Information-theoretic perspective on performance assessment and fundamental limit of quantum imaging [Invited]

Performance assessment of an imaging system is important for the optimization design with various technologies.The information-theoretic viewpoint based on communication theory or statistical inference theory can provide objective and operational measures on imaging performance. These approaches can be further developed by combining with the quantum statistical inference theory for optimizing imaging performance over measurements and analyze its quantum limits, which is demanded in order to improve an imaging system when the photon shot noise in the measurement is the dominant noise source. The aim of this review is to discuss and analyze the recent developments in this branch of quantum imaging.

Design of a novel hybrid quantum deep neural network in INEQR images classification

We redesign the parameterized quantum circuit in the quantum deep neural network, construct a three-layer structure as the hidden layer, and then use classical optimization algorithms to train the parameterized quantum circuit, thereby propose a novel hybrid quantum deep neural network(HQDNN) used for image classification. After bilinear interpolation reduces the original image to a suitable size, an improved novel enhanced quantum representation(INEQR) is used to encode it into quantum states as the input of the HQDNN. Multi-layer parameterized quantum circuits are used as the main structure to implement feature extraction and classification. The output results of parameterized quantum circuits are converted into classical data through quantum measurements and then optimized on a classical computer. To verify the performance of the HQDNN, we conduct binary classification and three classification experiments on the MNIST(Modified National Institute of Standards and Technology) data set. In the first binary classification, the accuracy of 0 and 4 exceeds98%. Then we compare the performance of three classification with other algorithms, the results on two datasets show that the classification accuracy is higher than that of quantum deep neural network and general quantum convolutional neural network.

Double quantum images encryption scheme based on chaotic system

This paper explores a double quantum images representation(DNEQR)model that allows for simultaneous storage of two digital images in a quantum superposition state.Additionally,a new type of two-dimensional hyperchaotic system based on sine and logistic maps is investigated,offering a wider parameter space and better chaotic behavior compared to the sine and logistic maps.Based on the DNEQR model and the hyperchaotic system,a double quantum images encryption algorithm is proposed.Firstly,two classical plaintext images are transformed into quantum states using the DNEQR model.Then,the proposed hyperchaotic system is employed to iteratively generate pseudo-random sequences.These chaotic sequences are utilized to perform pixel value and position operations on the quantum image,resulting in changes to both pixel values and positions.Finally,the ciphertext image can be obtained by qubit-level diffusion using two XOR operations between the position-permutated image and the pseudo-random sequences.The corresponding quantum circuits are also given.Experimental results demonstrate that the proposed scheme ensures the security of the images during transmission,improves the encryption efficiency,and enhances anti-interference and anti-attack capabilities.

Integer multiple quantum image scaling based on NEQR and bicubic interpolation

As a branch of quantum image processing,quantum image scaling has been widely studied.However,most of the existing quantum image scaling algorithms are based on nearest-neighbor interpolation and bilinear interpolation,the quantum version of bicubic interpolation has not yet been studied.In this work,we present the first quantum image scaling scheme for bicubic interpolation based on the novel enhanced quantum representation(NEQR).Our scheme can realize synchronous enlargement and reduction of the image with the size of 2^(n)×2^(n) by integral multiple.Firstly,the image is represented by NEQR and the original image coordinates are obtained through multiple CNOT modules.Then,16 neighborhood pixels are obtained by quantum operation circuits,and the corresponding weights of these pixels are calculated by quantum arithmetic modules.Finally,a quantum matrix operation,instead of a classical convolution operation,is used to realize the sum of convolution of these pixels.Through simulation experiments and complexity analysis,we demonstrate that our scheme achieves exponential speedup over the classical bicubic interpolation algorithm,and has better effect than the quantum version of bilinear interpolation.

