arXiv subjects

Lvzhou Li

Publications and source records attributed to Lvzhou Li.

At least 55 records · Page 3Linked to original sources

arXiv2023

Detection and evaluation of abnormal user behavior based on quantum generation adversarial network

Quantum computing holds tremendous potential for processing high-dimensional data, capitalizing on the unique capabilities of superposition and parallelism within quantum states. As we navigate the noisy intermediate-scale quantum (NISQ) era, the exploration of quantum computing applications has emerged as a compelling frontier. One area of particular interest within the realm of cyberspace security is Behavior Detection and Evaluation (BDE). Notably, the detection and evaluation of internal abnormal behaviors pose significant challenges, given their infrequent occurrence or even their concealed nature amidst vast volumes of normal data. In this paper, we introduce a novel quantum behavior detection and evaluation algorithm (QBDE) tailored for internal user analysis. The QBDE algorithm comprises a Quantum Generative Adversarial Network (QGAN) in conjunction with a classical neural network for detection and evaluation tasks. The QGAN is built upon a hybrid architecture, encompassing a Quantum Generator ($G_Q$) and a Classical Discriminator ($D_C$). $G_Q$, designed as a parameterized quantum circuit (PQC), collaborates with $D_C$, a classical neural network, to collectively enhance the analysis process. To address the challenge of imbalanced positive and negative samples, $G_Q$ is employed to generate negative samples. Both $G_Q$ and $D_C$ are optimized through gradient descent techniques. Through extensive simulation tests and quantitative analyses, we substantiate the effectiveness of the QBDE algorithm in detecting and evaluating internal user abnormal behaviors. Our work not only introduces a novel approach to abnormal behavior detection and evaluation but also pioneers a new application scenario for quantum algorithms. This paradigm shift underscores the promising prospects of quantum computing in tackling complex cybersecurity challenges.

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arXiv2023

Universal approach to deterministic spatial search via alternating quantum walks

Spatial search is an important problem in quantum computation, which aims to find a marked vertex on a graph. We propose a novel approach for designing deterministic quantum search algorithms on a variety of graphs via alternating quantum walks. Our approach is universal because it does not require an instance-specific analysis for different graphs. We highlight the flexibility of our approach by proving that for Johnson graphs, rook graphs, complete-square graphs and complete bipartite graphs, our quantum algorithms can find the marked vertex with $100\%$ success probability and achieve quadratic speedups over classical algorithms. This not only gives an alternative succinct way to prove the existing results, but also leads to new interesting findings on more general graphs.

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arXiv2023

Quantum and classical query complexities for determining connectedness of matroids

Connectivity is a fundamental structural property of matroids, and has been studied algorithmically over 50 years. In 1974, Cunningham proposed a deterministic algorithm consuming $O(n^{2})$ queries to the independence oracle to determine whether a matroid is connected. Since then, no algorithm, not even a random one, has worked better. To the best of our knowledge, the classical query complexity lower bound and the quantum complexity for this problem have not been considered. Thus, in this paper we are devoted to addressing these issues, and our contributions are threefold as follows: (i) First, we prove that the randomized query complexity of determining whether a matroid is connected is $Ω(n^2)$ and thus the algorithm proposed by Cunningham is optimal in classical computing. (ii) Second, we present a quantum algorithm with $O(n^{3/2})$ queries, which exhibits provable quantum speedups over classical ones. (iii) Third, we prove that any quantum algorithm requires $Ω(n)$ queries, which indicates that quantum algorithms can achieve at most a quadratic speedup over classical ones. Therefore, we have a relatively comprehensive understanding of the potential of quantum computing in determining the connectedness of matroids.\

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arXiv2023

Characterization, synthesis, and optimization of quantum circuits over multiple-control $\textit{Z}$-rotation gates: A systematic study

