Continuous Quantum Walks Research Articles

Quantum hash function based on continuous quantum walks

A method of constructing a quantum hash function (QHF) based on continuous-time quantum walk is proposed, in which the time of quantum walk is controlled by the binary string message, namely, a binary string message as the input of the constructed QHF and the probability value of finally walking to the nodes on the cycle is used as the output of the QHF. Numerical simulation and performance analysis show that our QHF has obvious advantages in the efficiency of the QHF compared with the existing QHF based on the discrete-time quantum walk, that is, the efficiency of our scheme is nearly 4 times faster in computing the hash value of the same size file compared with the most efficient scheme. Besides, the QHF has also better collision resistance compared with the existing QHF based on the discrete-time quantum walk.

Quantum 2-Player Games and Realizations with Circuits.

Game theory problems are widely applied in many research areas such as computer science and finance, with the key issue being how to quickly make decisions. Here, we present a novel quantum algorithm for game theory problems based on a continuous quantum walk. Our algorithm exhibits quantum advantage compared to classical game algorithms. Furthermore, we exploit the analogy between the wave function of the Schrödinger equation and the voltage in Kirchhoff's law to effectively translate the design of quantum game trees into classical circuit networks. We have theoretically simulated the quantum game trees and experimentally validated the quantum functionality speedup on classical circuit networks. Due to the robust scalability and stability inherent in classical circuit networks, quantum game trees implemented within this framework hold promise for addressing more intricate application scenarios.

Optimal sensor spacing in IoT network based on quantum computing technology

Quantum computing has gained an advantage in recent years due to its compatibility and usage in various fields. The IoT environment is huge, and almost the whole world relies on it to acquire efficient data transmissions. Combining quantum computing with the IoT improves the system’s performance to a drastic level. The proposed technique focuses on introducing quantum computing into the IoT to optimise the sensor space without degrading the sensing ability of the sensors. To optimise the sensor space, the Quantum Computing based Rider Optimisation (QCRO) is introduced with optimal global search behaviour inspired by the Rider Optimisation Algorithm (ROA). After optimising sensor space, the Glued Tree based on Continuous quantum Walks (GTCW) is used to minimise the energy loss in the network. The continuous quantum walks are introduced into the Glued Trees (GT) to find the optimal path that can provide a minimised error rate. The experimental results demonstrated the effectiveness of the proposed approach against the other existing techniques in terms of data accuracy, data temporal efficiency and data cost.

MODELING OF OPTICAL CONDUCTIVITY OF NETWORKS WITH FULL BINARY ENCODING

Optical conductivity is the most important characteristic of graphs. It is a special case of continuous quantum walks on a graph and depends both on the structure of the graph and on the characteristics of the waveguides that are its edges. We investigated the conductivity of optical networks whose vertices have binary encoding depending on the thickness and length of the waveguides. It is found that the conductivity drops to zero with increasing graph size at different distributions of waveguide characteristics, which contrasts with random walks on a straight line, where such an effect does not occur. This effect has a purely interference nature and manifests itself precisely for graphs whose encoding contains all binary sets. More subtle dependences of conductivity on the characteristics of waveguides are also established. This result can be useful when choosing the structure of optical nanodevices of large sizes, for example, for optical quantum computers.

Continuous quantum walk in a 1-dimensional plasmonic lattice structure based on metal strip waveguides.

We experimentally studied a continuous time evolution of a "plasmonic" walker in a 1-dimensional lattice structure based on long-range surface plasmon polariton waveguides. The plasmonic walker exhibited a typical time evolution of a 1-dimensional quantum walk, which indicates that the plasmonic system is a potential platform to construct quantum walk simulators. By comparing experimental results to numerical simulations, the fidelity of the plasmonic quantum walk simulator is estimated to be > 0.96, which demonstrates that the plasmonic system can be a feasible platform for large-scale and high dimensional quantum walk simulators.

Pair state transfer

Let L denote the Laplacian matrix of a graph G. We study continuous quantum walks on G defined by the transition matrix $$U(t)=\exp \left( itL\right) $$ . The initial state is of the pair state form, $$e_a-e_b$$ with a, b being any two vertices of G. We provide two ways to construct infinite families of graphs that have perfect pair state transfer. We study a “transitivity” phenomenon which cannot occur in vertex state transfer. We characterize perfect pair state transfer on paths and cycles. We also study the case when quantum walks are generated by the unsigned Laplacians of underlying graphs and the initial states are of the plus state form, $$e_a+e_b$$ . When the underlying graphs are bipartite, plus state transfer is equivalent to pair state transfer.

