Quantum Walks Research Articles

Entropy fluctuation and correlation transfer in tunable discrete-time quantum walk with fractional Gaussian noise.

We study the time correlation in the von Neumann entropy fluctuation of the tunable discrete-time quantum walk in one dimension, induced by the coin disorder arising from the temporal fractional Gaussian noise (fGn). The fGn is characterized by the Hurst exponent H, which provides three different correlation scenarios, namely antipersistent (0<H<0.5), memoryless (H=0.5), and persistent (0.5<H<1). We show the correlation of fGn is transferred to the coin's degree of entanglement and eventually transpires in the time correlation of the von Neumann entropy fluctuation. This study hints at the potential of using noise correlation as a resource to sustain information backflow via the interaction of quantum system with the noisy environment.

Steepest entropy ascent solution for a continuous-time quantum walker.

We consider the steepest entropy ascent (SEA) ansatz to describe the nonlinear thermodynamic evolution of a quantum system. Recently this principle has been dubbed the fourth law of thermodynamics [Beretta, Phil. Trans. R. Soc. A 378, 20190168 (2020)10.1098/rsta.2019.0168]. A unique global equilibrium state exists in this context, and any other state is driven by the maximum entropy generation principle towards this equilibrium. We study the SEA evolution of a continuous-time quantum walker (CTQW) on a cycle graph with N nodes. SEA solutions are difficult to find analytically. We provide an approximate scheme to find a general single-particle evolution equationgoverned by the SEA principle, whose solution produces dissipation dynamics. We call this scheme the fixed Lagrange's multiplier (FLM) method. In the Bloch sphere representation, we find trajectories traced out by the Bloch vector within the sphere itself. We have discussed these trajectories under various initial conditions for the case of a qubit. A similar dissipative motion is also observed in the case of CTQW, where probability amplitudes have been used to characterize decoherence. Our FLM scheme shows good agreement with numerical results. As we report, in CTQW, a strong delocalization exists for low system relaxation time.

Faster search of clustered marked states with lackadaisical quantum walks

The nature of discrete-time quantum walk in the presence of multiple marked states has been studied by Nahimovs and Rivosh. They introduced an exceptional configuration of clustered marked states $i.e.,$ if the marked states are arranged in a $\sqrt{k} \times \sqrt{k}$ cluster within a $\sqrt{N} \times \sqrt{N}$ grid, where $k=n^{2}$ and $n$ an odd integer. They showed that finding a single marked state among the multiple ones using quantum walk with AKR (Ambainis, Kempe and Rivosh) coin requires $\Omega(\sqrt{N} - \sqrt{k})$ time. Furthermore, Nahimov and Rivosh also showed that the Grover's coin can find the same configuration of marked state both faster and with higher probability compared to that with the AKR coin. In this article, we show that using lackadaisical quantum walk, a variant of a three-state discrete-time quantum walk on a line, the success probability of finding all the clustered marked states of this exceptional configuration is nearly 1 with smaller run-time. We also show that the weights of the self-loop suggested for multiple marked states in the state-of-the-art works are not optimal for this exceptional configuration of clustered mark states. We propose a range of weights of the self-loop from which only one can give the desired result for this configuration.

Image two-classification algorithm based on continuous-time quantum walk

To further improve the performance of image retrieval, we propose an image two-classification algorithm based on continuous-time quantum walk (CTQW) model. First, a weighted undirected complete graph is constructed based on a new weighted edge method, that is, the images in the image set are used to represent the nodes on the weighted complete graph, and every two images are connected to represent the edge on the weighted complete image, the weighted value of which represents the similarity between adjacent images. Then, the CTQW is executed to realize the image two-classification based on the quantum walk model of the weighted complete graph, that is, the CTQW is performed from two types of unrelated images, and the two-classification of images is realized that the relevant and unrelated labels are assigned to each image according to the obtained probability distribution two-classification. Finally, the experimental results show that the proposed two-classification algorithm based on CTQW has a good classification effect from the visual effect, and the evaluation indexes of image classification, such as the average precision rate, the recall rate, and the accuracy rate, are all higher than 0.85. In addition, the proposed image two-classification algorithm based on CTQW can be extended to image multiclassification if the quantum walk is performed using multiple irrelevant pictures as the starting point. Similarly, the extended image multiclassification method also has a better classification effect through the experiments and the evaluation indexes of image classification.

Improving quantum simulation efficiency of final state radiation with dynamic quantum circuits

Reference arXiv:1904.03196 recently introduced an algorithm (QPS) for simulating parton showers with intermediate flavor states using polynomial resources on a digital quantum computer. We make use of a new quantum hardware capability called dynamical quantum computing to improve the scaling of this algorithm to significantly improve the method precision. In particular, we modify the quantum parton shower circuit to incorporate mid-circuit qubit measurements, resets, and quantum operations conditioned on classical information. This reduces the computational depth from $\mathcal{O}(N^5\log_2(N)^2)$ to $\mathcal{O}(N^3\log_2(N)^2)$ and the qubit requirements are reduced from $\mathcal{O}(N\log_2(N))$ to $\mathcal{O}(N)$. Using "matrix product state" statevector simulators, we demonstrate that the improved algorithm yields expected results for 2, 3, 4, and 5-steps of the algorithm. We compare absolute costs with the original QPS algorithm, and show that dynamical quantum computing can significantly reduce costs in the class of digital quantum algorithms representing quantum walks (which includes the QPS).