We conduct a systematic study of quantum circuits composed of multiple-control $Z$-rotation (MCZR) gates as primitives, since they are widely-used components in quantum algorithms and also have attracted much experimental interest in recent years. Herein, we establish a circuit-polynomial correspondence to characterize the functionality of quantum circuits over the MCZR gate set with continuous parameters. An analytic method for exactly synthesizing such quantum circuit to implement any given diagonal unitary matrix with an optimal gate count is proposed, which also enables the circuit depth optimal for specific cases with pairs of complementary gates. Furthermore, we present a gate-exchange strategy together with a flexible iterative algorithm for effectively optimizing the depth of any MCZR circuit, which can also be applied to quantum circuits over any other commuting gate set. Besides the theoretical analysis, the practical performances of our circuit synthesis and optimization techniques are further evaluated by numerical experiments on two typical examples in quantum computing, including diagonal Hermitian operators and Quantum Approximate Optimization Algorithm (QAOA) circuits with tens of qubits, which can demonstrate a reduction in circuit depth by 33.40\% and 15.55\% on average over relevant prior works, respectively. Therefore, our methods and results provide a pathway for implementing quantum circuits and algorithms on recently developed devices.

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arXiv2023

GSQAS: Graph Self-supervised Quantum Architecture Search

Quantum Architecture Search (QAS) is a promising approach to designing quantum circuits for variational quantum algorithms (VQAs). However, existing QAS algorithms require to evaluate a large number of quantum circuits during the search process, which makes them computationally demanding and limits their applications to large-scale quantum circuits. Recently, predictor-based QAS has been proposed to alleviate this problem by directly estimating the performances of circuits according to their structures with a predictor trained on a set of labeled quantum circuits. However, the predictor is trained by purely supervised learning, which suffers from poor generalization ability when labeled training circuits are scarce. It is very time-consuming to obtain a large number of labeled quantum circuits because the gate parameters of quantum circuits need to be optimized until convergence to obtain their ground-truth performances. To overcome these limitations, we propose GSQAS, a graph self-supervised QAS, which trains a predictor based on self-supervised learning. Specifically, we first pre-train a graph encoder on a large number of unlabeled quantum circuits using a well-designed pretext task in order to generate meaningful representations of circuits. Then the downstream predictor is trained on a small number of quantum circuits' representations and their labels. Once the encoder is trained, it can apply to different downstream tasks. In order to better encode the spatial topology information and avoid the huge dimension of feature vectors for large-scale quantum circuits, we design a scheme to encode quantum circuits as graphs. Simulation results on searching circuit structures for variational quantum eigensolver and quantum state classification show that GSQAS outperforms the state-of-the-art predictor-based QAS, achieving better performance with fewer labeled circuits.

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arXiv2023

Playing Mastermind on quantum computers

From the 1970s up to now, Mastermind, a classic two-player game, has attracted plenty of attention, not only from the public as a popular game, but also from the academic community as a scientific issue. Mastermind with n positions and k colors is formally described as: the codemaker privately chooses a secret $s\in [k]^n$, and the coderbreaker want to determine $s$ in as few queries like $f_s(x)$ as possible to the codemaker, where $f_s(x)$ indicates how x is close to s. The complexity of a strategy is measured by the number of queries used. In this work we study playing Mastermind on quantum computers in both non-adaptive and adaptive settings, obtaining efficient quantum algorithms which are all exact (i.e., return the correct result with certainty) and show huge quantum speedups. Technically, we develop a three-step framework for designing quantum algorithms for the general string learning problem, which not only allows huge quantum speedups on playing Mastermind, but also may shed light on exploring quantum speedups for other string learning problems.

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arXiv2023

Concise and Efficient Quantum Algorithms for Distribution Closeness Testing

We study the impact of quantum computation on the fundamental problem of testing the property of distributions. In particular, we focus on testing whether two unknown classical distributions are close or far enough, and propose the currently best quantum algorithms for this problem under the metrics of $l^1$-distance and $l^2$-distance. Compared with the latest results given in \cite{gilyen2019distributional} which relied on the technique of quantum singular value transformation (QSVT), our algorithms not only have lower complexity, but also are more concise.

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arXiv2023

Deterministic quantum search with adjustable parameters: implementations and applications

Grover's algorithm provides a quadratic speedup over classical algorithms to search for marked elements in an unstructured database. The original algorithm is probabilistic, returning a marked element with bounded error. There are several schemes to achieve the deterministic version, by using the generalized Grover's iteration $G(α,β):=S_r(β)\, S_o(α)$ composed of phase oracle $S_o(α)$ and phase rotation $S_r(β)$. However, in all the existing schemes the value range of $α$ and $β$ is limited; for instance, in the three early schemes $α$ and $β$ are determined by the proportion of marked states $M/N$. In this paper, we break through this limitation by presenting a search framework with adjustable parameters, which allows $α$ or $β$ to be arbitrarily given. The significance of the framework lies not only in the expansion of mathematical form, but also in its application value, as we present two disparate problems which we are able to solve deterministically using the proposed framework, whereas previous schemes are ineffective.

