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Neural network is a powerful learning paradigm for data feature learning in the era of big data. However, most neural network models are deterministic models that ignore the uncertainty of data. Fuzzy neural networks are proposed to address this problem. FDNN is a hierarchical deep neural network that derives information from both fuzzy and neural representations, the representations are then fused to form representation to be classified. FDNN perform well on uncertain data classification tasks. In this paper, we proposed a novel hierarchical fused quantum fuzzy neural network (HQFNN). Different from classical FDNN, HQFNN uses quantum neural networks to learn fuzzy membership functions in fuzzy neural network. We conducted simulated experiment on two types of datasets (Dirty-MNIST and 15-Scene), the results show that the proposed model can outperform several existing methods. In addition, we demonstrate the robustness of the proposed quantum circuit.

The Quantum Alternating Operator Ansatz (QAOA+) is one of the Variational Quantum Algorithm (VQA) specifically developed to tackle combinatorial optimization problems by exploring the feasible space in search of a target solution. For constrained optimization problems with unconstrained variables, which we call Unconstrained-Variables Problems (UVPs), the mixed operators in the QAOA+ circuit are applied to the constrained variables, while the single-qubit rotating gates $R_X$ operate on the unconstrained variables. The expressibility of this circuit is limited by the shortage of two-qubit gates and the parameter sharing in the $R_X$, which consequently impacts the performance of QAOA+ for solving UVPs. Therefore, it is crucial to develop a suitable ansatz for UVPs. In this paper, we propose the Variational Quantum Algorithm-Preserving Feasible Space (VQA-PFS) ansatz, exemplified by the Uncapacitated Facility Location Problem (UFLP), that applies mixed operators on constrained variables while employing Hardware-Efficient Ansatz (HEA) on unconstrained variables. The numerical results demonstrate that VQA-PFS significantly enhances the success probability and exhibits faster convergence compared to QAOA+, Quantum Approximation Optimization Algorithm (QAOA), and HEA. Furthermore, VQA-PFS reduces the circuit depth dramatically in comparison to QAOA+ and QAOA. Our algorithm is general and instructive in tackling UVPs.

As a general framework, Matrix Exponential Dimensionality Reduction (MEDR) deals with the small-sample-size problem that appears in linear Dimensionality Reduction (DR) algorithms. High complexity is the bottleneck in this type of DR algorithm because one has to solve a large-scale matrix exponential eigenproblem. To address it, here we design a general quantum algorithm framework for MEDR based on the block-encoding technique. This framework is configurable, that is, by selecting suitable methods to design the block-encodings of the data matrices, a series of new efficient quantum algorithms can be derived from this framework. Specifically, by constructing the block-encodings of the data matrix exponentials, we solve the eigenproblem and then obtain the digital-encoded quantum state corresponding to the compressed low-dimensional dataset, which can be directly utilized as input state for other quantum machine learning tasks to overcome the curse of dimensionality. As applications, we apply this framework to four linear DR algorithms and design their quantum algorithms, which all achieve a polynomial speedup in the dimension of the sample over their classical counterparts., Comment: 14 pages, 6 figures

Recently, Zhou et al. have proposed a novel Interpolation-based (INTERP) strategy to generate the initial parameters for the Parameterized Quantum Circuit (PQC) in Quantum Approximate Optimization Algorithm (QAOA). INTERP produces the guess of the initial parameters at level $i+1$ by applying linear interpolation to the optimized parameters at level $i$, achieving better performance than random initialization (RI). Nevertheless, INTERP consumes extensive running costs for deep QAOA because it necessitates optimization at each level of the PQC. To address this problem, a Multilevel Leapfrogging Interpolation (MLI) strategy is proposed. MLI can produce the guess of the initial parameters from level $i+1$ to $i+l$ ($l>1$) at level $i$, omitting the optimization rounds from level $i+1$ to $(i+l-1)$. The final result is that MLI executes optimization at few levels rather than each level, and this operation is referred to as Multilevel Leapfrogging optimization (M-Leap). The performance of MLI is investigated on the Maxcut problem. Compared with INTERP, MLI reduces most optimization rounds. Remarkably, the simulation results demonstrate that MLI can achieve the same quasi-optima as INTERP while consuming only 1/2 of the running costs required by INTERP. In addition, for MLI, where there is no RI except for level $1$, the greedy-MLI strategy is presented. The simulation results suggest that greedy-MLI has better stability (i.e., a higher average approximation ratio) than INTERP and MLI beyond obtaining the same quasi-optima as INTERP. According to the efficiency of finding the quasi-optima, the idea of M-Leap might be extended to other training tasks, especially those requiring numerous optimizations, such as training adaptive quantum circuits.

