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 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.

Quantum computation is becoming an important and effective tool to overcome the high real-time computational requirements of classical digital image processing. In this paper, based on analysis of existing quantum image representations, a novel enhanced quantum representation (NEQR) for digital images is proposed, which improves the latest flexible representation of quantum images (FRQI). The newly proposed quantum image representation uses the basis state of a qubit sequence to store the gray-scale value of each pixel in the image for the first time, instead of the probability amplitude of a qubit, as in FRQI. Because different basis states of qubit sequence are orthogonal, different gray scales in the NEQR quantum image can be distinguished. Performance comparisons with FRQI reveal that NEQR can achieve a quadratic speedup in quantum image preparation, increase the compression ratio of quantum images by approximately 1.5X, and retrieve digital images from quantum images accurately. Meanwhile, more quantum image operations related to gray-scale information in the image can be performed conveniently based on NEQR, for example partial color operations and statistical color operations. Therefore, the proposed NEQR quantum image model is more flexible and better suited for quantum image representation than other models in the literature.

Ghost images are obtained by correlating the output of a single-pixel (bucket) photodetector—which collects light that has been transmitted through or reflected from an object—with the output from a high spatial-resolution scanning photodetector or photodetector array whose illumination has not interacted with that object. The term “ghost image” is apt because neither detector’s output alone can yield an image: the bucket detector has no spatial resolution, while the high spatial-resolution detector has not viewed the object. The first ghost imaging experiment relied on the entangled signal and idler outputs from a spontaneous parametric downconverter, and hence the image was interpreted as a quantum phenomenon. Subsequent theory and experiments showed, however, that classical correlations can be used to form ghost images. For example, ghost images can be formed with pseudothermal light, for which quantum mechanics is not required to characterize its photodetection statistics. This paper presents an overview of the physics of ghost imaging. It clarifies and unites two disparate interpretations of pseudothermal ghost imaging—two-photon interference and classical intensity-fluctuation correlations—that had previously been thought to be conflicting. It also reviews recent work on ghost imaging in reflection, ghost imaging through atmospheric turbulence, computational ghost imaging, and two-color ghost imaging.

A Flexible Representation of Quantum Images (FRQI) is proposed to provide a representation for images on quantum computers in the form of a normalized state which captures information about colors and their corresponding positions in the images. A constructive polynomial preparation for the FRQI state from an initial state, an algorithm for quantum image compression (QIC), and processing operations for quantum images are combined to build the whole process for quantum image processing on FRQI. The simulation experiments on FRQI include storing, retrieving of images and a detection of a line in binary images by applying quantum Fourier transform as a processing operation. The compression ratios of QIC between groups of same color positions range from 68.75 to 90.63% on single digit images and 6.67–31.62% on the Lena image. The FRQI provides a foundation not only to express images but also to explore theoretical and practical aspects of image processing on quantum computers.

Quantum image processing (QIMP) is devoted to utilizing the quantum computing technologies to capture, manipulate, and recover quantum images in different formats and for different purposes. Logically, percolating this requires that representations to encode images based on the quantum mechanical composition of any potential quantum computing hardware be conjured. This paper gathers the current mainstream quantum image representations (QIRs) and discusses the advances made in the area. Some similarities, differences, and likely applications for some of the available QIRs are reviewed. We believe this compendium will provide the readership an overview of progress witnessed in the area of QIMP while also simulating further interest to pursue more advanced research in it.

A quantum realization of the generalized Arnold transform is designed. A novel quantum image encryption algorithm based on generalized Arnold transform and double random-phase encoding is proposed. The pixels are scrambled by the generalized Arnold transform, and the gray-level information of images is encoded by the double random-phase operations. The keys of the encryption algorithm include the independent parameters of coefficients matrix, iterative times and classical binary sequences, and thus, the key space is extremely large. Numerical simulations and theoretical analyses demonstrate that the proposed algorithm with good feasibility and effectiveness has lower computational complexity than its classical counterpart.

A new quantum dialogue protocol is designed by using the continuous-variable two-mode squeezed vacuum states due to its entanglement property. The two communication parties encode their own secret information into the entangled optical modes with the translation operations. Each communication party could deduce the secret information of their counterparts with the help of his or her secret information and the Bell-basis measurement results. The security of the proposed quantum dialogue protocol is guaranteed by the correlation between two-mode squeezed vacuum states and the decoy states performed with translation operations in randomly selected time slots. Compared with the discrete variable quantum dialogue protocols, the proposed continuous-variable quantum dialogue protocol is easy to realize with perfect utilization of quantum bits.

