infinite teeth and show that the return probability to the starting point decays with time t as t −1. We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spine with a finite probability. The walk along the spine and into the teeth behaves qualitatively as a quantum walk on a line. This behaviour is quite different from that of classical random walk on the comb.

Deterministic Search on Star Graphs via Quantum Walks.

We propose a novel algorithm for quantum spatial search on a star graph using interleaved continuous-time quantum walks and marking oracle queries. Initializing the system in the star's central vertex, we determine the optimal quantum walk times to reach full overlap with the marked state using ⌈(π/4)sqrt[N]-(1/2)⌉ oracle queries, matching the well-known lower bound of Grover's search. We implement the deterministic search in a database of size seven on photonic quantum hardware, and demonstrate the effective scaling of the approach up to size 115. This is the first experimental demonstration of quantum walk-based search on the highly noise-resistant star graph, which provides new evidence for the applications of quantum walk in quantum algorithms and quantum information processing.

Spatial search on Johnson graphs by continuous-time quantum walk

Spatial search on graphs is one of the most important algorithmic applications of quantum walks. To show that a quantum-walk-based search is more efficient than a random-walk-based search is a difficult problem, which has been addressed in several ways. Usually, graph symmetries aid in the calculation of the algorithm's computational complexity, and Johnson graphs are an interesting class regarding symmetries because they are regular, Hamilton-connected, vertex- and distance-transitive. In this work, we show that spatial search on Johnson graphs by continuous-time quantum walk achieves the Grover lower bound $\pi\sqrt{N}/2$ with success probability $1$ asymptotically for every fixed diameter, where $N$ is the number of vertices. The proof is mathematically rigorous and can be used for other graph classes.

Simplifying continuous-time quantum walks on dynamic graphs

A continuous-time quantum walk on a dynamic graph evolves by Schr\"odinger's equation with a sequence of Hamiltonians encoding the edges of the graph. This process is universal for quantum computing, but in general, the dynamic graph that implements a quantum circuit can be quite complicated. In this paper, we give six scenarios under which a dynamic graph can be simplified, and they exploit commuting graphs, identical graphs, perfect state transfer, complementary graphs, isolated vertices, and uniform mixing on the hypercube. As examples, we simplify dynamic graphs, in some instances allowing single-qubit gates to be implemented in parallel.

Unstructured search by random and quantum walk

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

Entanglement Transfer Via Chiral and Continuous-Time Quantum Walks on a Triangular Chain

Entanglement Transfer Via Chiral and Continuous-Time Quantum Walks on a Triangular Chain

Quantum dynamics and correlations of indistinguishable anyons in one-dimensional lattices

Anyons, namely particles obeying fractional quantum statistics that interpolate between bosons and fermions, possess a lot of new and exotic physical properties related to the particle exchange statistics. In this work, we explore the few-body quantum dynamics and quantum correlations of indistinguishable anyons with on-site interactions in one-dimensional lattices within the scheme of three-body continuous-time quantum walks. By employing a time-evolving block decimation (TEBD) algorithm, we numerically calculate the dynamical evolution process of the local density distribution of anyons among the whole lattice. Numerical simulations shown in the main text mainly focus on a three-body initial state as <inline-formula><tex-math id="M1">\begin{document}$ \left|\psi(t=0)\right>=\hat{a}_{-1}^{\dagger}\hat{a}_{0}^{\dagger}\hat{a}_{1}^{\dagger}\left|0\right>$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20220188_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20220188_M1.png"/></alternatives></inline-formula>, in which three particles are located on neighbouring sites at lattice centre. This choice of initial state features that the three particles influence one another most strongly, while we have also implemented numerical simulations on other choices of three-body initial states as are discussed in appendix. It is shown that the local density distribution of anyons is dramatically altered by fractional particle statistics with new dynamical structure showing up during the time evolution. For free anyons, an inner cone emerges as the statistical parameter increases, while the outer cone remains robust all along. When the on-site interaction joins in, the structure of the inner cone is further modified with new features. Specifically, for interaction of finite strength, an exotic dynamical asymmetry in real space, is clearly demonstrated during the time evolution of the local density distribution for particles within the fractional statistics regime. However, for boson limit and pseudofermion limit, the time evolution of the local density distribution keeps symmetric as the three-body initial state. And remarkably, the dynamical asymmetry is interaction-dependent manifested as the local density distribution of anyons favors opposite side of the lattice for repulsive and attractive interaction, respectively. Moreover, when the on-site interaction is further increased to hard-core limit, the dynamical asymmetry will then be largely suppressed. We also calculate the density-density correlations for anyons before they reach the lattice boundary to reveal the interesting effect of fractional statistics on quantum correlations. It is shown that the inner cone corresponds to co-walking of anyons, while the outer cone is related to individual walking and is immune to the variation of statistical parameter. Furthermore, the exotic real-space asymmetry originated from the interplay of fractional statistics and finite interaction is also shown up in the density-density correlations.