The Quantum Alternating Operator Ansatz (QAOA+) is an extension of the Quantum Approximate Optimization Algorithm (QAOA), where the search space is smaller in solving constrained combinatorial optimization problems. However, QAOA+ requires a trivial feasible solution as the initial state, so it cannot be used for problems that are difficult to find a trivial feasible solution. For simplicity, we call them as Non-Trivial-Feasible-Solution Problems (NTFSP). In this paper, we take the Minimum Exact Cover (MEC) problem as an example, studying how to apply QAOA+ to NTFSP. As we know, exact covering (EC) is the feasible space of MEC problem, which has no trivial solutions. To overcome the above problem, the EC problem is divided into two steps to solve. First, disjoint sets are obtained, which is equivalent to solving independent sets. Second, on this basis, the sets covering all elements (i.e., EC) are solved. In other words, we transform MEC into a multi-objective constrained optimization problem, where feasible space consists of independent sets that are easy to find. Finally, we also verify the feasibility of the algorithm from numerical experiments. Our method provides a feasible way for applying QAOA+ to NTFSP, and is expected to expand its application significantly.

Anomaly detection of sequences is a hot topic in data mining. Anomaly Detection using Piecewise Aggregate approximation in the Amplitude Domain (called ADPAAD) is one of the widely used methods in anomaly detection of sequences. The core step in the classical algorithm for performing ADPAAD is to construct an approximate representation of the subsequence, where the elements of each subsequence are divided into several subsections according to the amplitude domain and then the average of the subsections is computed. It is computationally expensive when processing large-scale sequences. In this paper, we propose a quantum algorithm for ADPAAD, which can divide the subsequence elements and compute the average in parallel. Our quantum algorithm can achieve polynomial speedups on the number of subsequences and the length of subsequences over its classical counterpart.

Mean Centering (MC) is an important data preprocessing technique, which has a wide range of applications in data mining, machine learning, and multivariate statistical analysis. When the data set is large, this process will be time-consuming. In this paper, we propose an efficient quantum MC algorithm based on the block-encoding technique, which enables the existing quantum algorithms can get rid of the assumption that the original data set has been classically mean-centered. Specifically, we first adopt the strategy that MC can be achieved by multiplying by the centering matrix $C$, i.e., removing the row means, column means and row-column means of the original data matrix $X$ can be expressed as $XC$, $CX$ and $CXC$, respectively. This allows many classical problems involving MC, such as Principal Component Analysis (PCA), to directly solve the matrix algebra problems related to $XC$, $CX$ or $CXC$. Next, we can employ the block-encoding technique to realize MC. To achieve it, we first show how to construct the block-encoding of the centering matrix $C$, and then further obtain the block-encodings of $XC$, $CX$ and $CXC$. Finally, we describe one by one how to apply our MC algorithm to PCA and other algorithms.

Discriminative Canonical Correlation Analysis (DCCA) is a powerful supervised feature extraction technique for two sets of multivariate data, which has wide applications in pattern recognition. DCCA consists of two parts: (i) mean-centering that subtracts the sample mean from the sample; (ii) solving the generalized eigenvalue problem. The cost of DCCA is expensive when dealing with a large number of high-dimensional samples. To solve this problem, here we propose a quantum DCCA algorithm. Specifically, we devise an efficient method to compute the mean of all samples, then use block-Hamiltonian simulation and quantum phase estimation to solve the generalized eigenvalue problem. Our algorithm achieves a polynomial speedup in the dimension of samples under certain conditions over its classical counterpart.

Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix operations. Here, we propose an efficient quantum algorithm to solve it based on a assumption that the element of each vertex and its norms can be effectively accessed via a quantum random access memory data structure. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework to implement the quantum simulation of the Laplacian matrix. Then, the eigenvalues and eigenvectors of the Laplacian matrix are extracted by the quantum phase estimation algorithm. The core of our entire algorithm is to construct the block-encoding of the Laplacian matrix. To achieve this, we propose in detail how to construct the block-encodings of operators containing the information of the weight matrix and the degree matrix respectively, and further obtain the block-encoding of the Laplacian matrix. Compared with its classical counterpart, our algorithm has a polynomial speedup on the number of vertices and an exponential speedup on the dimension of each vertex. We also show that our algorithm can be extended to solve the eigenproblem of symmetric (non-symmetric) normalized Laplacian matrix.

