Quantum Search Algorithm Research Articles

An overview of quantum computing and quantum communication systems

An overview of quantum computing and quantum communication systems

Optimization of 16-Element Quantum Search on IBMQ

Unstructured searching, which is to find the marked element from a given unstructured data set, is a widely studied problem in computer science. It is well known that Grover algorithm provides a quadratic speedup to solve unstructured search problem compared with the classical algorithm. This algorithm has received a lot of attention due to the strong versatility. In this manuscript, we report experimental results of searching a unique target from 16 elements on five different quantum devices of IBM quantum Experience (IBMQ). We first implement the original Grover algorithm on these devices. However, the experiment probability of success of finding the correct target is almost the same as random choice. We then optimize the quantum circuit size of the search algorithm. The oracle operator and diffusion operator are two of the most costly operators in Grover algorithm. For the 16-element quantum search algorithm, both the oracle operator and diffusion operator consist of a triple controlled [Formula: see text] gate ([Formula: see text]) and some single-qubit gates. So we optimize the implementation of the [Formula: see text] gate according to the qubits layout of different quantum devices. On the ibmq_santiago, the experimental success rate of the 16-element quantum search algorithm is increased to [Formula: see text] by the optimization, which is better than all the published experiments implemented on IBMQ devices. For other IBMQ devices, the experimental success rate of 16-element quantum search also has been significantly improved. We then try to further reduce the size of the quantum circuit by modifying the Grover algorithm, with a tolerable loss of the theoretical success probability. On ibmq_quito, the experimental success rate is further improved from 25.23% to 27.56% after optimization. These experimental results show the importance of circuit optimization and algorithm optimization in the Noisy-Intermediate-Scale Quantum (NISQ) era.

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

Implementation of efficient quantum search algorithms on NISQ computers

Despite the advent of Grover's algorithm for the unstructured search, its successful implementation on near-term quantum devices is still limited. We apply three strategies to reduce the errors associated with implementing quantum search algorithms. Our improved search algorithms have been implemented on the IBM quantum processors. Using them, we demonstrate three- and four-qubit search algorithm with higher average success probabilities compared to previous works. We present the successful execution of the five-qubit search on the IBM quantum processor for the first time. The results have been benchmarked using degraded ratio, which is the ratio between the experimental and the theoretical success probabilities. The fast decay of the degraded ratio supports our divide-and-conquer strategy. Our proposed strategies are also useful for implementation of quantum search algorithms in the post-NISQ era.

Multiplicative inverse with quantum search algorithm under π/18 phase rotation

Multiplicative inverse with quantum search algorithm under π/18 phase rotation

Quantum Search Algorithm for Exceptional Vertexes in Regular Graphs and its Circuit Implementation

Quantum Search Algorithm for Exceptional Vertexes in Regular Graphs and its Circuit Implementation

Information geometric aspects of probability paths with minimum entropy production for quantum state evolution

We present an information geometric analysis of both entropic speeds and entropy production rates arising from geodesic evolution on manifolds parametrized by pure quantum states. In particular, we employ pure states that emerge as outputs of suitably chosen [Formula: see text] time-dependent Hamiltonian operators that characterize analog quantum search algorithms of specific types. The [Formula: see text] Hamiltonian models under consideration are specified by external time-dependent magnetic fields within which spin-[Formula: see text] test particles are immersed. The positive definite Riemannian metrization of the parameter manifold is furnished by the Fisher information function. The Fisher information function is evaluated along parametrized squared probability amplitudes obtained from the temporal evolution of these spin-[Formula: see text] test particles. A minimum action approach is then utilized to induce the transfer of the quantum system from its initial state to its final state on the parameter manifold over a finite temporal interval. We demonstrate in an explicit manner that the minimal (that is, optimum) path corresponds to the shortest (that is, geodesic) path between the initial and final states. Furthermore, we show that the minimal path serves also to minimize the total entropy production occurring during the transfer of states. Finally, upon evaluating the entropic speed as well as the total entropy production along optimal transfer paths within several scenarios of physical interest in analog quantum searching algorithms, we demonstrate in a transparent quantitative manner a correspondence between a faster transfer and a higher rate of entropy production. We therefore conclude that higher entropic speed is associated with lower entropic efficiency within the context of quantum state transfer.