An anti-aliasing filtering of quantum images in spatial domain using a pyramid structure

As a part of quantum image processing,quantum image filtering is a crucial technology in the development of quantum computing.Low-pass filtering can effectively achieve anti-aliasing effects on images.Currently,most quantum image filterings are based on classical domains and grayscale images,and there are relatively fewer studies on anti-aliasing in the quantum domain.This paper proposes a scheme for anti-aliasing filtering based on quantum grayscale and color image scaling in the spatial domain.It achieves the effect of anti-aliasing filtering on quantum images during the scaling process.First,we use the novel enhanced quantum representation(NEQR)and the improved quantum representation of color images(INCQI)to represent classical images.Since aliasing phenomena are more pronounced when images are scaled down,this paper focuses only on the anti-aliasing effects in the case of reduction.Subsequently,we perform anti-aliasing filtering on the quantum representation of the original image and then use bilinear interpolation to scale down the image,achieving the anti-aliasing effect.The constructed pyramid model is then used to select an appropriate image for upscaling to the original image size.Finally,the complexity of the circuit is analyzed.Compared to the images experiencing aliasing effects solely due to scaling,applying anti-aliasing filtering to the images results in smoother and clearer outputs.Additionally,the anti-aliasing filtering allows for manual intervention to select the desired level of image smoothness.

Quantum metrology

The statistical error is ineluctable in any measurement. Quantum techniques, especially with the development of quantum information, can help us squeeze the statistical error and enhance the precision of measurement. In a quantum system, there are some quantum parameters, such as the quantum state, quantum operator, and quantum dimension, which have no classical counterparts. So quantum metrology deals with not only the traditional parameters, but also the quantum parameters. Quantum metrology includes two important parts： measuring the physical parameters with a precision beating the classical physics limit and measuring the quantum parameters precisely. In this review, we will introduce how quantum characters （e.g., squeezed state and quantum entanglement） yield a higher precision, what the research areas are scientists most interesting in, and what the development status of quantum metrology and its perspectives are.

Quantum color image scaling based on bilinear interpolation

As a part of quantum image processing, quantum image scaling is a significant technology for the development of quantum computation. At present, most of the quantum image scaling schemes are based on grayscale images, with relatively little processing for color images. This paper proposes a quantum color image scaling scheme based on bilinear interpolation, which realizes the 2^(n_(1)) × 2^(n_(2)) quantum color image scaling. Firstly, the improved novel quantum representation of color digital images(INCQI) is employed to represent a 2^(n_(1)) × 2^(n_(2)) quantum color image, and the bilinear interpolation method for calculating pixel values of the interpolated image is presented. Then the quantum color image scaling-up and scaling-down circuits are designed by utilizing a series of quantum modules, and the complexity of the circuits is analyzed.Finally, the experimental simulation results of MATLAB based on the classical computer are given. The ultimate results demonstrate that the complexities of the scaling-up and scaling-down schemes are quadratic and linear, respectively, which are much lower than the cubic function and exponential function of other bilinear interpolation schemes.

Quantum watermarking based on threshold segmentation using quantum informational entropy

We propose a new quantum watermarking scheme based on threshold selection using informational entropy of quantum image.The core idea of this scheme is to embed information into object and background of cover image in different ways.First,a threshold method adopting the quantum informational entropy is employed to determine a threshold value.The threshold value can then be further used for segmenting the cover image to a binary image,which is an authentication key for embedding and extraction information.By a careful analysis of the quantum circuits of the scheme,that is,translating into the basic gate sequences which show the low complexity of the scheme.One of the simulation-based experimental results is entropy difference which measures the similarity of two images by calculating the difference in quantum image informational entropy between watermarked image and cover image.Furthermore,the analyses of peak signal-to-noise ratio,histogram and capacity of the scheme are also provided.

Quantum Steganography Application in the Electrical Network for Quantum Image and Watermark with Self-Adaptive

With the development of Globe Energy Internet,quantum steganography has been used for information hiding to improve copyright protection.Based on secure quantum communication protocol,and flexible steganography,secret information is embedded in quantum images in covert communication.Under the premise of guaranteeing the quality of the quantum image,the secret information is transmitted safely with virtue of good imperceptibility.A novel quantum watermark algorithm is proposed in the paper,based on the shared group key value of the communication parties and the transmission of the selected carrier map pixel gray higher than 8 bits.According to the shared group key value of the communication parties,the two effective Bell state qubits of the carried quantum streak image are replaced with secret information.Compared with the existing algorithms,the new algorithm improves the robustness of the secret information itself and the execution efficiency of its embedding and extraction.Experimental simulation and performance analysis also show that the novel algorithm has an excellent performance in transparency,robustness and embedded capacity.

