Unsorted Database Research Articles

Are Quantum Computers Threat to Privacy?

Abstract: Quantum computers work in a fundamentally different way from classical computers. Each qubit is in a superposition of 0 and 1 and can be used to store information. Quantum computers have the potential to revolutionize many industries. They can solve exponentially more difficult problems than conventional computers in seconds. Encryption is the process of turning plain text into ciphertext using a key. By measuring the quantum register we can find the marked number. Grover's algorithm can be used to solve a variety of problems related to finding an item in an unsorted database, such as cryptography, optimization, or machine learning. The main limitation of Grover's algorithm is that it is probabilistic. This means that there is a chance that the algorithm will not find the highlighted item after a certain number of iterations. However, the probability that the algorithm does not find the highlighted element decreases exponentially with the number of iterations. The Grover algorithm is a powerful quantum algorithm with many potential applications. However, it is still only a theoretical algorithm and has not been implemented on actual quantum computers. .. Many challenges must be overcome before the Grover algorithm can be used to solve real-world problems

Simulation of Grover Search with Polar CH3CN Molecules by Optimal Control Fields

AbstractTheoretical schemes for the implementation of the Grover search algorithm are proposed based on ultracold polar molecules in an electric field. The molecular qubits and qudit are chosen as the field‐dressed states formed by the rotational modes of CH3CN. With the help of multi‐target optimal control theory, the microwave pulses for the elementary logic operations including the one‐qubit Hadamard gates and the conditional phase gates in a dipole‐dipole system are designed and the factorized Grover algorithm demonstrated using two coupled CH3CN molecules. To reduce the accumulation of imprecision and decoherence resulting from the combination of multiple elementary gates, the optimal pulses that can achieve the two‐qubit Hadamard gate and diffusion operation in one step are designed successfully. In this way, the probabilities to find the states corresponding to the desired elements in an unsorted database are enhanced to some extent. Moreover, the optimal pulse sequence suitable for a four‐level molecular qudit, which can be used to simulate the quantum search on a single CH3CN molecule with high fidelity, is designed. These results can shed some light on the physical realization of quantum computing based on ultracold polyatomic molecules.

Inverse problems, constraint satisfaction, reversible logic, invertible logic and Grover quantum oracles for practical problems

It is well-known that the “Unsorted Database” quantum algorithm by Grover gives quadratic speedup to several practically important combinatorial and enumerative problems, such as: SAT, Graph Coloring, Maximum Cliques, Traveling Salesman and many others. Recently, quantum programming languages such as QISKIT, QSHARP or Quipper start to be used to design, verify and simulate practical quantum algorithms for important problems in Quantum Machine Learning or optimization. So far, however, no methodologies have been created to program Grover's Oracles for particular classes of problems. In contrast, such methodologies have been already created for classical Constraint Satisfaction Problems (CSP). Invertible Logic was recently introduced by Supriyo Datta and his team. The goal of this paper is to present results of our research towards creating a systematic methodology to solve search problems in Artificial Intelligence, Logic Design and Machine Learning by repeated applications of modified hardware oracles. These oracles can use: (1) classical Boolean logic, (2) quantum logic, or (3) invertible logic. Our methodology to design and exercise all types of general oracles is based on bottom-up synthesis and technology transformations. For CSP problems we apply unified blocks which use various data representations and encodings. • Introduces a methodology of solving Constraint Satisfaction and Optimization Problem based on general oracles. • This methodology applies to classical binary hardware, quantum computing and invertible logic. • Illustrates how to solve optimization problems in quantum as repeated calls of Grover Algorithm with modified oracles. • Discusses the bottom-up design of oracles.

Spatial search on Johnson graphs by discrete-time quantum walk

The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking steps, where N is the number of vertices of the Johnson graph.

Quantum Speedup for Inferring the Value of Each Bit of a Solution State in Unsorted Databases Using a Bio-Molecular Algorithm on IBM Quantum's Computers.