Neighborhood Preserving Embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101, 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality $n$. However, the algorithm has two disadvantages: (1) It is incomplete in the sense that the input of the third sub-algorithm cannot be obtained by the second sub-algorithm. (2) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points $m$ and an exponential speedup on the dimensionality $n$ under certain conditions over the classical NPE algorithm, and achieve significant speedup compared to Liang et al.'s algorithm even without considering the complexity of the VQA., Comment: 12 pages

Anomaly detection plays a critical role in fraud detection, health care, intrusion detection, military surveillance, etc. Anomaly detection algorithm based on density estimation (called ADDE algorithm) is one of widely used algorithms. Liang et al. proposed a quantum version of the ADDE algorithm [Phys. Rev. A 99, 052310 (2019)] and it is believed that the algorithm has exponential speedups on both the number and the dimension of training data point over the classical algorithm. In this paper, we find that Liang et al.'s algorithm doesn't actually execute. Then we propose a new quantum ADDE algorithm based on amplitude estimation. It is shown that our algorithm can achieves exponential speedup on the number $M$ of training data points compared with the classical counterpart. Besides, the idea of our algorithm can be applied to optimize the anomaly detection algorithm based on kernel principal component analysis (called ADKPCA algorithm). Different from the quantum ADKPCA proposed by Liu et al. [Phys. Rev. A 97, 042315 (2018)], compared with the classical counterpart, which offer exponential speedup on the dimension $d$ of data points, our algorithm achieves exponential speedup on $M$.

Unsharp measurements play an increasingly important role in quantum information theory. In this paper, we study a three-party prepare-transform-measure experiment with unsharp measurements based on $ 3 \rightarrow 1 $ sequential random access codes (RACs). We derive optimal trade-off between the two correlation witnesses in $ 3 \rightarrow 1 $ sequential quantum random access codes (QRACs), and use the result to complete the self-testing of quantum preparations, instruments and measurements for three sequential parties. We also give the upper and lower bounds of the sharpness parameter to complete the robustness analysis of the self-testing scheme. In addition, we find that classical correlation witness violation based on $3 \rightarrow 1 $ sequential RACs cannot be obtained by both correlation witnesses simultaneously. This means that if the second party uses strong unsharp measurements to overcome the classical upper bound, the third party cannot do so even with sharp measurements. Finally, we give the analysis and comparison of the random number generation efficiency under different sharpness parameters based on the determinant value, $2 \rightarrow 1 $ and $3 \rightarrow 1 $ QRACs separately. This letter sheds new light on generating random numbers among multi-party in semi-device independent framework., Comment: arXiv admin note: text overlap with arXiv:1905.06726 by other authors

Most quantum information tasks based on Bell tests relie on the assumption of measurement independence. However, it is difficult to ensure that the assumption of measurement independence is always met in experimental operations, so it is crucial to explore the effects of relaxing this assumption on Bell tests. In this paper, we discuss the effects of relaxing the assumption of measurement independence on 1-parameter family of Bell (1-PFB) tests. For both general and factorizable input distributions, we establish the relationship among measurement dependence, guessing probability, and the maximum value of 1-PFB correlation function that Eve can fake. The deterministic strategy when Eve fakes the maximum value is also given. We compare the unknown information rate of Chain inequality and 1-PFB inequality, and find the range of the parameter in which it is more difficult for Eve to fake the maximum quantum violation in 1-PFB inequality than in Chain inequality.

Dimensionality reduction (DR) algorithms, which reduce the dimensionality of a given data set while preserving the information of the original data set as well as possible, play an important role in machine learning and data mining. Duan \emph{et al}. proposed a quantum version of the A-optimal projection algorithm (AOP) for dimensionality reduction [Phys. Rev. A 99, 032311 (2019)] and claimed that the algorithm has exponential speedups on the dimensionality of the original feature space $n$ and the dimensionality of the reduced feature space $k$ over the classical algorithm. In this paper, we correct the time complexity of Duan \emph{et al}.'s algorithm to $O(\frac{\kappa^{4s}\sqrt{k^s}} {\epsilon^{s}}\mathrm{polylog}^s (\frac{mn}{\epsilon}))$, where $\kappa$ is the condition number of a matrix that related to the original data set, $s$ is the number of iterations, $m$ is the number of data points and $\epsilon$ is the desired precision of the output state. Since the time complexity has an exponential dependence on $s$, the quantum algorithm can only be beneficial for high dimensional problems with a small number of iterations $s$. To get a further speedup, we propose an improved quantum AOP algorithm with time complexity $O(\frac{s \kappa^6 \sqrt{k}}{\epsilon}\mathrm{polylog}(\frac{nm}{\epsilon}) + \frac{s^2 \kappa^4}{\epsilon}\mathrm{polylog}(\frac{\kappa k}{\epsilon}))$ and space complexity $O(\log_2(nk/\epsilon)+s)$. With space complexity slightly worse, our algorithm achieves at least a polynomial speedup compared to Duan \emph{et al}.'s algorithm. Also, our algorithm shows exponential speedups in $n$ and $m$ compared with the classical algorithm when both $\kappa$, $k$ and $1/\epsilon$ are $O(\mathrm{polylog}(nm))$., Comment: 11 pages, 2 figures

Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper, we present efficient and memory-reduced quantum algorithms for solving linear systems with such structures by devising a new approach to implement the block-encodings of these structured matrices. More specifically, by decomposing $n\times n$ dense matrices into linear combinations of displacement matrices, we first deduce the parameterized representations of the matrices with displacement structures so that they can be treated similarly. With such representations, we then construct $\epsilon$-approximate block-encodings of these structured matrices in two different data access models, i.e., the black-box model and the QRAM data structure model. It is shown the quantum linear system solvers based on the proposed block-encodings provide a quadratic speedup with respect to the dimension over classical algorithms in the black-box model and an exponential speedup in the QRAM data structure model. In particular, these linear system solvers subsume known results with significant improvements and also motivate new instances where there was no specialized quantum algorithm before. As an application, one of the quantum linear system solvers is applied to the linear prediction of time series, which justifies the claimed quantum speedup is achievable for problems of practical interest., Comment: 19 pages

Logistic regression (LR) is an important machine learning model for classification, with wide applications in text classification, image analysis and medicine diagnosis, etc. However, training LR generally entails an iterative gradient descent method, and is quite time consuming when processing big data sets. To solve this problem, we present a quantum algorithm for LR to implement the key task of the gradient descent method, obtaining the classical gradients in each iteration. It is shown that our algorithm achieves exponential speedup over its classical counterpart in each iteration when the dimension of each data point M grows polylogarithmically with the number of data points N, i.e.,M=O(polylog N). It is worth noting that the optimal model parameters are finally derived by performing simple calculations on the obtained gradients. So once the optimal model parameters are determined, one can use them to classify new data at little cost., Comment: We have just found that our core algorithm is similar to the algorithm of a published paper (Shao C. Physical Review A, 2019, 99(4): 042325.),which makes our algorithm less innovative, so we apply for retraction

Ridge regression (RR) is an important machine learning technique which introduces a regularization hyperparameter $\alpha$ to ordinary multiple linear regression for analyzing data suffering from multicollinearity. In this paper, we present a quantum algorithm for RR, where the technique of parallel Hamiltonian simulation to simulate a number of Hermitian matrices in parallel is proposed and used to develop a quantum version of $K$-fold cross-validation approach, which can efficiently estimate the predictive performance of RR. Our algorithm consists of two phases: (1) using quantum $K$-fold cross-validation to efficiently determine a good $\alpha$ with which RR can achieve good predictive performance, and then (2) generating a quantum state encoding the optimal fitting parameters of RR with such $\alpha$, which can be further utilized to predict new data. Since indefinite dense Hamiltonian simulation has been adopted as a key subroutine, our algorithm can efficiently handle non-sparse data matrices. It is shown that our algorithm can achieve exponential speedup over the classical counterpart for (low-rank) data matrices with low condition numbers. But when the condition numbers of data matrices is large to be amenable to full or approximately full ranks of data matrices, only polynomial speedup can be achieved., Comment: 12 pages, 3 figures

Solving the Toeplitz systems, which is to find the vector $x$ such that $T_nx = b$ given an $n\times n$ Toeplitz matrix $T_n$ and a vector $b$, has a variety of applications in mathematics and engineering. In this paper, we present a quantum algorithm for solving the linear equations of Toeplitz matrices, in which the Toeplitz matrices are generated by discretizing a continuous function. It is shown that our algorithm's complexity is nearly $O(\kappa\textrm{log}^2 n)$, where $\kappa$ and $n$ are the condition number and the dimension of $T_n$ respectively. This implies our algorithm is exponentially faster than the best classical algorithm for the same problem if $\kappa=O(\textrm{poly}(\textrm{log}\,n))$. Since no assumption on the sparseness of $T_n$ is demanded in our algorithm, it can serve as an example of quantum algorithms for solving non-sparse linear systems., Comment: 10 pages