Solving Linear Systems of Equations by Using the Concept of Grover’s Search Algorithm: an IBM Quantum Experience

Quantum algorithm, as compared to classical algorithm, plays a notable role in solving linear systems of equations with an exponential speedup. Here, we demonstrate a method for solving a particular system of equations by using the concept of well-known Grover’s quantum search algorithm. The algorithm finds the solution by rotating the initial state vector in the Hilbert space to get the target solution state. It mainly involves finding particular matrices that solve the set of equations and constructing corresponding quantum circuits using the basic quantum gates. We explicitly illustrate the whole process by taking 48 different set of equations and solving them by using the concept of Grover’s algorithm. We propose new quantum circuits for each set of equations and design those on the IBM quantum simulator. We run the quantum circuit for one set of equations and obtain the desired results, and hence verify the working of the algorithm.

A Little Bit of Classical Magic to Achieve (Super-)Quantum Speedup

We introduce nebit, a classical bit with a signed probability distribution. We study its properties and basic transformations that can be applied to it. Then, we introduce a simple dynamical model – a classical random walk supplemented with nebits. We show that such a model exhibits some counterintuitive non-classical properties and that it can achieve or even exceed the speedup of Grover’s quantum search algorithm. The proposed classical dynamics never reveals negativity of nebits and thus we do not need any operational interpretation of negative probabilities. We argue that nebits can be useful as a measure of non-classicality as well as a tool to find new quantum algorithms.

Several Remarks on Grover's Quantum Search Algorithm with a Single Marked Element

Several Remarks on Grover's Quantum Search Algorithm with a Single Marked Element

Implementing a quantum search algorithm with nonorthogonal states

The search algorithm is one of the most widely studied quantum algorithms, while its complete implementation in experiments is difficult and only a few experimental works have been reported. Based on Grover's algorithm, we propose a scheme of quantum search using nonorthogonal states and experimentally implement the entire scheme in a linear optical system. With this scheme, a large-sized search space can be encoded using only one qubit and the search probability of the target state is $100%$ in theory. The experimental setup and the operations for implementing this scheme are extremely simple. In our linear optical experiment, different target states are successfully found from a database of nine elements with all fidelities $>0.99$, the state evolution from the initial state to a target state is studied, and the medium states and the final state are shown in a Bloch sphere. The scheme is general and can be easily realized in a wide range of quantum systems. This work may pave an avenue to implement quantum algorithms with nonorthogonal states.

Mermin polynomials for non-locality and entanglement detection in Grover’s algorithm and Quantum Fourier Transform

The entanglement of a quantum system can be valuated using Mermin polynomials. This gives us a means to study entanglement evolution during the execution of quantum algorithms. We first consider Grover's quantum search algorithm, noticing that states during the algorithm are maximally entangled in the direction of a single constant state, which allows us to search for a single optimal Mermin operator and use it to evaluate entanglement through the whole execution of Grover's algorithm. Then the Quantum Fourier Transform is also studied with Mermin polynomials. A different optimal Mermin operator is searched at each execution step, since in this case there is no single direction of evolution. The results for the Quantum Fourier Transform are compared to results from a previous study of entanglement with Cayley hyperdeterminant. All our computations can be replayed thanks to a structured and documented open-source code that we provide.

