Vector quantization (VQ) is an important data compression method. The key of the encoding of VQ is to find the closest vector among N vectors for a feature vector. Many classical linear search algorithms take O(N) steps of distance computing between two vectors. The quantum VQ iteration and corresponding quantum VQ encoding algorithm that takes O(N1/2) steps are presented in this paper. The unitary operation of distance computing can be performed on a number of vectors simultaneously because the quantum state exists in a superposition of states. The quantum VQ iteration comprises three oracles, by contrast many quantum algorithms have only one oracle, such as Shor's factorization algorithm and Grover's algorithm. Entanglement state is generated and used, by contrast the state in Grover's algorithm is not an entanglement state. The quantum VQ iteration is a rotation over subspace, by contrast the Grover iteration is a rotation over global space. The quantum VQ iteration extends the Grover iteration to the more complex search that requires more oracles. The method of the quantum VQ iteration is universal.
Vector quantization (VQ) is an important data compression method. The key of the encoding of VQ is to find the closest vector among N vectors for a feature vector. Many classical linear search algorithms take O(N) steps of distance computing between two vectors. The quantum VQ iteration and corresponding quantum VQ encoding algorithm that takes O(N1/2) steps are presented in this paper. The unitary operation of distance computing can be performed on a number of vectors simultaneously because the quantum state exists in a superposition of states. The quantum VQ iteration comprises three oracles, by contrast many quantum algorithms have only one oracle, such as Shor's factorization algorithm and Grover's algorithm. Entanglement state is generated and used, by contrast the state in Grover's algorithm is not an entanglement state. The quantum VQ iteration is a rotation over subspace, by contrast the Grover iteration is a rotation over global space. The quantum VQ iteration extends the Grover iteration to the more complex search that requires more oracles. The method of the quantum VQ iteration is universal.
Many classical encoding algorithms of vector quantization (VQ) of image compression that can obtain global optimal solution have computational complexity O ( N ). A pure quantum VQ encoding algorithm with probability of success near 100% has been proposed, that performs operations 45 N 1/2 times approximately. In this paper, a hybrid quantum VQ encoding algorithm between the classical method and the quantum algorithm is presented. The number of its operations is less than N 1/2 for most images, and it is more efficient than the pure quantum algorithm.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [Childs-linear-system,Clader,Long11], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. %Since in iteration algorithms, m often refers to the iteration steps, which is small if the iteration algorithms are efficient. Moreover, as an application of the new LCU, three new quantum algorithms to the searching problem will be proposed, which will provide us new insights into Grover's searching algorithm. We also show that the problem of LCU is closely related to the problem of if we can efficiently implement Ut for $0
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
Grover's quantum algorithm was developed to solve the problem of searching an unstructured database for a certain unique element. In general, this problem can be formulated as follows: an unstructured database consists of elements and contains one unique element that has a certain property that can be tested with polynomial complexity, and which must be found with minimal time and complexity. Classical methods require querying the database to find the desired element, while Grover's algorithm allows you to perform only approximately steps, which are iterations of the procedure, to the database and be sure that the resulting element will be exactly the desired element with a probability close to 1. As in quantum algorithms aimed at cryptanalysis of symmetric transformations, Grover's algorithm repeats the procedure (Grover's iteration) to increase the probability of obtaining the correct result. Similarly to such algorithms in Grover's method, when iterations are continued after reaching the required number of iterations, the probability of obtaining the correct result decreases. This is due to the fact that during the execution of Grover's iteration, a rotation is performed in the complex space. Thus, each iteration, making a turn, brings the register closer and closer to the desired state, but at a certain point a maximum closeness is reached, at which the continued use of iterations will lead to a turn past the desired state, which will move the state of the system away from the desired state. The database for this algorithm can be any search space consisting of a certain number of elements. So, for example, it can be applied to find a secret key for a symmetric cryptograms formation or to find a collision for a hashing function.
Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.