In this paper, we propose a bio-molecular algorithm with O( n 2) biological operations, O( 2n-1 ) DNA strands, O( n ) tubes and the longest DNA strand, O( n ), for inferring the value of a bit from the only output satisfying any given condition in an unsorted database with 2n items of n bits. We show that the value of each bit of the outcome is determined by executing our bio-molecular algorithm n times. Then, we show how to view a bio-molecular solution space with 2n-1 DNA strands as an eigenvector and how to find the corresponding unitary operator and eigenvalues for inferring the value of a bit in the output. We also show that using an extension of the quantum phase estimation and quantum counting algorithms computes its unitary operator and eigenvalues from bio-molecular solution space with 2n-1 DNA strands. Next, we demonstrate that the value of each bit of the output solution can be determined by executing the proposed extended quantum algorithms n times. To verify our theorem, we find the maximum-sized clique to a graph with two vertices and one edge and the solution b that satisfies b2 ≡ 1 (mod 15) and using IBM Quantum's backend.

Robust Quantum Search with Uncertain Number of Target States.

The quantum search algorithm is one of the milestones of quantum algorithms. Compared with classical algorithms, it shows quadratic speed-up when searching marked states in an unsorted database. However, the success rates of quantum search algorithms are sensitive to the number of marked states. In this paper, we study the relation between the success rate and the number of iterations in a quantum search algorithm of given , where M is the number of marked state and N is the number of items in the dataset. We develop a robust quantum search algorithm based on Grover–Long algorithm with some uncertainty in the number of marked states. The proposed algorithm has the same query complexity as the Grover’s algorithm, and shows high tolerance of the uncertainty in the ratio . In particular, for a database with an uncertainty in the ratio , our algorithm will find the target states with a success rate no less than .

Implementation and analysis of quantum computing application to Higgs boson reconstruction at the large Hadron Collider

With the advent of the High-Luminosity Large Hadron Collider (HL-LHC) era, high energy physics (HEP) event selection will require new approaches to rapidly and accurately analyze vast databases. The current study addresses the enormity of HEP databases in an unprecedented manner—a quantum search using Grover’s Algorithm (GA) on an unsorted database, ATLAS Open Data, from the ATLAS detector. A novel method to identify rare events at 13 TeV in CERN’s LHC using quantum computing (QC) is presented. As indicated by the Higgs boson decay channel Hrightarrow ZZ^*rightarrow 4l, the detection of four leptons in one event may be used to reconstruct the Higgs boson and, more importantly, evince Higgs boson decay to some new phenomena, such as Hrightarrow ZZ_d rightarrow 4l. Searching the dataset for collisions resulting in detection of four leptons using a Jupyter Notebook, a classical simulation of GA, and several quantum computers with multiple qubits, the current application was found to make the proper selection in the unsorted dataset. Quantum search efficacy was analyzed for the incoming HL-LHC by implementing the QC method on multiple classical simulators and IBM’s quantum computers with the IBM Qiskit Open Source Software. The current QC application provides a novel, high-efficiency alternative to classical database searches, demonstrating its potential utility as a rapid and increasingly accurate search method in HEP.

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.

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 search for unknown number of target items by hybridizing fixed-point method with trail-and-error method**Project supported by the National Natural Science Foundation of China (Grant Nos. 11504430 and 61502526) and the National Basic Research Program of China (Grant No. 2013CB338002).

For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in rather than , where λ0 is a known lower bound of λ. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.

Entanglement, and Unsorted Database Search in Noise-Based Logic

We explore the collapse of “wavefunction” and the measurement of entanglement in the superpositions of hyperspace vectors in classical physical instantaneous-noise-based logic (INBL). We find both similarities with and major differences from the related properties of quantum systems. Two search algorithms utilizing the observed features are introduced. For the first one we assume an unsorted names database set up by Alice that is a superposition (unknown by Bob) of up to n = 2N strings; those we call names. Bob has access to the superposition wave and to the 2N reference noises of the INBL system of N noise bits. For Bob, to decide if a given name x is included in the superposition, once the search has begun, it takes N switching operations followed by a single measurement of the superposition wave. Thus, the time and hardware complexity of the search algorithm is O[log(n)], which indicates an exponential speedup compared to Grover’s quantum algorithm in a corresponding setting. An extra advantage is that the error probability of the search is zero. Moreover, the scheme can also check the existence of a fraction of a string, or several separate string fractions embedded in an arbitrarily long, arbitrary string. In the second algorithm, we expand the above scheme to a phonebook with n names and s phone numbers. When the names and numbers have the same bit resolution, once the search has begun, the time and hardware complexity of this search algorithm is O[log(n)]. In the case of one-to-one correspondence between names and phone numbers (n = s), the algorithm offers inverse phonebook search too. The error probability of this search algorithm is also zero.