ML Criterion Based Signal Detection of a MIMO-OFDM System Using Quantum and Semi-Quantum Assisted Modified DHA/BBHT Search Algorithm

The quantum search algorithm (QSA) is one of the most recent developments in the era of quantum signal processing (QSP) and quantum information theory (QIT). It shows a faster performance in finding the solutions of system such as multiple-input multiple-output (MIMO) communication system, quantum error correction and others with a polynomial order over the classical one. Grover's algorithm is one of the first proposed QSA in identifying the index number of the solution of a system, whose solutions are known to the oracle. A further improved QSA called Boyer-Brassard-Høyer-Tapp (BBHT) is followed by the Durr-Hoyer algorithm (DHA), which proposes to find the solutions, while the exact number of the solutions is not known. In this treatise, we propose two modifications to the existing DHA and BBHT algorithms. In DHA, we propose a semi-quantum framework, where after each iteration in DHA, the non-solution points are removed in classical framework. This makes the data set iteratively shorter. In the case of BBHT algorithm, we propose to choose the Grover search number based on the Gamma distribution instead of the uniform one, which gives more chances to the new larger values to improve probability of success. The performance of the proposed QSAs demonstrates plausible improvement in search success performance as compared to their existing BBHT and DHA counterparts. Numerical simulations are conducted with the proposed QSA algorithms on a MIMO orthogonal frequency division multiplexing (OFDM) system to demonstrate the performance improvement.

Design and Analysis of Pattern Matching Algorithms Based on QuRAM Processing

The pattern matching problem remains in survival since past decades and becomes more sophisticated due to exponential increase in size of text databases. An effective deterministic classical algorithm is always expected to be at least $${\rm O}\left( N \right)$$ time. Quantum computations are enough capable of performing exponential operations in single step of execution, so the quantum algorithms are effective. In general, the quantum pattern matching solution is possible in $${\rm O}\left( {\sqrt N } \right)$$ time as its design is based on Grover’s quantum search algorithm. To our knowledge, quantum algorithms for single pattern matching are available with limitations, and no algorithm has designed for multiple pattern matching. The main objective is to design quantum algorithm for both single and multiple patterns on a processing architecture of quantum random access memory $$\left( {QuRAM} \right)$$ . This gives a significant advantage to process large text databases in an efficient manner. Our complexity analysis justifies that the quantum algorithmic solutions achieve computational speedup over classical methods. We summarize the emergence of quantum-based pattern matching algorithms to process biological applications. The simulation is additionally done to validate and analyze the performance of proposed quantum algorithms. Lastly, we justify that our algorithms outperform the classical and quantum solutions and they are competent for implementing over quantum computer.

Quantum Key Agreement Protocol Based on Quantum Search Algorithm

Quantum key agreement (QKA) and quantum search algorithm (QSA) are two important branches in quantum topic. The former asks all participants to negotiate the agreement key equally, and none of them can fully determine the agreement key. The latter is used to provide an acceleration in searching the marked item in an unsorted database. To date, several QKA schemes based on either BB84 or entangled states have been proposed, the QKA protocol based on QSA is rare, only exists one multi-party QKA (MQKA) protocol based on QSA. In this paper, on the basis of the properties of Grover’s QSA, the first two-party quantum key agreement protocol based on quantum search algorithm known as Grover’s algorithm is proposed. The participants transmit a two-particle quantum state sequence directly by inserting decoy photons randomly. The initial-prepared two-particle quantum states are easy to be prepared with current technology. Moreover, there is no swapping entanglement technology used in this protocol, it only needs unitary operations and single-particle measurements. Compared with the existing two-party QKA protocol, the proposed QKA protocol is more efficient. In addition, the security analysis shows that the proposed protocol is secure against both external attacks and internal ones. Finally, the proposed protocol is generated to MQKA protocol based on quantum search algorithm.

Improving Quantum Search on Simple Graphs by Pretty Good Structured Oracles

Quantum search algorithms provide a way to speed up combinatorial search, and have found several applications in modern quantum technology. In particular, spatial search on graphs, based on continuous-time quantum walks (CTQW), represents a promising platform for the implementation of quantum search in condensed matter systems. CTQW-based algorithms, however, work exactly on complete graphs, while they are known to perform poorly on realistic graphs with low connectivity. In this paper, we put forward an alternative search algorithm, based on structuring the oracle operator, which allows one to improve the localization properties of the walker by tuning only the on-site energies of the graph, i.e., without altering its topology. As such, the proposed algorithm is suitable for implementation in systems with low connectivity, e.g., rings of quantum dots or superconducting circuits. Oracle parameters are determined by Hamiltonian constraints, without the need for numerical optimization.