Quantum digital spiral imaging

We demonstrate that the combination of digital spiral imaging with high-dimensional orbital angular momentum(OAM)entanglement can be used for efficiently probing and identifying pure phase objects,where the probing light does not necessarily touch the object,via the experimental,non-local decomposition of non-integer pure phase vortices in OAM-entangled photon pairs.The entangled photons are generated by parametric downconversion and then measured with spatial light modulators and single-mode fibers.The fractional phase vortices are defined in the idler photons,while their corresponding spiral spectra are obtained non-locally by scanning the measured OAM states in the signal photons.We conceptually illustrate our results with the biphoton Klyshko picture and the effective dimensionality to demonstrate the high-dimensional nature of the associated quantum OAM channels.Our result is a proof of concept that quantum imaging techniques exploiting high-dimensional entanglement can potentially be used for remote sensing.

Improving the resolution temporal in quantum and classical imaging

The point-spread function of an optical system determines its optical resolution for both spatial and temporal imaging. For spatial imaging, it is given by a Fourier transform of the pupil function of the system. For temporal imaging based on nonlinear optical processes, such as sum-frequency generation or four-wave mixing, the pointspread function is related to the waveform of the pump wave by a nonlinear transformation. We compare the point-spread functions of three temporal imaging schemes： sum-frequency generation, co-propagating four-wave mixing, and counter-propagating four-wave mixing, and demonstrate that the last scheme provides the best temporal resolution. Our results are valid for both quantum and classical temporal imaging.

Quantum advantages for image filtering on images with efficient encoding and lower-bounded signal-to-noise ratio

Quantum image processing has long been a fascinating field,but establishing the existence of quantum speedup for all images remains challenging.In this study,we aim to identify a subset of images for which a quantum algorithm can be developed with a guaranteed advantage.Specifically,we present a quantum image filtering algorithm that exhibits an exponential speedup for efficiently encoded images with a lower-bounded signal-to-noise ratio.Our approach relies on a fixed-point Grover's search to emulate the effect of Hadamard multiplication with the filtering function.To demonstrate its effectiveness,we apply our method to three typical filtering problems.Additionally,we emphasize the significance of the efficient-encoding assumption by illustrating that the quantum speedup may diminish for images lacking efficient encoding.Our work underscores the importance of exploring image types and features to realize potential quantum advantages in image processing.

Quantum-limited resolution of partially coherent sources

Discriminating two spatially separated sources is one of the most fundamental problems in imaging.Recent research based on quantum parameter estimation theory shows that the resolution limit of two incoherent point sources given by Rayleigh can be broken.However,in realistic optical systems,there often exists coherence in the imaging light field,and there have been efforts to analyze the optical resolution in the presence of partial coherence.Nevertheless,how the degree of coherence between two point sources affects the resolution has not been fully understood.Here,we analyze the quantum-limited resolution of two partially coherent point sources by explicitly relating the state after evolution through the optical systems to the coherence of the sources.In particular,we consider the situation in which coherence varies with the separation.We propose a feasible experiment scheme to realize the nearly optimal measurement,which adaptively chooses the binary spatial-mode demultiplexing measurement and direct imaging.Our results will have wide applications in imaging involving coherence of light.

Efficient quantum arithmetic operation circuits for quantum image processing

Efficient quantum circuits for arithmetic operations are vital for quantum algorithms.A fault-tolerant circuit is required for a robust quantum computing in the presence of noise.Quantum circuits based on Clifford+T gates are easily rendered faulttolerant.Therefore,reducing the T-depth and T-Count without increasing the qubit number represents vital optimization goals for quantum circuits.In this study,we propose the fault-tolerant implementations for TR and Peres gates with optimized T-depth and T-Count.Next,we design fault-tolerant circuits for quantum arithmetic operations using the TR and Peres gates.Then,we implement cyclic and complete translations of quantum images using quantum arithmetic operations,and the scalar matrix multiplication.Comparative analysis and simulation results reveal that the proposed arithmetic and image operations are efficient.For instance,cyclic translations of a quantum image produce 50%T-depth reduction relative to the previous best-known cyclic translation.

Chromatic framework for quantum movies and applications in creating montages

A framework that introduces chromatic considerations to earlier descriptions of movies on quantum computers is proposed. This chromatic framework for quantum movies （CFQM） integrates chromatic components of individual frames （each a multi-channel quantum image - MCQI state） that make up the movie, while each frame is tagged to a time component of a quantum register （i.e., a movie strip）. The formulation of the CFQM framework and properties inherent to the MCQI images facilitate the execution of a cortege of carefully formulated transformations including the frame-to-frame （FTF）, color of interest （COI）, and subblock swapping （SBS） operations that are not realizable on other quantum movie formats. These innovative transformations are deployed in the creation of digital movie-like montages on the CFQM framework. Future studies could explore additional MCQI-related operations and their use to execute more advanced montage applications.

