Quantum Walks Research Articles

Parrondo’s game of quantum search based on quantum walk

Parrondo’s game of quantum search based on quantum walk

Instability in the quantum restart problem.

Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.

Phase transition of a continuous-time quantum walk on the half line

Phase transition of a continuous-time quantum walk on the half line

Apportionable matrices and gracefully labelled graphs

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labellings of graphs, Hadamard matrices, and equiangular lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labellings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labellings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are 2×2 matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.

Continuous-Time Quantum Walk in Glued Trees: Localized State-Mediated Almost Perfect Quantum-State Transfer.

In this work, the dynamics of a quantum walker on glued trees is revisited to understand the influence of the architecture of the graph on the efficiency of the transfer between the two roots. Instead of considering regular binary trees, we focus our attention on leafier structures where each parent node could give rise to a larger number of children. Through extensive numerical simulations, we uncover a significant dependence of the transfer on the underlying graph architecture, particularly influenced by the branching rate (M) relative to the root degree (N). Our study reveals that the behavior of the walker is isomorphic to that of a particle moving on a finite-size chain. This chain exhibits defects that originate in the specific nature of both the roots and the leaves. Therefore, the energy spectrum of the chain showcases rich features, which lead to diverse regimes for the quantum-state transfer. Notably, the formation of quasi-degenerate localized states due to significant disparities between M and N triggers a localization process on the roots. Through analytical development, we demonstrate that these states play a crucial role in facilitating almost perfect quantum beats between the roots, thereby enhancing the transfer efficiency. Our findings offer valuable insights into the mechanisms governing quantum-state transfer on trees, with potential applications for the transfer of quantum information.

IoMT-Based Smart Healthcare Detection System Driven by Quantum Blockchain and Quantum Neural Network.

Electrocardiogram (ECG) is the main criterion for arrhythmia detection. As a means of identification, ECG leakage seems to be a common occurrence due to the development of the Internet of Medical Things. The advent of the quantum era makes it difficult for classical blockchain technology to provide security for ECG data storage. Therefore, from the perspective of safety and practicality, this article proposes a quantum arrhythmia detection system called QADS, which achieves secure storage and sharing of ECG data based on quantum blockchain technology. Furthermore, a quantum neural network is used in QADS to recognize abnormal ECG data, which contributes to further cardiovascular disease diagnosis. Each quantum block stores the hash of the current and previous block to construct a quantum block network. The new quantum blockchain algorithm introduces a controlled quantum walk hash function and a quantum authentication protocol to guarantee legitimacy and security while creating new blocks. In addition, this article constructs a hybrid quantum convolutional neural network called HQCNN to extract the temporal features of ECG to detect abnormal heartbeats. The simulation experimental results show that HQCNN achieves an average training and testing accuracy of 94.7% and 93.6%. And the detection stability is much higher than classical CNN with the same structure. HQCNN also has certain robustness under the perturbation of quantum noise. Besides, this article demonstrates through mathematical analysis that the proposed quantum blockchain algorithm has strong security and can effectively resist various quantum attacks, such as external attacks, Entanglement-Measure attack and Interception-Measurement-Repeat attack.

Quantum hitting time according to a given distribution

In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.

Modeling urban traffic volume incorporating driver stochasticity and traffic correlation with waveguide-based quantum walk

Urban traffic involves two stochastic factors, complicating accurate modeling: (1) Driver stochasticity, arising from driver heterogeneity and random environment perturbations, introduces unpredictable fluctuations in urban traffic volume. (2) Traffic correlation, referring to the interaction in traffic states across different regions. This paper proposes a waveguide-based quantum walk urban traffic volume model (UTVDC-QW), incorporating driver stochasticity and traffic correlation. The model uses waveguide arrays to map urban traffic networks, incorporates random environment perturbations via noise, and traffic correlation via waveguide coupling. UTVDC-QW develops measures of environment perturbations and traffic correlations. Experiments with Chengdu traffic datasets demonstrate that UTVDC-QW improves the simulation accuracy compared to reference models. In addition, the model reveals the characteristics of environment perturbations and traffic correlation in the study area. Overall, this study effectively incorporates driver stochasticity and traffic correlation by waveguide-based quantum walk, is a pioneering attempt to apply quantum computing to urban traffic modeling.

Failing to Hash Into Supersingular Isogeny Graphs

Abstract An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of ‘hard supersingular curves’ that is equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular $\ell $-isogeny graph, which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd’s of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces, and applying Kummer surfaces and (v) using quantum random walks.

Quantum walk of two particles on the graphene structure graph with interaction

In this paper, we discuss the quantum walk of two interacting particles on graphene structure graph. Based on the localized graphene structure graph, we get three cases for the position of the two particles, that is ∂(i, j) = 0, ∂(i, j) = 1, and ∂(i, j) = 2. Subsequently, we investigate the quantum walk of two interacting particles from a new perspective, namely, based on the view of quantum probability, a localized quantum decomposition was used to get the corresponding creation operators and annihilation operators for the first time. Finally, we get the Hamiltonian and the time evolution operator of two interacting particles on the graphene structure graph.

Quantum ratchet with Lindblad rate equations.

A quantum random-walk model is established on a one-dimensional periodic lattice that fluctuates between two possible states. This model is defined by Lindblad rate equationsthat incorporate the transition rates between the two lattice states. Leveraging the system's symmetries, the particle velocity can be described using a finite set of equations, even though the state space is of infinite dimension. These equationsyield an analytical expression for the velocity in the long-time limit, which is employed to analyze the characteristics of directed motion. Notably, the velocity can exhibit multiple inversions, and to achieve directed motion, distinct, nonzero transition rates between lattice states are required.