Electric-Circuit Realization of Fast Quantum Search

Quantum search algorithm, which can search an unsorted database quadratically faster than any known classical algorithms, has become one of the most impressive showcases of quantum computation. It has been implemented using various quantum schemes. Here, we demonstrate both theoretically and experimentally that such a fast search algorithm can also be realized using classical electric circuits. The classical circuit networks to perform such a fast search have been designed. It has been shown that the evolution of electric signals in the circuit networks is analogies of quantum particles randomly walking on graphs described by quantum theory. The searching efficiencies in our designed classical circuits are the same to the quantum schemes. Because classical circuit networks possess good scalability and stability, the present scheme is expected to avoid some problems faced by the quantum schemes. Thus, our findings are advantageous for information processing in the era of big data.

Quantum algorithm for solving the test suite minimization problem

Test-suite minimization problem is an essential problem in software engineering as its application helps to improve the software quality. This paper proposes a quantum algorithm to solve the test-suite minimization problem with high probability in O2n, where n is the number of test cases. It generates an incomplete superposition to find the best solution. It also handles the non-uniform amplitudes’ distribution case for the system with multisolutions. The proposed algorithm uses amplitude amplification techniques to search for the minimum number of test cases required to test all the requirements. The proposed algorithm employs two quantum search algorithms, Younes et al. algorithm for quantum searching via entanglement and partial diffusion to prepare incomplete superpositions that represent different search spaces such that the number of test cases is incremented in each search space, and updated Arima’s algorithm to handle the multisolutions case. The updated Arima’s algorithm searches for a quantum state that satisfies an oracle that represent the instance of the test-suite minimization problem.

Accelerated adiabatic quantum search algorithm via pulse control in a non-Markovian environment

Adiabatic quantum computation requires that the system remains in its ground state. However, an adiabatic process often loses partly or entirely its quantumness, such as entanglement in its long runtime, due to the system-environment interactions. Here we put forward an effective quantum control technique to realize the adiabatic quantum search algorithm in a nonadiabatic regime and in the presence of environment. Using the non-Markovian quantum state diffusion equation approach, we numerically study the system dynamics characterized by the success probability. The results show that the probability increases with decreasing coupling strength and temperature. In particular, non-Markovianity from the environment can help to enhance the success probability. By choosing a suitable pulse intensity and period, a high success probability can be obtained in a short runtime for weak system-bath coupling, low temperature, and strong non-Markovian heat baths.

Modification of the quantum grover algorithm by using the inversion method around the middle

The article proposes a modification of Grover’s quantum oracle quantum search algorithm, which makes it easier to search the database. The algorithm is implemented in the Python programming language using the Reggeti Forest cloud quantum service. The authors of the article use the mean flipping method, which solves the search problem during the iterative order √ (2 ^ n). Development offers great potential for the practical application of the Grover quantum algorithm, as it is characterized by higher performance and speed when performing research. Theoretically, the algorithm provides quadratic acceleration compared to conventional computers. It is not an exponential acceleration, but it remains important for large data carriers. The quantum parallelism of the Grover search algorithm is based on a simultaneous change in the amplitudes of all the inputs. This is done through a superposition of states, which is a purely quantum concept. In addition, the research is carried out globally, which indicates a significant improvement in optimization procedures. Grover’s algorithm, on the other hand, is sensitive to the number of iterations. The more iterations, the smaller the amplitude of the correct answer, so the wrong choice of this parameter can digest the solution. In addition, the operation of the algorithm is limited in the case of the introduction of noise into a quantum system, which is real in modern quantum computers.