We present a robust watermark strategy for quantum images. The watermark image is embedded into the fourier coefficients of the quantum carrier image, which will not affect the carrier image’s visual effect. Before being embedded into the carrier image, the watermark image is preprocessed to be seemingly meaningless using quantum circuit, which further ensures the security of the watermark image. The properties of fourier transform ensure that the watermark embedded in the carrier image resists the unavoidable noise and cropping.

Based on EPR pairs, this paper proposes a different quantum private comparison (QPC) protocol enabling two parties to compare the equality of their information without revealing the information content. Due to the use of quantum entanglement of Bell state as well as one-way quantum transmission, the new protocol provides easier implementation as well as better qubit efficiency (near 50%) than the other QPCs. It is secure against Trojan horse attack and other well-known attacks.

A theoretical scheme is proposed to implement bidirectional quantum controlled teleportation (BQCT) by using a nine-qubit entangled state as a quantum channel, where Alice may transmit an arbitrary two-qubit state called qubits $$A_1$$ A 1 and $$A_2$$ A 2 to Bob; and at the same time, Bob may also transmit an arbitrary two-qubit state called qubits $$B_1$$ B 1 and $$B_2$$ B 2 to Alice via the control of the supervisor Charlie. Based on our channel, we explicitly show how the bidirectional quantum controlled teleportation protocol works. And we show this bidirectional quantum controlled teleportation scheme may be determinate and secure. Taking the amplitude-damping noise and the phase-damping noise as typical noisy channels, we analytically derive the fidelities of the BQCT process and show that the fidelities in these two cases only depend on the amplitude parameter of the initial state and the decoherence noisy rate.

With the overwhelming success in the field of quantum information in the last decades, the ‘quest’ for a Quantum Neural Network (QNN) model began in order to combine quantum computing with the striking properties of neural computing. This article presents a systematic approach to QNN research, which so far consists of a conglomeration of ideas and proposals. Concentrating on Hopfield-type networks and the task of associative memory, it outlines the challenge of combining the nonlinear, dissipative dynamics of neural computing and the linear, unitary dynamics of quantum computing. It establishes requirements for a meaningful QNN and reviews existing literature against these requirements. It is found that none of the proposals for a potential QNN model fully exploits both the advantages of quantum physics and computing in neural networks. An outlook on possible ways forward is given, emphasizing the idea of Open Quantum Neural Networks based on dissipative quantum computing.

The power of quantum mechanics has been extensively exploited to meet the high computational requirement of classical image processing. However, existing quantum image models can only represent the images sampled in Cartesian coordinates. In this paper, quantum log-polar image (QUALPI), a novel quantum image representation is proposed for the storage and processing of images sampled in log-polar coordinates. In QUALPI, all the pixels of a QUALPI are stored in a normalized superposition and can be operated on simultaneously. A QUALPI can be constructed from a classical image via a preparation whose complexity is approximately linear in the image size. Some common geometric transformations, such as symmetry transformation, rotation, etc., can be performed conveniently with QUALPI. Based on these geometric transformations, a fast rotation-invariant quantum image registration algorithm is designed for log-polar images. Performance comparison with classical brute-force image registration method reveals that our quantum algorithm can achieve a quartic speedup.

The quantum Fourier transform, the quantum wavelet transform, etc., have been shown to be a powerful tool in developing quantum algorithms. However, in classical computing, there is another kind of transforms, image scrambling, which are as useful as Fourier transform, wavelet transform, etc. The main aim of image scrambling, which is generally used as the preprocessing or postprocessing in the confidentiality storage and transmission, and image information hiding, was to transform a meaningful image into a meaningless or disordered image in order to enhance the image security. In classical image processing, Arnold and Fibonacci image scrambling are often used. In order to realize these two image scrambling in quantum computers, this paper proposes the scrambling quantum circuits based on the flexible representation for quantum images. The circuits take advantage of the plain adder and adder modulo $$N$$ N to factor the classical transformations into basic unitary operators such as Control-NOT gates and Toffoli gates. Theoretical analysis indicates that the network complexity grows linearly with the size of the number to be operated.