Three-dimensional quantum wavelet transforms

Wavelet transform is being widely used in the field of information processing.One-dimension and two-dimension quantum wavelet transforms have been investigated as important tool algorithms.However,three-dimensional quantum wavelet transforms have not been reported.This paper proposes a multi-level three-dimensional quantum wavelet transform theory to implement the wavelet transform for quantum videos.Then,we construct the iterative formulas for the multi-level three-dimensional Haar and Daubechies D4 quantum wavelet transforms,respectively.Next,we design quantum circuits of the two wavelet transforms using iterative methods.Complexity analysis shows that the proposed wavelet transforms offer exponential speed-up over their classical counterparts.Finally,the proposed quantum wavelet transforms are selected to realize quantum video compression as a primary application.Simulation results reveal that the proposed wavelet transforms have better compression performance for quantum videos than two-dimension quantum wavelet transforms.

A New Quantum Gray-Scale Image Encoding Scheme

In this paper, a new quantum images encoding scheme is proposed. The proposed scheme mainly consists of four different encoding algorithms. The idea behind of the scheme is a binary key generated randomly for each pixel of the original image. Afterwards, the employed encoding algorithm is selected corresponding to the qubit pair of the generated randomized binary key. The security analysis of the proposed scheme proved its enhancement through both randomization of the generated binary image key and altering the gray-scale value of the image pixels using the qubits of randomized binary key. The simulation of the proposed scheme assures that the final encoded image could not be recognized visually. Moreover, the histogram diagram of encoded image is flatter than the originM one. The Shannon entropies of the final encoded images are significantly higher than the original one, which indicates that the attacker can not gain any information about the encoded images.

A Spatial Domain Quantum Watermarking Scheme

This paper presents a spatial domain quantum watermarking scheme. For a quantum watermarking scheme,a feasible quantum circuit is a key to achieve it. This paper gives a feasible quantum circuit for the presented scheme. In order to give the quantum circuit, a new quantum multi-control rotation gate, which can be achieved with quantum basic gates, is designed. With this quantum circuit, our scheme can arbitrarily control the embedding position of watermark images on carrier images with the aid of auxiliary qubits. Besides reversely acting the given quantum circuit, the paper gives another watermark extracting algorithm based on quantum measurements. Moreover, this paper also gives a new quantum image scrambling method and its quantum circuit. Differ from other quantum watermarking schemes, all given quantum circuits can be implemented with basic quantum gates. Moreover, the scheme is a spatial domain watermarking scheme, and is not based on any transform algorithm on quantum images. Meanwhile, it can make sure the watermark be secure even though the watermark has been found. With the given quantum circuit, this paper implements simulation experiments for the presented scheme. The experimental result shows that the scheme does well in the visual quality and the embedding capacity.

Quantum Color Image Scaling on QIRHSI Model

Scaling operations are widely used in traditional image processing.Therefore, in this paper, an improved quantum image representation based on HSIcolor space (IQIRHSI) is proposed, which extends the original 2n × 2n size togeneral 2n1 × 2n2 size. Then, the quantum algorithms and circuits were designedto implement quantum image scaling. Interpolation was introduced to recover thelost information in the scaled image. The nearest neighbor interpolation methodwas researched on scaled IQIRHSI to make the interpolation method easy toimplement. Finally, the complexity of the quantum circuit for image scaling wasanalyzed and the process of quantum image scaling was described in detail byexamples.

Joint optimal measurement for locating two incoherent optical point sources near the Rayleigh distance

Simultaneously optimizing the estimation of the centroid and separation of two incoherent optical point sources is constrained by a tradeoff relation determined by an incompatibility coefficient.At the Rayleigh distance,the incompatibility coefficient vanishes and thus the tradeoff relation no longer restricts the simultaneous optimization of measurement for a joint estimation.We construct such a joint optimal measurement by an elaborated analysis on the operator algebra of the symmetric logarithmic derivative.Our work not only confirms the existence of a joint optimal measurement for this specific imaging model,but also gives a promising method to characterize the condition on measurement compatibility for general multiparameter estimation problems.