Quantum Dynamics and Correlations of Indistinguishable Anyons in One-Dimensional Lattices

Anyons, namely particles obeying fractional quantum statistics that interpolate between bosons and fermions, possess a lot of new and exotic physical properties related to the particle exchange statistics. In this work, we explore the few-body quantum dynamics and quantum correlations of indistinguishable anyons with on-site interactions in onedimensional lattices within the scheme of three-body continuous-time quantum walks. By employing a time-evolving block decimation (TEBD) algorithm, we numerically calculate the dynamical evolution process of the local density distribution of anyons among the whole lattice. Numerical simulations shown in the main text mainly focus on a threebody initial state as ψ|(t=0)＞=â<sup>†</sup><sub>-1</sub>â<sup>†</sup><sub>0</sub>â<sup>†</sup><sub>1</sub>|0＞, in which three particles are located on neighbouring sites at lattice centre. This choice of initial state features that the three particles influence one another most strongly, while we have also implemented numerical simulations on other choices of three-body initial states as are discussed in appendix. It is shown that the local density distribution of anyons is dramatically altered by fractional particle statistics with new dynamical structure showing up during the time evolution. For free anyons, an inner cone emerges as the statistical parameter increases, while the outer cone remains robust all along. When the on-site interaction joins in, the structure of the inner cone is further modified with new features. Specifically, for interaction of finite strength, an exotic dynamical asymmetry in real space, is clearly demonstrated during the time evolution of the local density distribution for particles within the fractional statistics regime. However, for boson limit and pseudofermion limit, the time evolution of the local density distribution keeps symmetric as the three-body initial state. And remarkably, the dynamical asymmetry is interactiondependent manifested as the local density distribution of anyons favors opposite side of the lattice for repulsive and attractive interaction, respectively. Moreover, when the on-site interaction is further increased to hard-core limit, the dynamical asymmetry will then be largely suppressed. We also calculate the density-density correlations for anyons before they reach the lattice boundary to reveal the interesting effect of fractional statistics on quantum correlations. It is shown that the inner cone corresponds to co-walking of anyons, while the outer cone is related to individual walking and is immune to the variation of statistical parameter. Furthermore, the exotic real-space asymmetry originated from the interplay of fractional statistics and finite interaction is also shown up in the density-density correlations.

Equivalent Laplacian and adjacency quantum walks on irregular graphs

The continuous-time quantum walk is a particle evolving by Schr\"odinger's equation in discrete space. Encoding the space as a graph of vertices and edges, the Hamiltonian is proportional to the discrete Laplacian. In some physical systems, however, the Hamiltonian is proportional to the adjacency matrix instead. It is well known that these quantum walks are equivalent when the graph is regular, i.e., when each vertex has the same number of neighbors. If the graph is irregular, however, the quantum walks evolve differently. In this paper, we show that for some irregular graphs, if the particle is initially localized at a certain vertex, the probability distributions of the two quantum walks are identical, even though the amplitudes differ. We analytically prove this for a graph with five vertices and a graph with six vertices. By simulating the walks on all 1 018 689 568 simple, connected, irregular graphs with 11 vertices or less, we found 64 graphs with this notion of equivalence. We also give eight infinite families of graphs supporting these equivalent walks.