Implementation of a discrete-time quantum walk with a circulant matrix on a graph by optical polarizing elements

In this paper, we introduce a quantum walk whose local scattering at each vertex is denoted by a unitary circulant matrix, namely, the circulant quantum walk. We also introduce another quantum walk induced by the circulant quantum walk, namely, the optical quantum walk, whose underlying graph is a 2-regular directed graph and obtained by blowing up the original graph in some way. We propose a design of an optical circuit which implements the stationary state of the optical quantum walk. We show that if the induced optical quantum walk does not have $+1$ eigenvalue, then the stationary state of the optical quantum walk gives that of the original circulant quantum walk. From this result, we give a useful condition for the setting of the circulant quantum walks which can be implemented by this optical circuit.

Invisible non-Hermitian potentials in discrete-time photonic quantum walks

Discrete-time photonic quantum walks on a synthetic lattice, where both spatial and temporal evolution of light are discretized, have recently provided a fascinating platform for the observation of a wealth of non-Hermitian physical phenomena and for the control of light scattering in complex media. A rather open question is whether invisible potentials, analogous to the ones known for continuous optical media, do exist in such discretized systems. Here it is shown that, under certain conditions, slowly drifting Kramers–Kronig potentials behave as invisible potentials in discrete-time photonic quantum walks.

Decoherence in the three-state quantum walk

Quantum walks are dynamic systems with a wide range of applications in quantum computation and quantum simulation of analog systems, therefore it is of common interest to understand what changes from an isolated process to one embedded in an environment. In the present work, we analyze the decoherence in a three-state uni-dimensional quantum walk. The approaches taken into consideration to account for the environment effects are phase and amplitude damping Kraus operators, unitary noise on the coin space, and broken links. The results are compared with the two-state quantum walk.

Demonstrating quantum transport enhancement in time-reversal asymmetric quantum walks

Demonstrating quantum transport enhancement in time-reversal asymmetric quantum walks

Implementation of quantum hitting times of cubelike graphs on IBM’s Qiskit platform

In this paper, we give a procedure to construct quantum circuits for implementing discrete-time quantum walks on a family of Cayley graphs called cubelike graphs. We construct these circuits on IBM’s Qiskit platform and demonstrate an implementation of the quantum hitting times on cubelike graphs. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. This conjecture, if proved, will generalize the result of hitting times of discrete-time quantum walks on hypercubes to general family of cubelike graphs.

Optical image encryption scheme based on quantum s-box and meaningful ciphertext generation algorithm

With the development of quantum computers, difficult computational problems of classical cryptosystems could potentially become solvable. Hence, due to the powerful computing power of quantum computers, all current cryptographic algorithms based on computational complexity may become ineffective. Furthermore, most of the ciphertexts generated by traditional encryption schemes are noise-like, and such ciphertexts are easily suspected by cryptanalysts during transmission. To solve these problems, a novel image encryption scheme is proposed by visually meaningful ciphertext generation and quantum walks (QWs). The proposed scheme is composed of three phases: (I) Permutation, (II) Diffusion, and (III) Visually meaningful ciphertext generation. In phase I, the quantum substitution box (s-box) is firstly constructed by QWs, and the original image is permutated by the quantum s-box to reduce the correlation between adjacent pixels. In phase II, the permutated quantum image is diffused by a key image, which is constructed by the probability matrix generated by the same QWs. In phase III, the diffused image is disguised as a meaningful image to avoid the suspicions of cryptanalysts. Compared to traditional encryption algorithms, the introduction of quantum technology effectively improves the security of the cryptosystem. Compared to previous quantum encryption methods, our algorithm generates meaningful ciphertexts that can further enhance security by preventing cryptanalysts from suspecting noise-like ciphertexts. Thus, our encryption scheme is secure even in the future quantum computer age. Besides, the decryption time of our proposed scheme can also be effectively reduced since the decryption process can be optically implemented, partially. Therefore, real-time scenarios can be satisfied. Simulation results and numerical analysis show the effectiveness of the proposed scheme.