Controlled quantum search

Quantum searching for one of N marked items in an unsorted database of n items is solved in $$\mathcal {O}(\sqrt{n/N})$$ steps using Grover’s algorithm. Using nonlinear quantum dynamics with a Gross–Pitaevskii-type quadratic nonlinearity, Childs and Young (Phys Rev A 93:022314, 2016, https://doi.org/10.1103/PhysRevA.93.022314 ) discovered an unstructured quantum search algorithm with a complexity $$\mathcal {O}( \min \{ 1/g \, \log (g n), \sqrt{n} \}) $$ , which can be used to find a marked item after $$o(\log (n))$$ repetitions, where g is the nonlinearity strength. In this work we develop an quantum search on a complete graph using a time-dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime $$\mathcal {O}(n /(g \sqrt{N(n-N)}))$$ , where g is related to the time-dependent nonlinearity. We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 4.

Dissipative environment may improve the quantum annealing performances of the ferromagnetic p -spin model

We investigate the quantum annealing of the ferromagnetic $ p $-spin model in a dissipative environment ($ p = 5 $ and $ p = 7 $). This model, in the large $ p $ limit, codifies the Grover's algorithm for searching in an unsorted database. The dissipative environment is described by a phonon bath in thermal equilibrium at finite temperature. The dynamics is studied in the framework of a Lindblad master equation for the reduced density matrix describing only the spins. Exploiting the symmetries of our model Hamiltonian, we can describe many spins and extrapolate expected trends for large $ N $, and $ p $. While at weak system bath coupling the dissipative environment has detrimental effects on the annealing results, we show that in the intermediate coupling regime, the phonon bath seems to speed up the annealing at low temperatures. This improvement in the performance is likely not due to thermal fluctuation but rather arises from a correlated spin-bath state and persists even at zero temperature. This result may pave the way to a new scenario in which, by appropriately engineering the system-bath coupling, one may optimize quantum annealing performances below either the purely quantum or classical limit.

Remedies for the Inconsistences in the Times of Execution of the Unsorted Database Search Algorithm within the Wave Approach

The typical semiclassical wave version of the unsorted database search algorithm based on a system of coupled simple harmonic oscillators does not consider an important ingredient of Grover’s original algorithm as it is quantum entanglement. The role of entanglement in the wave version of the unsorted database search algorithm is explored and contradictions with the time of execution of Grover’s algorithm are found. We remedy the contradictions by employing two arguments, one of them qualitative and the other quantitative. For the qualitative argument we employ the probabilistic nature of a legitimate quantum algorithm and remedy the above inconsistence. Within the quantitative argument we identify a parameter in the wave version of the unsorted database search algorithm which is related to entanglement. The contradiction with the time of execution of Grover’s algorithm is solved by choosing an appropriate values of such a parameter which incorporates entanglement to the wave version of the unsorted database search algorithm. The utility of the present arguments are evident if the wave version of the unsorted data base search algorithm is experimentally implemented through a system of N quantum dots with a harmonic oscillator potential as a confinement potential for each of the quantum dots. Each of the above N vibrating quantum dots must be coupled∗vlkmanuel@uaemex.mx1to an extra single vibrating quantum dot which entangles to all of them. In order to obtain optimal results, the coupling constants of the mentioned quantum dots should be adjusted in the way described in the present work.

Complete 3-Qubit Grover search on a programmable quantum computer

The Grover quantum search algorithm is a hallmark application of a quantum computer with a well-known speedup over classical searches of an unsorted database. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state marking scheme required to perform a classical search. We also report the deterministic implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.