Although image scaling algorithms in classical image processing have been extensively studied and widely used as basic image transformation methods, the quantum versions do not exist. Therefore, this paper proposes quantum algorithms and circuits to realize the quantum image scaling based on the improved novel enhanced quantum representation (INEQR) for quantum images. It is necessary to use interpolation in image scaling because there is an increase or a decrease in the number of pixels. The interpolation method used in this paper is nearest neighbor which is simple and easy to realize. First, NEQR is improved into INEQR to represent images sized $$2^{n_{1}} \times 2^{n_{2}}$$ 2 n 1 × 2 n 2 . Based on it, quantum circuits for image scaling using nearest neighbor interpolation from $$2^{n_{1}} \times 2^{n_{2}}$$ 2 n 1 × 2 n 2 to $$2^{m_{1}} \times 2^{m_{2}}$$ 2 m 1 × 2 m 2 are proposed. It is the first time to give the quantum image processing method that changes the size of an image. The quantum strategies developed in this paper initiate the research about quantum image scaling.

In this work, we investigate the dynamic features of the entropic uncertainty for two incompatible measurements under local unital and nonunital channels. Herein, we choose Pauli operators $$\sigma _x $$ σ x and $$\sigma _z $$ σ z as a pair of observables of interest measuring on particle A, and the uncertainty can be predicted when particle A is entangled with quantum memory B. We explore the dynamics of the uncertainty for the measurement under local unitary (phase-damping) and nonunitary (amplitude-damping) channels, respectively. Remarkably, we derive the entropic uncertainty relation under three different kinds of measurements of Pauli-observable pair under various realistic noisy environments; it has been found that the entropic uncertainty has the same tendency of its evolution during the AD and PD channel when we choose $$\sigma _x $$ σ x and $$\sigma _y $$ σ y measurement. Besides, we find out that the entropic uncertainty will have an optimal value if one chooses $$\sigma _x $$ σ x and $$\sigma _z $$ σ z as the measurement incompatibility, comparing with others. Furthermore, in order to reduce the entropic uncertainty in noisy environment, we propose an effective strategy to steer the amount by means of implementing a filtering operation on the particle under the two types of channels, respectively. It turns out that this operation can greatly reduce the entropic uncertainty by modulation of the operation strength. Thus, our investigations might offer an insight into the dynamics and steering of the entropic uncertainty in an open system.

In this paper, a novel watermarking scheme based on quantum wavelet transform (QWT) is proposed. Firstly, the wavelet coefficients are extracted by executing QWT on quantum image. Then, we utilize a dynamic vector for controlling embedding strength instead of a fixed parameter for embedding process in other schemes. Analysis and results show that the proposed dynamic watermarking scheme has better visual quality under a higher embedding capacity and outperforms the existing schemes in the literature.

Image translation, which maps the position of each picture element into a new position, is a basic image transformation. Although it has been deeply researched and widely used in classical image processing, its quantum version is a vacancy. This paper studies the quantum image translation (QIT) for the first time to promote the development of quantum image processing. Two types of QIT: entire translation and cyclic translation are proposed by giving the quantum translation circuits. The translation in $$X$$ X -direction and $$Y$$ Y -direction is separable, and the circuits for translating right or left are different.

We study the perfectly local indistinguishability of multipartite product states. Firstly, we follow the method of Zhang et al. (Phys Rev A 93:012314, 2016) to give another more concise set of $$2n-1$$ 2 n - 1 orthogonal product states in $${\mathbb {C}}^m\otimes {\mathbb {C}}^n$$ C m ⊗ C n $$(4\le m\le n)$$ ( 4 ≤ m ≤ n ) which can not be distinguished by local operations and classical communication. Then we use the three-dimensional cubes to present some product states which give us an intuitive view on how to construct locally indistinguishable product states in tripartite quantum systems. At last, we give an explicit construction of locally indistinguishable orthogonal product states for general multipartite systems.

It is shown that a realistic controlled bidirectional remote state preparation is possible using a large class of entangled quantum states having a particular structure. Existing protocols of probabilistic, deterministic and joint remote state preparation are generalized to obtain the corresponding protocols of controlled bidirectional remote state preparation (CBRSP). A general way of incorporating the effects of two well-known noise processes, the amplitude-damping and phase-damping noise, on the probabilistic CBRSP process is studied in detail by considering that noise only affects the travel qubits of the quantum channel used for the probabilistic CBRSP process. Also indicated is how to account for the effect of these noise channels on deterministic and joint remote state CBRSP protocols.

A quantum key agreement (QKA) protocol by utilizing a four-photon cluster state is proposed in this paper. The proposed QKA protocol extends the two-party QKA protocol with four-qubit cluster state (Shen et al. in Quantum Inf Process 13:2313–2324, 2014) into a multi-party case. The block transmission technique and decoy photons method are used in the presented protocol. Meanwhile, the qubit efficiency of the presented protocol is also improved by using the dense coding method. Security analysis shows that the proposed protocol is secure against both participant and outside attacks.