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arXiv2022

A brief introduction to quantum algorithms

Quantum algorithms are demonstrated to outperform classical algorithms for certain problems and thus are promising candidates for efficient information processing. Herein we aim to provide a brief and popular introduction to quantum algorithms for both the academic community and the general public with interest. We start from elucidating quantum parallelism, the basic framework of quantum algorithms and the difficulty of quantum algorithm design. Then we mainly focus on a historical overview of progress in quantum algorithm research over the past three to four decades. Finally, we clarify two common questions about the study of quantum algorithms, hoping to stimulate readers for further exploration.

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arXiv2022

Automatic Depth-Optimized Quantum Circuit Synthesis for Diagonal Unitary Matrices with Asymptotically Optimal Gate Count

Current noisy intermediate-scale quantum (NISQ) devices can only execute small circuits with shallow depth, as they are still constrained by the presence of noise: quantum gates have error rates and quantum states are fragile due to decoherence. Hence, it is of great importance to optimize the depth/gate-count when designing quantum circuits for specific tasks. Diagonal unitary matrices are well-known to be key building blocks of many quantum algorithms or quantum computing procedures. Prior work has discussed the synthesis of diagonal unitary matrices over the primitive gate set $\{\text{CNOT}, R_Z\}$. However, the problem has not yet been fully understood, since the existing synthesis methods have not optimized the circuit depth. In this paper, we propose a depth-optimized synthesis algorithm that automatically produces a quantum circuit for any given diagonal unitary matrix. Specially, it not only ensures the asymptotically optimal gate-count, but also nearly halves the total circuit depth compared with the previous method. Technically, we discover a uniform circuit rewriting rule well-suited for reducing the circuit depth. The performance of our synthesis algorithm is both theoretically analyzed and experimentally validated by evaluations on two examples. First, we achieve a nearly 50\% depth reduction over Welch's method for synthesizing random diagonal unitary matrices with up to 16 qubits. Second, we achieve an average of 22.05\% depth reduction for resynthesizing the diagonal part of specific quantum approximate optimization algorithm (QAOA) circuits with up to 14 qubits.

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arXiv2022

Query complexity of unitary operation discrimination

Discrimination of unitary operations is fundamental in quantum computation and information. A lot of quantum algorithms including the well-known Deutsch-Jozsa algorithm, Simon's algorithm, and Grover's algorithm can essentially be regarded as discriminating among individual, or sets of unitary operations (oracle operators). The problem of discriminating between two unitary operations $U$ and $V$ can be described as: Given $X\in\{U, V\}$, determine which one $X$ is. If $X$ is given with multiple copies, then one can design an adaptive procedure that takes multiple queries to $X$ to output the identification result of $X$. In this paper, we consider the problem: How many queries are required for achieving a desired failure probability $ε$ of discrimination between $U$ and $V$. We prove in a uniform framework: (i) if $U$ and $V$ are discriminated with bound error $ε$ , then the number of queries $T$ must satisfy $T\geq \left\lceil\frac{2\sqrt{1-4ε(1-ε)}}{Θ(U^\dagger V)}\right\rceil$, and (ii) if they are discriminated with one-sided error $ε$, then there is $T\geq \left\lceil\frac{2\sqrt{1-ε^2}}{Θ(U^\dagger V)}\right\rceil$, where $\lceil k\rceil$ denotes the minimum integer not less than $k$ and $Θ(W)$ denotes the length of the smallest arc containing all the eigenvalues of $W$ on the unit circle.