Association rules mining (ARM) is one of the most important problems in knowledge discovery and data mining. Given a transaction database that has a large number of transactions and items, the task of ARM is to acquire consumption habits of customers by discovering the relationships between itemsets (sets of items). In this paper, we address ARM in the quantum settings and propose a quantum algorithm for the key part of ARM, finding out frequent itemsets from the candidate itemsets and acquiring their supports. Specifically, for the case in which there are $M_f^{(k)}$ frequent $k$-itemsets in the $M_c^{(k)}$ candidate $k$-itemsets ($M_f^{(k)} \leq M_c^{(k)}$), our algorithm can efficiently mine these frequent $k$-itemsets and estimate their supports by using parallel amplitude estimation and amplitude amplification with complexity $\mathcal{O}(\frac{k\sqrt{M_c^{(k)}M_f^{(k)}}}{\epsilon})$, where $\epsilon$ is the error for estimating the supports. Compared with the classical counterpart, classical sampling-based algorithm, whose complexity is $\mathcal{O}(\frac{kM_c^{(k)}}{\epsilon^2})$, our quantum algorithm quadratically improves the dependence on both $\epsilon$ and $M_c^{(k)}$ in the best case when $M_f^{(k)}\ll M_c^{(k)}$ and on $\epsilon$ alone in the worst case when $M_f^{(k)}\approx M_c^{(k)}$., Comment: 8 pages

Quantum coherence is important in quantum mechanics, and its essence is from superposition principle. We study the coherence of any two pure states and that of their arbitrary superposition, and obtain the relationship between them. In the case that the two states have support on orthogonal subspaces, the relationship is simple, that is, the difference between the coherence of their superposition state and the average coherence of them is smaller than 1. In other cases, we obtain different and a little more complicated relationships. Furthermore, we also obtain the lower bound of coherence of superpositions., Comment: 4 papers

Recently, Zhang et al [Phys. Rev. A 92, 012332 (2015)] presented $4d-4$ orthogonal product states that are locally indistinguishable and completable in a $d\otimes d$ quantum system. Later, Zhang et al. [arXiv: 1509.01814v2 (2015)] constructed $2n-1$ orthogonal product states that are locally indistinguishable in $m\otimes n$ ($3\leq m \leq n$). In this paper, we construct a locally indistinguishable and completable orthogonal product basis with $4p-4$ members in a general $m\otimes n$ ($3\leq m \leq n$) quantum system, where $p$ is an arbitrary integer from $3$ to $m$, and give a very simple but quite effective proof for its local indistinguishability. Specially, we get a completable orthogonal product basis with $8$ members that cannot be locally distinguished in $m\otimes n$ ($3\leq m \leq n$) when $p=3$. It is so far the smallest completable orthogonal product basis that cannot be locally distinguished in a $m\otimes n$ quantum system. On the other hand, we construct a small locally indistinguishable orthogonal product basis with $2p-1$ members, which is maybe uncompletable, in $m\otimes n$ ($3\leq m \leq n$ and $p$ is an arbitrary integer from $3$ to $m$). We also prove its local indistinguishability. As a corollary, we give an uncompletable orthogonal product basis with $5$ members that are locally indistinguishable in $m\otimes n$ ($3\leq m \leq n$). All the results can lead us to a better understanding of the structure of a locally indistinguishable product basis in $m \otimes n$.

Association rules mining is one of the most important problems in knowledge discovery and data mining. The goal of it is to acquire consumption habits of customers by discovering the relationships between items from a transaction database that has a large number of transactions and items. The most compute intensive process for ARM is to find out the frequent 1-itemsets and 2-itemsets. In this paper, we propose a quantum algorithm for finding out the frequent 1-itemsets and 2-itemsets. In our algorithm, to mine the frequent 1-itemsets efficiently, we use the technique of amplitude amplification. To mine the frequent 2-itemsets efficiently, we propose a new tomography scheme, i.e., pure-state-based quantum state tomography. It is shown that our algorithm is potential to achieve exponential speedup in the number of transactions and polynomial speedup in the number of items over the classical algorithm., Comment: 5 pages

In this paper, we present a quantum-key-distribution (QKD)-based quantum private query (QPQ) protocol utilizing single-photon signal of multiple optical pulses. It maintains the advantages of the QKD-based QPQ, i.e., easy to implement and loss tolerant. In addition, different from the situations in the previous QKD-based QPQ protocols, in our protocol, the number of the items an honest user will obtain is always one and the failure probability is always zero. This characteristic not only improves the stability (in the sense that, ignoring the noise and the attack, the protocol would always succeed), but also benefits the privacy of the database (since the database will no more reveal additional secrets to the honest users). Furthermore, for the user's privacy, the proposed protocol is cheat sensitive, and for security of the database, we obtain an upper bound for the leaked information of the database in theory., Comment: 7 pages, 1 figure

In the general bipartite quantum system $m \otimes n$, Wang \emph{et al.} [Y.-L Wang \emph{et al.}, Phys. Rev. A \textbf{92}, 032313 (2015)] presented $3(m+n)-9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\otimes n $. First, in $3\otimes n (3< n)$ quantum system, we construct $3n-2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\otimes n (4\leq m\leq n)$, we present $3n+m-4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\otimes n(3\leq m\leq n)$, we show a smaller set with $2n-1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.