Quantum hub and authority centrality measures for directed networks based on continuous-time quantum walks

Abstract In this article, we introduce, test and discuss three quantum methods for computing hub and authority centrality scores in directed networks. The methods are based on unitary, continuous-time quantum walks; the construction of a suitable Hermitian Hamiltonian is achieved by performing a quantum walk on the associated bipartite graph. Two methods, called CQAu and CQAw, use the same evolution operator, inspired by the classical Hyperlink-Induced Topic Search (HITS) algorithm, but with different initial states; the computation of hub and authority scores is performed simultaneously. The third method, called CQG and inspired by classical PageRank, requires instead two separate runs with different evolution operators, one for hub and one for authority scores. The methods are tested on several directed graphs with different sizes and properties; a comparison with other well-established ranking algorithms is provided. CQAw emerges as the most reliable of the three methods and yields rankings that are largely compatible with results from HITS, although CQAu and CQG also present interesting features and potential for applications.

Implementing quantum gates using length-3 dynamic quantum walks

It is well-known that any quantum gate can be decomposed into the universal gate set {T, H, CNOT}, and recent results have shown that each of these gates can be implemented using a dynamic quantum walk, which is a continuous-time quantum walk on a sequence of graphs. This procedure for converting a quantum gate into a dynamic quantum walk, however, can result in long sequences of graphs. To alleviate this, in this paper, we develop a length-3 dynamic quantum walk that implements any single-qubit gate. Furthermore, we extend this result to give length-3 dynamic quantum walks that implement any single-qubit gate controlled by any number of qubits. Using these, we implement Draper's quantum addition circuit, which is based on the quantum Fourier transform, using a dynamic quantum walk.

A framework for optimal quantum spatial search using alternating phase-walks

We present a novel methodological framework for quantum spatial search, generalising the Childs & Goldstone () algorithm via alternating applications of marked-vertex phase shifts and continuous-time quantum walks. We determine closed form expressions for the optimal walk time and phase shift parameters for periodic graphs. These parameters are chosen to rotate the system about subsets of the graph Laplacian eigenstates, amplifying the probability of measuring the marked vertex. The state evolution is asymptotically optimal for any class of periodic graphs having a fixed number of unique eigenvalues. We demonstrate the effectiveness of the algorithm by applying it to obtain search on a variety of graphs. One important class is the n × n 3 rook graph, which has N = n 4 vertices. On this class of graphs the algorithm performs suboptimally, achieving only overlap after time . Using the new alternating phase-walk framework, we show that overlap is obtained in phase-walk iterations.

Experimentally Detecting Quantized Zak Phases without Chiral Symmetry in Photonic Lattices.

Symmetries play a major role in identifying topological phases of matter and in establishing a direct connection between protected edge states and topological bulk invariants via the bulk-boundary correspondence. One-dimensional lattices are deemed to be protected by chiral symmetry, exhibiting quantized Zak phases and protected edge states, but not for all cases. Here, we experimentally realize an extended Su-Schrieffer-Heeger model with broken chiral symmetry by engineering one-dimensional zigzag photonic lattices, where the long-range hopping breaks chiral symmetry but ensures the existence of inversion symmetry. By the averaged mean displacement method, we detect topological invariants directly in the bulk through the continuous-time quantum walk of photons. Our results demonstrate that inversion symmetry protects the quantized Zak phase but edge states can disappear in the topological nontrivial phase, thus breaking the conventional bulk-boundary correspondence. Our photonic lattice provides a useful platform to study the interplay among topological phases, symmetries, and the bulk-boundary correspondence.

Improved upper bounds for the hitting times of quantum walks

Continuous-time quantum walks have proven to be an extremely useful framework for the design of several quantum algorithms. Often, the running time of quantum algorithms in this framework is characterized by the quantum hitting time: the time required by the quantum walk to find a vertex of interest with a high probability. In this article, we provide improved upper bounds for the quantum hitting time that can be applied to several CTQW-based quantum algorithms. In particular, we apply our techniques to the glued-trees problem, improving their hitting time upper bound by a polynomial factor: from $O(n^5)$ to $O(n^2\log n)$. Furthermore, our methods also help to exponentially improve the dependence on precision of the continuous-time quantum walk based algorithm to find a marked node on any ergodic, reversible Markov chain by Chakraborty et al. [PRA 102, 022227 (2020)].

Generalized quantum-classical correspondence for random walks on graphs

We introduce a minimal set of physically motivated postulates that the Hamiltonian H of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e. the diagonal elements of H, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential energy landscape for the quantum walk, and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of H may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electro-magnetic vector potential for a charged walker.

Role of Symmetry in Quantum Search via Continuous-Time Quantum Walk

For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.