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arXiv2022

Deterministic Algorithms for the Hidden Subgroup Problem

We consider deterministic algorithms for the well-known hidden subgroup problem ($\mathsf{HSP}$): for a finite group $G$ and a finite set $X$, given a function $f:G \to X$ and the promise that for any $g_1, g_2 \in G, f(g_1) = f(g_2)$ iff $g_1H=g_2H$ for a subgroup $H \le G$, the goal of the decision version is to determine whether $H$ is trivial or not, and the goal of the identification version is to identify $H$. An algorithm for the problem should query $f(g)$ for $g\in G$ at least as possible. Nayak asked whether there exist deterministic algorithms with $O(\sqrt{\frac{|G|}{|H|}})$ query complexity for $\mathsf{HSP}$. We answer this problem by proving the following results, which also extend the main results of Ref. [30], since here the algorithms do not rely on any prior knowledge of $H$. (i)When $G$ is a general finite Abelian group, there exist an algorithm with $O(\sqrt{\frac{|G|}{|H|}})$ queries to decide the triviality of $H$ and an algorithm to identify $H$ with $O(\sqrt{\frac{|G|}{|H|}\log |H|}+\log |H|)$ queries. (ii)In general there is no deterministic algorithm for the identification version of $\mathsf{HSP}$ with query complexity of $O(\sqrt{\frac{|G|}{|H|}})$, since there exists an instance of $\mathsf{HSP}$ that needs $ω(\sqrt{\frac{|G|}{|H|}})$ queries to identify $H$. $f(x)$ is said to be $ω(g(x))$ if for every positive constant $C$, there exists a positive constant $N$ such that for $x>N$, $f(x)\ge C\cdot g(x)$, which means $g$ is a strict lower bound for $f$. On the other hand, there exist instances of $\mathsf{HSP}$ with query complexity far smaller than $O(\sqrt{\frac{|G|}{|H|}})$, whose query complexity is $O(\log \frac{|G|}{|H|})$ and even $O(1)$.

cs.DS↗

arXiv2022

Quantum Algorithms for Identifying Hidden Strings with Applications to Matroid Problems

In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of identifying a pair of $n$-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string $s, s'\in\{0, 1\}^n$ satisfying the above constraints, for any $x\in\{0, 1\}^n$ the max inner product oracle $O_{max}(x)$ returns the max value between $s\cdot x$ and $s'\cdot x$, and the sub-set oracle $O_{sub}(x)$ indicates whether the index set of the 1s in $x$ is a subset of that in $s$ or $s'$. We present a quantum algorithm consuming $O(1)$ queries to the max inner product oracle for identifying the pair $\{s, s'\}$, and prove that any classical algorithm requires $Ω(n/\log_{2}n)$ queries. Also, we present a quantum algorithm consuming $\frac{n}{2}+O(\sqrt{n})$ queries to the subset oracle, and prove that any classical algorithm requires at least $n+Ω(1)$ queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called $k$-bases if it has $k$ bases.

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arXiv2022

Robust Quantum Walk Search Without Knowing the Number of Marked Vertices

There has been a very large body of research on searching a marked vertex on a graph based on quantum walks, and Grover's algorithm can be regarded as a quantum walk-based search algorithm on a special graph. However, the existing quantum walk-based search algorithms suffer severely from the soufflé problem which mainly means that the success probability of finding a marked vertex could shrink dramatically even to zero when the number of search steps is greater than the right one, thus heavily reducing the robustness and practicability of the algorithm. Surprisingly, while the soufflé problem of Grover's algorithm has attracted enough attention, how to address this problem for general quantum walk-based search algorithms is missing in the literature. Here we initiate the study of overcoming the soufflé problem for quantum walk-based search algorithms by presenting a new quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup. In this framework, for any adjustable parameter $ε$, the quantum algorithm can find a marked vertex on an $N$-vertex {\it complete bipartite graph} with probability at least $ 1-ε$, whenever the number of search steps $h$ satisfies $h \geq \ln(\frac{2}{\sqrtε})\sqrt{N} + 1$. Note that the algorithm need not know the exact number of marked vertices. Consequently, we obtain quantum search algorithms with stronger robustness and practicability.