Quantum walks with memory(QWM) are a type of modified quantum walks that record the walker's latest path. As we know, only two kinds of QWM are presented up to now. It is desired to design more QWM for research, so that we can explore the potential of QWM. In this work, through presenting the one-to-one correspondence between QWM on a regular graph and quantum walks without memory(QWoM) on line digraph of the regular graph, we construct a generic model of QWM on regular graphs. This construction gives a general scheme for building all possible standard QWM on regular graphs and makes it possible to study properties of different kinds of QWM. Here, by taking the simplest example which is QWM with 1 memory on the line, we analyze some properties of QWM, such as variance, occupancy rate and localization., Comment: 8 pages, 5 figures

The determination of many special types of quantum states has been studied thoroughly, such as the generalized |GHZ> states, |W> states equivalent under stochastic local operations and classical communication and Dicke states. In this paper, we are going to study another special entanglement states which is stabilizer states. The stabilizer states and their subset graph states play an important role in quantum error correcting codes, multipartite purification and so on. We show that all n- qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for systems which are the supports of n independent generators of the corresponding stabilizer formalisms.

By proposing device-independent protocols, S. Pironio et al. [Nature 464, 1021-1024 (2010)] and R. Colbeck et al. [Nature Physics 8, 450-453 (2012)] proved that new randomness can be generated by using perfectly free random sources or partially free ones as seed. Subsequently, Li et al. [Phys. Rev. A 84, 034301 (2011)] studied this topic in the framework of semi-device-independent and proved that new randomness can be obtained from perfectly free random sources. Here we discuss whether and how partially free random sources bring us new randomness in semi-device-independent scenario. We propose a semi-device-independent randomness expansion protocol with partially free random sources, and obtain the condition that the partially free random sources should satisfy to generate new randomness. In the process of analysis, we acquire a new 2-dimensional quantum witness. Furthermore, we get the analytic relationship between the generated randomness and the 2-dimensional quantum witness violation., Comment: 8 pages, 3 figures

For any bipartite state, how strongly can one subsystem be quantum correlated with another? Using the Koashi-Winter relation, we study the upper bound of purified quantum discord, which is given by the sum of the von Neumann entropy of the unmeasured subsystem and the entanglement of formation shared between the unmeasured subsystem with the environment. In particular, we find that the Luo et al.'s conjecture on the quantum correlations and the Lindblad conjecture are all ture, when the entanglement of formation vanishes. Let the difference between the left discord and the right discord be captured by the discord distance. If the Lindblad conjecture is true, we show that the joint entropy is a tight upper bound for the discord distance. Further, we obtain a necessary and sufficient condition for saturating upper bounds of purified quantum discord and discord distance separately with the equality conditions for the Araki-Lieb inequality and the Lindblad conjecture. Furthermore, we show that the subadditive relation holds for any bipartite quantum discord., Comment: 7 pages, 0 figures

For any $n$-partite state $\rho_{A_{1}A_{2}\cdot\cdot\cdot A_{n}}$, we define its quantum mutual information matrix as an $n$ by $n$ matrix whose $(i,j)$-entry is given by quantum mutual information $I(\rho_{A_{i}A_{j}})$. Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we obtain necessary and sufficient conditions for the positive semidefinite quantum mutual information matrix. We further define the \emph{genuine} $n$-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states., Comment: 5 pages, no figures

Quantum private query (QPQ) is a kind of quantum protocols to protect both users' privacy in their communication. There is an interesting example, that is, Alice wants to buy one item from Bob's database, which is composed of a quantity of valuable messages. QPQ protocol is the communication procedure ensuring that Alice can get only one item from Bob, and at the same time, Bob cannot know which one was taken by Alice. Owing to its practicability, quantum-key-distribution-based QPQ has draw much attention in recent years. However, the post-processing of the key in such protocols, called oblivious key, remains far from being satisfactorily known. Especially, the error correction method for such special key is still missing. Here we focus on the post-processing of the oblivious key, including both dilution and error correction. On the one hand, we demonstrate that the previous dilution method, which greatly reduces the communication complexity, will bring Alice the chance to illegally obtain much additional information about Bob's database. Simulations show that by very limited queries Alice can obtain the whole database. On the other hand, we present an effective error-correction method for the oblivious key, which completes its post-processing and makes such QPQ more practical., Comment: 11 pages, 5 figures, 3 tables