A Hadamard walk model and its application in identification of important edges in complex networks

A Hadamard coin driven quantum walk (i.e., Hadamard walk) model is proposed to identify the important edges of undirected complex networks. In this proposed model, the importance of an edge is scored through the observed probabilities on a pair of nodes with a common edge, based on which the rankings of all important edges are obtained according to the score of each edge. By considering the robustness index of the complex network as estimation standard, experimental results indicate that the capability of the proposed Hadamard walk model to identify important edges is 4.59%∼20.03% higher than existing algorithms in static complex networks. Moreover, to further establish its utility, the proposed model was deployed in a dynamic complex network involving a typical communication scenario found in unmanned aerial vehicle (UAV) swarms. Specifically, we implemented the proposed model in simulations to select significant UAV nodes in a dynamic network and outcomes indicate that our model outperforms various algorithms in the verification of epidemic dynamics model.

A novel image cryptosystem using Gray code, quantum walks, and Henon map for cloud applications

Cloud computing plays a vital task in our daily lives, in which an enormous amount of data is stored daily on cloud storage. The highest priority for cloud data storage is guaranteeing the security of confidential data. The security of confidential data can be realised through utilising one of the cryptographic mechanisms including encryption and data hiding. With the rapid development for the realization of quantum computers, modern cryptosystems may be cracked including cloud systems. Accordingly, it is a crucial task for achieving confidentiality of data stored on cloud storage before the availability of quantum computers. Therefore, this study aims to utilise one of the quantum computational models, as a quantum-inspired system, to layout a new data confidentiality technique that can be applied in digital devices to have the capability for resisting the potential attacks from quantum and digital computers. In this paper, a new image security algorithm for real-time cloud applications using Gray code, quantum walks (QW), and Henon map is proposed. In the proposed image cryptosystem, the generated key streams from QW and Henon map is related to the plain image with high sensitivity of slight bit changes on the plain image. The outcomes based on deep analysis proves that the presented algorithm is efficient with high security for real-time application.

Signatures of Nontrivial Pairing in the Quantum Walk of Two-Component Bosons.

Nearest neighbor bosons possessing only on-site interactions do not form on-site bound pairs in their quantum walk due to fermionization. We obtain signatures of nontrivial on-site pairing in the quantum walk of strongly interacting two component bosons in a one dimensional lattice. By considering an initial state with particles from different components located at the nearest-neighbor sites in the central region of the lattice, we show that in the dynamical evolution of the system, competing intra- and intercomponent on-site repulsion leads to the formation of on-site intercomponent bound states. We find that when the total number of particles is three, an intercomponent pair is favored in the limit of equal intra- and intercomponent interaction strengths. However, when two bosons from each species are considered, intercomponent pairs and trimer are favored depending on the ratios of the intra- and intercomponent interactions. In both cases, we find that the quantum walks exhibit a reentrant behavior as a function of intercomponent interaction.

Quantum walks do not like bridges

We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing theorem, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.

Topological Spin Texture of Chiral Edge States in Photonic Two-Dimensional Quantum Walks.

Topological insulators host topology-linked boundary states, whose spin and charge degrees of freedom could be exploited to design topological devices with enhanced functionality. We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk. Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-spin texture has nontrivial winding. This topological spin winding is protected by a chiral-like symmetry emerging for the low-energy Hamiltonian. Our experiment confirms that two-dimensional anomalous Floquet topological systems exhibit topological spin texture on the boundary, which could inspire novel topology-based spintronic phenomena and devices.

Quantum walk in a reinforced free-energy landscape: Quantum annealing with reinforcement

Providing an optimal path to a quantum annealing algorithm is key to finding good approximate solutions to computationally hard optimization problems. Reinforcement is one of the strategies that can be used to circumvent the exponentially small energy gaps of the system in the annealing process. Here a time-dependent reinforcement term is added to the Hamiltonian in order to give lower energies to the most probable states of the evolving system. In this study, we take a local entropy in the configuration space for the reinforcement and apply the algorithm to a number of easy and hard optimization problems. The reinforced algorithm performs better than the standard quantum annealing algorithm in the quantum search problem, where the optimal parameters behave very differently depending on the number of solutions. Moreover, the reinforcements can change the discontinuous phase transitions of the mean-field p-spin model ($p>2$) to a continuous transition. The algorithm's performance in the binary perceptron problem is also superior to that of the standard quantum annealing algorithm, which already works better than a classical simulated annealing algorithm.

Rogue waves in discrete-time quantum walks

Rogue waves are rapid and unpredictable events of exceptional amplitude reported in various fields, such as oceanography and optics, with much of the interest being targeted towards their physical origins and likelihood of occurrence. Here, we use the all-round framework of discrete-time quantum walks to study the onset of those events due to a random phase modulation, unveiling its long-tailed statistics, survival time, and dependence upon the degree of randomness. We find the minimal disorder strength allowing for their occurrence to scale $\ensuremath{\propto}{N}^{\ensuremath{-}1/2}, N$ being the number of sites. Moreover, an extreme-value analysis converges to the Gumbel class of limiting distributions.

Search Algorithm Based on Permutation Group by Quantum Walk on Hypergraphes

Search Algorithm Based on Permutation Group by Quantum Walk on Hypergraphes

Comfortable place for quantum walker on finite path

Comfortable place for quantum walker on finite path