Sedentary quantum walks

Let X be a graph with adjacency matrix A. The continuous quantum walk on X is determined by the unitary matrices U(t)=exp⁡(itA) (for t∈R). If X is the complete graph Kn and a∈V(X), then1−|U(t)a,a|≤2/n. Roughly speaking, this means that a quantum walk on a complete graph stays home with high probability. We say that a family of graphs is sedentary if there is a constant c such that 1−|U(t)a,a|≤c/|V(X)| for all t. In this paper we investigate this condition, and produce further examples of sedentary graphs.A cone over a graph X is the graph we get by adjoining a new vertex and making it adjacent to each vertex of X. We prove that if X is the cone over an ℓ-regular graph on n vertices, then |U(t)a,a|≤ℓ2/(ℓ2+4n). It follows that if we choose ℓ and n such that n/ℓ2→0, then a continuous quantum walk starting on the “conical” vertex will remain there with probability close to 1. On the other hand, if ℓ≤2, we show there is a time t such that all entries in the a-column of U(t)ea have absolute value 1/n. We show that there are large classes of strongly regular graphs such that 1−|U(t)a,a|≤c/V(X) for some constant c. On the other hand, for Paley graphs on n vertices, we prove that if t=π/n, then |U(t)a,a|≤1/n.

Quantum walks on graphene nanoribbons using quantum gates as coins

Both discrete and continuous quantum walks on graphs are universal for quantum computation. We define and use discrete quantum walks on the graphene honeycomb lattice to investigate the possibility of using graphene armchair and zigzag nanoribbons to implement quantum gates. The probability distribution of the quantum walker location represents the particle (electron) density distribution on the graphene lattice. We use a universal set of quantum gates as coins that drive the quantum walk and show that different quantum gates result in distinguishable particle distributions on the graphene lattice.

Average mixing of continuous quantum walks

If X is a graph with adjacency matrix A, then we define H(t) to be the operator exp(itA). The Schur (or entrywise) product H(t)∘H(−t) is a doubly stochastic matrix and because of work related to quantum computing, we are concerned with the average mixing matrixMˆX, defined byMˆX=limT→∞1T∫0TH(t)∘H(−t)dt. In this paper we establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We see that in a number of cases its form is surprisingly simple. Thus for the path on n vertices it is equal to12n+2(2J+I+T) where T is the permutation matrix that swaps j and n+1−j for each j. If X is an odd cycle or, more generally, if X is one of the graphs in a pseudocyclic association scheme on n vertices with d classes, each of valency m, then its average mixing matrix isn−m+1n2J+m−1nI. (One reason this is interesting is that a graph in a pseudocyclic scheme may have trivial automorphism group, and then the mixing matrix is more symmetric than the graph itself.)

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.

Complete characterization of mixing time for the continuous quantum walk on the hypercube with markovian decoherence model

The n-dimensional hypercube quantum random walk (QRW) is a particularilyappealing example of a quantum walk because it has a natural implementationon a register on n qubits. However, any real implementation will encounterdecoherence effects due to interactions with uncontrollable degrees of freedom.We present a complete characterization of the mixing properties of the hypercubeQRW under a physically relevant Markovian decoherence model. In the localdecoherence model considered the non-unitary dynamics are modeled as a sum ofprojections on individual qubits to an arbitrary direction on the Bloch sphere. Weprove that there is always classical (asymptotic) mixing in this model and specifythe conditions under which instantaneous mixing always exists. And we show thatthe latter mixing property, as well as the classical mixing time, depend heavilyon the exact environmental interaction and its strength. Therefore, algorithmicapplications of the QRW on the hypercube, if they intend to employ mixingproperties, need to consider both the walk dynamics and the precise decoherencemodel.

Complete characterization of mixing time for the continuous quantum walk on the hypercube with Markovian decoherence model

The n-dimensional hypercube quantum random walk (QRW) is a particularily appealing example of a quantum walk because it has a natural implementation on a register on n qubits. However, any real implementation will encounter decoherence effects due to interactions with uncontrollable degrees of freedom. We present a complete characterization of the mixing properties of the hypercube QRW under a physically relevant Markovian decoherence model. In the local decoherence model considered the non-unitary dynamics are modeled as a sum of projections on individual qubits to an arbitrary direction on the Bloch sphere. We prove that there is always classical (asymptotic) mixing in this model and specify the conditions under which instantaneous mixing always exists. And we show that the latter mixing property, as well as the classical mixing time, depend heavily on the exact environmental interaction and its strength. Therefore, algorithmic applications of the QRW on the hypercube, if they intend to employ mixing properties, need to consider both the walk dynamics and the precise decoherence model.

Hitting time for the continuous quantum walk

We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $\lambda$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.