Recently, counterfactual quantum cryptography proposed by T. G. Noh [Phys. Rev. Lett. 103, 230501 (2009)] becomes an interesting direction in quantum cryptography, and has been realized by some researchers (such as Y. Liu et al's [Phys. Rev. Lett. 109, 030501 (2012)]). However, we find out that it is insecure in practical high lossy channel setting. We analyze the secret key rates in lossy channel under a polarization-splitting-measurement attack. Analysis indicates that the protocol is insecure when the loss rate of the one-way channel exceeds $50%$., Comment: In this version, we changed some description

Recently, Sun et al. [Quant Inf Proc DOI: 10.1007/s11128-013-0569-x] presented an efficient multi-party quantum key agreement (QKA) protocol by employing single particles and unitary operations. The aim of this protocol is to fairly and securely negotiate a secret session key among $N$ parties with a high qubit efficiency. In addition, the authors claimed that no participant can learn anything more than his/her prescribed output in this protocol, i.e., the sub-secret keys of the participants can be kept secret during the protocol. However, here we points out that the sub-secret of a participant in Sun et al.'s protocol can be eavesdropped by the two participants next to him/her. In addition, a certain number of dishonest participants can fully determine the final shared key in this protocol. Finally, we discuss the factors that should be considered when designing a really fair and secure QKA protocol., Comment: 7 pages

Based on the instantaneous nonlocal quantum computation (INQC), Buhrman et al. proposed an excellent attack strategy to quantum position verification (QPV) protocols in 2011, and showed that, if the colluding adversaries are allowed to previously share unlimited entangled states, it is impossible to design an unconditionally secure QPV protocol in the previous model. Here, trying to overcome this no-go theorem, we find some assumptions in the INQC attack, which are implicit but essential for the success of this attack, and present three different QPV protocols where these assumptions are not satisfied. We show that for the general adversaries, who execute the attack operations at every common time slot or the time when they detect the arrival of the challenge signals from the verifiers, secure QPV is achievable. This implies practically secure QPV can be obtained even if the adversaries is allowed to share unlimited entanglement previously. Here by "practically" we mean that in a successful attack the adversaries need launch a new round of attack on the coming qubits with extremely high frequency so that none of the possible qubits, which may be sent at random time, will be missed. On the other side, using such Superdense INQC (SINQC) attack, the adversaries can still attack the proposed protocols successfully in theory. The particular attack strategies to our protocols are presented respectively. On this basis, we demonstrate the impossibility of secure QPV with looser assumptions, i.e. the enhanced no-go theorem for QPV., Comment: 19 pages, single column, 3 tables, 6 figures

In this paper, we present a semi-loss-tolerant strong quantum coin-flipping (QCF) protocol with the best bias of 0.3536. Our manuscript applies Quantum non-demolition (QND) measurement to quantum coin-flipping protocol. Furthermore, a single photon as a single qubit is used to avoid the difficult implementation of EPR resources. We also analyze the security of our protocol obtaining the best result among all coin-flipping protocols considering loss. A semi-loss-tolerant Quantum Dice Rolling (QDR) protocol is first proposed, and the security of corresponding three-party QDR is analyzed to better demonstrate the security of our QCF., Comment: 5 pages, 1 figure

A multi-user quantum key distribution protocol is proposed with single particles and the collective eavesdropping detection strategy on a star network. By utilizing this protocol, any two users of the network can accomplish quantum key distribution with the help of a serving center. Due to the utilization of collective eavesdropping detection strategy, the users of the protocol just need have the ability of performing certain unitary operations. Furthermore, we present three fault-tolerant versions of the proposed protocol, which can combat with the errors over different collective-noise channels. The security of all the proposed protocols is guaranteed by the theorems on quantum operation discrimination., Comment: 10 pages, 3 figures,1 table / Revised according to the reviewers' comments / Changed title / Near to the published version

Recently, Yang and Tan proposed a certificateless key exchange protocol without pairing, and claimed their scheme satisfies forward secrecy, which means no adversary could derive an already-established session key unless the full user secret keys (including a private key and an ephemeral secret key) of both communication parties are compromised. However, in this paper, we point out their protocol is actually not secure as claimed by presenting an attack launched by an adversary who has learned the private key of one party and the ephemeral secret key of the other, but not the full user secret keys of both parties. Furthermore, to make up this flaw, we also provide an improved protocol in which the private key and the ephemeral secret key are closely intertwined with each other for generating the session key, thus above attack can be efficiently resisted.

We present a flexible quantum-key-distribution-based protocol for quantum private queries. Similar to M. Jakobi et al's protocol [Phys. Rev. A 83, 022301 (2011)], it is loss tolerant, practical and robust against quantum memory attack. Furthermore, our protocol is more flexible and controllable. We show that, by adjusting the value of $\theta$, the average number of the key bits Alice obtains can be located on any fixed value the users wanted for any database size. And the parameter $k$ is generally smaller (even $k=1$ can be achieved) when $\theta<\pi/4$, which implies lower complexity of both quantum and classical communications. Furthermore, the users can choose a smaller $\theta$ to get better database security, or a larger $\theta$ to obtain a lower probability with which Bob can correctly guess the address of Alice's query., Comment: 6 pages, 5 figures

In this paper, we present a quantum strong coin flipping protocol. In this protocol, an EPR pair and a quantum memory storage are made use of, and losses in the quantum communication channel and quantum memory storage are all analyzed. We obtain the bias in the fair scenario as a function of $p$, where $p$ is the probability that the particle in Bob's quantum memory storage is lost, which means our bias varies as the degree of losses in the quantum memory storage changes. Therefore we call our protocol semi-loss-tolerant. We also show that the bias decreases with decreasing $p$. When $p$ approaches 0, the bias approaches 0.3536, which is less than that of all the previous loss-tolerant protocols. Details of both parties' optimal cheating strategies are also given and analyzed. What's more, experimental feasibility is discussed and demonstrated. Compared with previous qubit-based loss-tolerant SCF protocols, we introduce the EPR pair to keep our protocol loss-tolerant while trying to push down the bias. In addition, a quantum memory storage is used and the losses in it has been taken into account. We obtain the bias in the fair scenario as a function of $p$, where $p$ is the probability that the particle in Bob's quantum memory storage is lost, which means our bias varies as the degree of losses in the quantum memory storage changes. We also show that the bias decreases with decreasing $p$. When $p$ approaches 0, the bias approaches 0.3536, which is less than that of all the previous loss-tolerant protocols. Details of both parties' optimal cheating strategies are also given and analyzed. Besides, experimental feasibility is discussed and demonstrated., Comment: 12 pages, 1 figure; Quantum Info. Comput. (2012)

As a new model for signing quantum message, arbitrated quantum signature (AQS) has recently received a lot of attention. In this paper we study the cryptanalysis of previous AQS protocols from the aspects of forgery and disavowal. We show that in these protocols the receiver Bob can realize existential forgery of the sender's signature under known message attack. Bob can even achieve universal forgery when the protocols are used to sign a classical message. Furthermore, the sender Alice can successfully disavow any of her signatures by simple attack. The attack strategies are described in detail and some discussions about the potential improvements of the protocols are given. Finally we also present several interesting topics in future study on AQS protocols., Comment: 7 pages, no figures

A family of rank-n (n=5,6,7,8) three-qubit mixed states are constructed. The explicit expressions for the three-tangle and optimal decompositions for all these states are given. The CKW relations for these states are also discussed., Comment: 10 pages, 2 eps figures

Cryptanalysis is an important branch in the study of cryptography, including both the classical cryptography and the quantum one. In this paper we analyze the security of two three-party quantum key distribution protocols (QKDPs) proposed recently, and point out that they are susceptible to a simple and effective attack, i.e. the dense-coding attack. It is shown that the eavesdropper Eve can totally obtain the session key by sending entangled qubits as the fake signal to Alice and performing collective measurements after Alice's encoding. The attack process is just like a dense-coding communication between Eve and Alice, where a special measurement basis is employed. Furthermore, this attack does not introduce any errors to the transmitted information and consequently will not be discovered by Alice and Bob. The attack strategy is described in detail and a proof for its correctness is given. At last, the root of this insecurity and a possible way to improve these protocols are discussed., Comment: 6 pages, 3 figures

Based on quantum encryption, we present a new idea for quantum public-key cryptography (QPKC) and construct a whole theoretical framework of a QPKC system. We show that the quantum-mechanical nature renders it feasible and reasonable to use symmetric keys in such a scheme, which is quite different from that in conventional public-key cryptography. The security of our scheme is analyzed and some features are discussed. Furthermore, the state-estimation attack to a prior QPKC scheme is demonstrated., Comment: 8 pages, 1 figure, Revtex4