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arXiv2022

Optimal exact quantum algorithm for the promised element distinctness problem

The element distinctness problem is to determine whether a string $x=(x_1,\ldots,x_N)$ of $N$ elements contains two elements of the same value (a.k.a colliding pair), for which Ambainis proposed an optimal quantum algorithm. The idea behind Ambainis' algorithm is to first reduce the problem to the promised version in which $x$ is promised to contain at most one colliding pair, and then design an algorithm $\mathcal{A}$ requiring $O(N^{2/3})$ queries based on quantum walk search for the promise problem. However, $\mathcal{A}$ is probabilistic and may fail to give the right answer. We thus, in this work, design an exact quantum algorithm for the promise problem which never errs and requires $O(N^{2/3})$ queries. This algorithm is proved optimal. Technically, we modify the quantum walk search operator on quasi-Johnson graph to have arbitrary phases, and then use Jordan's lemma as the analyzing tool to reduce the quantum walk search operator to the generalized Grover's operator. This allows us to utilize the recently proposed fixed-axis-rotation (FXR) method for exact quantum search, and hence achieve 100\% success.

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arXiv2022

Verifiable blind quantum computation with identity authentication for different types of clients

Quantum computing has considerable advantages in solving some problems over its classical counterpart. Currently various physical systems are developed to construct quantum computers but it is still challenging and the first use of quantum computers may adopt the cloud style. Blind quantum computing (BQC) provides a solution for clients with limited quantum capabilities to delegate their quantum computation to remote quantum servers while keeping input, output, and even algorithm private. In this paper, we propose three multi-party verifiable blind quantum computing (VBQC) protocols with identity authentication to handle clients with varying quantum capabilities in quantum networks, such as those who can just make measurements, prepare single qubits, or perform a few single-qubit gates. They are client-friendly and flexible since the clients can achieve BQC depending on their own quantum devices and resist both insider outsider attacks in quantum networks. Furthermore, all the three proposed protocols are verifiable, namely that the clients can verify the correctness of their calculations.

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arXiv2022

Quantum algorithm for learning secret strings and its experimental demonstration

In this paper, we consider the secret-string-learning problem in the teacher-student setting: the teacher has a secret string $s\in {\{0,1\}^{n}}$, and the student wants to learn the secret $s$ by question-answer interactions with the teacher, where at each time, the student can ask the teacher with a pair $(x, q) \in {\{0,1\}^{n}}\times\{0,1,\cdots, n-1\}$ and the teacher returns a bit given by the oracle $f_{s}(x,q)$ that indicates whether the length of the longest common prefix of $s$ and $x$ is greater than $q$ or not. Our contributions are as follows. (i) We prove that any classical deterministic algorithm needs at least $n$ queries to the oracle $f_{s}$ to learn the $n$-bit secret string $s$ in both the worst case and the average case, and also present an optimal classical deterministic algorithm learning any $s$ using $n$ queries. (ii) We obtain a quantum algorithm learning the $n$-bit secret string $s$ with certainty using $\left\lceil n/2\right\rceil$ queries to the oracle $f_s$, thus proving a double speedup over classical counterparts. (iii) Experimental demonstrations of our quantum algorithm on the IBM cloud quantum computer are presented, with average success probabilities of $85.3\%$ and $82.5\%$ for all cases with $n=2$ and $n=3$ , respectively.

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arXiv2022

Quantum continual learning of quantum data realizing knowledge backward transfer

For the goal of strong artificial intelligence that can mimic human-level intelligence, AI systems would have the ability to adapt to ever-changing scenarios and learn new knowledge continuously without forgetting previously acquired knowledge. When a machine learning model is consecutively trained on multiple tasks that come in sequence, its performance on previously learned tasks may drop dramatically during the learning process of the newly seen task. To avoid this phenomenon termed catastrophic forgetting, continual learning, also known as lifelong learning, has been proposed and become one of the most up-to-date research areas of machine learning. As quantum machine learning blossoms in recent years, it is interesting to develop quantum continual learning. This paper focuses on the case of quantum models for quantum data where the computation model and the data to be processed are both quantum. The gradient episodic memory method is incorporated to design a quantum continual learning scheme that overcomes catastrophic forgetting and realizes knowledge backward transfer. Specifically, a sequence of quantum state classification tasks is continually learned by a variational quantum classifier whose parameters are optimized by a classical gradient-based optimizer. The gradient of the current task is projected to the closest gradient, avoiding the increase of the loss at previous tasks, but allowing the decrease. Numerical simulation results show that our scheme not only overcomes catastrophic forgetting, but also realize knowledge backward transfer, which means the classifier's performance on previous tasks is enhanced rather than compromised while learning a new task.

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