A review on quantum search algorithms

The use of superposition of states in quantum computation, known as quantum parallelism, has significant advantage in terms of speed over the classical computation. It can be understood from the early invented quantum algorithms such as Deutsch's algorithm, Deutsch-Jozsa algorithm and its variation as Bernstein-Vazirani algorithm, Simon algorithm, Shor's algorithms etc. Quantum parallelism also significantly speeds up the database search algorithm, which is important in computer science because it comes as a subroutine in many important algorithms. Quantum database search of Grover achieves the task of finding the target element in an unsorted database in a time quadratically faster than the classical computer. We review the Grover quantum search algorithms for a singe and multiple target elements in a database. The partial search algorithm of Grover and Radhakrishnan and its optimization by Korepin, called GRK algorithm are also discussed.

Image Retrieval Relevance Feedback Based on Grover Quantum Searching Algorithm

In order to improving the retrieval rate and speed for a large number of images, an algorithm of image retrieval relevance feedback based on Grover quantum search is proposed. Grover quantum searching algorithm solves the unsorted database search problem with computation complexity ofO( N / M ) , where the size of database is N and there is M target solutions. Making use of the image features re-weighted feedback to make up the deficiency of computer’s understanding of user’s needs, so that the system can effectively retrieve the image library and improve the retrieval precision. The experiment proves the feasibility and effectiveness of the algorithm.

Adiabatic quantum optimization in the presence of discrete noise: Reducing the problem dimensionality

Adiabatic quantum optimization is a procedure to solve a vast class of optimization problems by slowly changing the Hamiltonian of a quantum system. The evolution time necessary for the algorithm to be successful scales inversely with the minimum energy gap encountered during the dynamics. Unfortunately, the direct calculation of the gap is strongly limited by the exponential growth in the dimensionality of the Hilbert space associated to the quantum system. Although many special-purpose methods have been devised to reduce the effective dimensionality, they are strongly limited to particular classes of problems with evident symmetries. Moreover, little is known about the computational power of adiabatic quantum optimizers in real-world conditions. Here, we propose and implement a general purposes reduction method that does not rely on any explicit symmetry and which requires, under certain general conditions, only a polynomial amount of classical resources. Thanks to this method, we are able to analyze the performance of "non-ideal" quantum adiabatic optimizers to solve the well-known Grover problem, namely the search of target entries in an unsorted database, in the presence of discrete local defects. In this case, we show that adiabatic quantum optimization, even if affected by random noise, is still potentially faster than any classical algorithm.

Dynamical analysis of Grover’s search algorithm in arbitrarily high-dimensional search spaces

We discuss at length the dynamical behavior of Grover's search algorithm for which all the Walsh---Hadamard transformations contained in this algorithm are exposed to their respective random perturbations inducing the augmentation of the dimension of the search space. We give the concise and general mathematical formulations for approximately characterizing the maximum success probabilities of finding a unique desired state in a large unsorted database and their corresponding numbers of Grover iterations, which are applicable to the search spaces of arbitrary dimension and are used to answer a salient open problem posed by Grover (Phys Rev Lett 80:4329---4332, 1998).

First experimental demonstration of an exact quantum search algorithm in nuclear magnetic resonance system

The success probability of searching an objective item from an unsorted database using standard Grovers algorithm is usually not exactly 1. It is exactly 1 only when it is used to find the target state from a database with four items. Exact search is always important in theoretical and practical applications. The failure rate of Grovers algorithm becomes big when the database is small, and this hinders the use of the commonly used divide-and-verify strategy. Even for large database, the failure rate becomes considerably large when there are many marked items. This has put a serious limitation on the usability of the Grovers algorithm. An important improved version of the Grovers algorithm, also known as the improved Grover algorithm, solves this problem. The improved Grover algorithm searches arbitrary number of target states from an unsorted database with full success rate. Here, we give the first experimental realization of the improved Grover algorithm, which finds a marked state with certainty, in a nuclear magnetic resonance system. The optimal control theory is used to obtain an optimized control sequence. The experimental results agree well with the theoretical predictions.