Self acceleration from spectral geometry in dissipative quantum-walk dynamics

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are significantly enriched, whereas the identification and comprehension of the underlying connections are challenging. Here we experimentally demonstrate the correspondence between the transient self-acceleration of local excitations and the non-Hermitian spectral topology using lossy photonic quantum walks. Focusing first on one-dimensional quantum walks, we show that the measured short-time acceleration of the wave function is proportional to the area enclosed by the eigenspectrum. We then reveal a similar correspondence in two-dimension quantum walks, where the self-acceleration is proportional to the volume enclosed by the eigenspectrum in the complex parameter space. In both dimensions, the transient self-acceleration crosses over to a long-time behavior dominated by a constant flow at the drift velocity. Our results unveil the universal correspondence between spectral topology and transient dynamics, and offer a sensitive probe for phenomena in non-Hermitian systems that originate from spectral geometry.

Quantum walk on simplicial complexes for simplicial community detection

Quantum walk on simplicial complexes for simplicial community detection

Optical Color Image Encryption Algorithm Based on Two-Dimensional Quantum Walking

The double random phase encoding (DRPE) image encryption method has garnered significant attention in color image processing and optical encryption thanks to its R, G, and B parallel encryption. However, DRPE-based color image encryption faces two challenges. Firstly, it disregards the correlation of R, G, and B, compromising the encrypted image’s robustness. Secondly, DRPE schemes relying on Discrete Fourier Transform (DFT) and Discrete Fractional Fourier Transform (DFRFT) are vulnerable to linear attacks, such as Known Plaintext Attack (KPA) and Chosen Plaintext Attack (CPA). Quantum walk is a powerful tool for modern cryptography, offering robust resistance to classical and quantum attacks. Therefore, this study presents an optical color image encryption algorithm that combines two-dimensional quantum walking (TDQW) with 24-bit plane permutation, dubbed OCT. This approach employs pseudo-random numbers generated by TDQW for phase modulation in DRPE and scrambles the encrypted image’s real and imaginary parts using the generalized Arnold transform. The 24-bit plane permutation helps reduce the R, G, and B correlation, while the generalized Arnold transform bolsters DRPE’s resistance to linear attacks. By incorporating TDQW, the key space is significantly expanded. The experimental results validate the effectiveness and security of the proposed method.

Quantum walks as thermalisations, with application to fullerene graphs

We consider to what extent quantum walks can constitute models of thermalisation, analogously to how classical random walks can be models for classical thermalisation. In a quantum walk over a graph, a walker moves in a superposition of node positions via a unitary time evolution. We show a quantum walk can be interpreted as an equilibration of a kind investigated in the literature on thermalisation in unitarily evolving quantum systems. This connection implies that recent results concerning the equilibration of observables can be applied to analyse the node position statistics of quantum walks. We illustrate this in the case of a family of graphs known as fullerenes. We find that a bound from Short et al., implying that certain expectation values will at most times be close to their time-averaged value, applies tightly to the node position probabilities. Nevertheless, the node position statistics do not thermalise in the standard sense. In particular, quantum walks over fullerene graphs constitute a counter-example to the hypothesis that subsystems equilibrate to the Gibbs state. We also exploit the bridge created to show how quantum walks can be used to probe the universality of the eigenstate thermalisation hypothesis relation. We find that whilst in C60 with a single walker, the eigenstate thermalisation hypothesis relation does not hold for node position projectors, it does hold for the average position, enforced by a symmetry of the Hamiltonian. The findings suggest a unified study of quantum walks and quantum self-thermalisations is natural and feasible.

Observation of Non-Hermitian Edge Burst Effect in One-Dimensional Photonic Quantum Walk.

Non-Hermitian systems can exhibit unique quantum phases without any Hermitian counterparts. For example, the latest theoretical studies predict a new surprising phenomenon that bulk bands can localize and dissipate prominently at the system boundary, which is dubbed the non-Hermitian edge burst effect. Here we realize a one-dimensional non-Hermitian Su-Schrieffer-Heeger lattice with bulk translation symmetry implemented with a photonic quantum walk. Employing time-resolved single-photon detection to characterize the chiral motion and boundary localization of bulk bands, we determine experimentally that the dynamics underlying the non-Hermitian edge burst effect is due to the interplay of non-Hermitian skin effect and imaginary band gap closing. This new non-Hermitian physical effect deepens our understanding of quantum dynamics in open quantum systems.

Generation of true quantum random numbers with on-demand probability distributions via single-photon quantum walks

Random numbers are at the heart of diverse fields, ranging from simulations of stochastic processes to classical and quantum cryptography. The requirement for true randomness in these applications has motivated various proposals for generating random numbers based on the inherent randomness of quantum systems. The generation of true random numbers with arbitrarily defined probability distributions is highly desirable for applications, but it is very challenging. Here we show that single-photon quantum walks can generate multi-bit random numbers with on-demand probability distributions, when the required “coin” parameters are found with the gradient descent (GD) algorithm. Our theoretical and experimental results exhibit high fidelity for various selected distributions. This GD-enhanced single-photon system provides a convenient way for building flexible and reliable quantum random number generators. Multi-bit random numbers are a necessary resource for high-dimensional quantum key distribution.

The spectrum of asymptotic Cayley trees

We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.

Floquet chiral quantum walk in a quantum computer

Floquet chiral quantum walk in a quantum computer

Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks

Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks