Conclusions

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.

Quantum cryptography and quantum search algorithm are considered as two important research topics in quantum information science. An asymmetrical quantum encryption protocol based on the properties of quantum one-way function and quantum search algorithm is proposed. Depending on the no-cloning theorem and trapdoor one-way functions of the public-key, the eavesdropper cannot extract any private-information from the public-keys and the ciphertext. Introducing key-generation randomized logarithm to improve security of our proposed protocol, i.e., one private-key corresponds to an exponential number of public-keys. Using unitary operations and the single photon measurement, secret messages can be directly sent from the sender to the receiver. The security of the proposed protocol is proved that it is information-theoretically secure. Furthermore, compared the symmetrical Quantum key distribution, the proposed protocol is not only efficient to reduce additional communication, but also easier to carry out in practice, because no entangled photons and complex operations are required.

A quantum computer encodes information in quantum states and runs quantum algorithms to surpass the classical counterparts by exploiting quantum superposition and quantum correlation. Grover's quantum search algorithm is a typical quantum algorithm that proves the superiority of quantum computing over classical computing. It has a quadratic reduction in the query complexity of a database search and is known to be optimal when no a priori information about the elements of the database is provided. In this work we consider a probabilistic Grover search algorithm allowing nonzero probability of failure for a database with a general a priori probability distribution of the elements and minimize the number of oracle calls by optimizing the initial state of the quantum system and the reflection axis of the diffusion operator. The initial state and the reflection axis are allowed to not coincide and thus the quantum search algorithm rotates the quantum system in a three-dimensional subspace spanned by the initial state, the reflection axis, and the search target state in general. The number of oracle calls is minimized by a variational method, and formal results are obtained with the assumption of low failure probability. The results show that for a nonuniform a priori distribution of the database elements, the number of oracle calls can be significantly reduced given a small decrease in the success probability of the quantum search algorithm, leading to a lower average query complexity to find the solution of the search problem. The results are applied to a simple but nontrivial database model with two-value a priori probabilities to show the power of the optimized quantum search algorithm. The paper concludes with a discussion about the generalization to higher-order results that allows for a larger failure probability for the quantum search algorithm.

This paper presents a quantum search algorithm (QSA) for weighted solutions. In a QSA, the final quantum state has an equal probability of solutions. However, solutions should have different probabilities in specific problems, such as dynamic spectrum management (DSM). We propose a method to make differences with probability in accordance with rewards for each solution, and present an application to DSM. The proposed algorithm has some error terms for the final quantum state, and we provide the error terms in accordance with the number of iterations. In addition, we present the simulation results of the proposed algorithm compared with a classical exact algorithm and QSA. The proposed algorithm has a lower complexity and relative difference than the classical exact algorithm. Furthermore, the proposed algorithm has a higher performance than the QSA in specific cases.

This chapter reviews the general problems and the high-level operation of the major quantum algorithms. It describes some of the quantum algorithms in more detail. These algorithms include: Deutsch algorithm, Deutsch–Jozsa algorithm, Simon's algorithm, Shor's algorithm, quantum phase estimation algorithm, Grover's Quantum Search Algorithm (QSA), Boyer–Brassard–Høyer–Tapp QSA, Dürr–Høyer QSA, quantum counting algorithm, quantum heuristic algorithm, quantum genetic algorithm, Harrow–Hassidim–Lloyd algorithm, quantum mean algorithm, and quantum weighted sum algorithm. The chapter provides more physical insights into some of the algorithms. Quantum computation is based on quantum interference, which is a dynamical process that allows one to evolve initial quantum states (inputs) into final states (outputs) by modifying intermediate multi-particle superpositions in some prescribed way. The chapter illustrates how interference patterns lead to computational problems that are well suited to quantum computations, by presenting the first such problem that was proposed by David Deutsch.

Modern quantum computing architectures are built to provide low-error environments within which to perform the unitary operations that form the bases of quantum computations. These low-error environments necessary for successful quantum computation require use of complex structures that make large-scale implementation difficult, leading to the investigation of coupled magnon-photon systems [1-6] that can achieve strong coupling rates (> 100 MHz) as a potential platform for hybrid quantum computing architectures. Such coupled magnon-photon systems benefit from the magnetic field tunability of the resonant magnon frequency, where dynamic tuning of resonant magnon frequencies with a pulsed magnetic field at rates comparable to or less than the order of the magnon-photon energy exchange time can realize unique operations not possible with static tuning [7-8]. Previously, we analyzed this problem numerically and found that a parabolically shaped pulsed magnetic field profile can realize the coherent exchange of quantum data and the generation of entanglement by changing the curvature λ and maximum frequency Δ of the parabolic profile [8]. That is, in the same physical structure, depending on the shape of the pulsed magnetic field, different unitary magnon-mediated quantum gates can be realized [8].Here, we consider an analytical approach to study this system. Namely, we consider a hybrid magnon-photon system, consisting of two photonic resonators having degenerate frequencies ωp and one magnonic resonator with a dynamically-tuned resonance frequency ωm(t). The photonic resonators are not directly coupled with each other but are both coupled with the magnonic mode with the coupling rate κ.We consider a simple analytically-solvable interaction protocol in this hybrid system, which can be experimentally realized with rectangular magnetic field pulses of sufficiently large magnitude. We assume that initially (t<0) the frequency difference │ωm–ωp│>>κ and interaction between the subsystems can be ignored. At t=0 the magnonic mode frequency ωm(t) is brought within the efficient interaction interval Δ=ωm–ωp∼κ and is held constant for the duration T. After t=T, the magnonic frequency is once again moved out of the photonic resonance region and any interaction between the subsystems stops. Depending on the control pulse parameters Δ and T, different behaviors of the hybrid system can be realized.Of particular interest is the situation, in which the magnonic and photonic subsystems stay completely decoupled after the action of the control pulse. Such a case corresponds to a unitary operation on the photonic modes, realized through the indirect magnon-mediated coupling, and is of great importance for practical applications of magnonic resonators in quantum computing systems.Using our model, we derived a simple analytical condition of realizing unitary magnon-mediated operations, namely, for each detuning Δ there exist an infinite series of pulse durationsTn(Δ)=2nπ/(8κ2+Δ2)½providing unitary conditions (here, n is a positive integer). The resulting effect of the magnon interaction on the photonic subsystem, in this case, can be described by 2×2 unitary matrix Un=Uei(nπ/2–Tn(Δ)Δ/4) relating photonic states before and after the pulse, which has components:U11=ei2πcos(θn), U12=eiπ/2sin(θn), U21=–e–iπ/2sin(θn), U22=e–i2πcos(θn)Here θn=nπ/2–Tn(Δ)Δ/4 has the meaning of inter-mode energy exchange angle. For example, for θn=π/2+mπ (m is an integer number) a complete exchange of two photonic states is achieved, while θn=π/4+mπ describes 50% mixing behavior that allows one to entangle the two photonic modes.The obtained results allow one to select pulse parameters realizing the desired unitary operation and, also, to analyze the limitations on the performance of the proposed magnon-mediated gates. Thus, it is clear that the simple rectangular pulse cannot realize all possible unitary gates since the matrix Un depends only on one continuous parameter (θn), while the general 2×2 unitary matrix has 4 independent parameters. On the other hand, this pulse profile is sufficient to achieve an arbitrary exchange angle θn. The shortest pulse duration for which a complete exchange (θn=π/2) is possible is Tmin=π/(κ√2)and is realized at exact resonance Δ=0.Our results, also, show a possibility to realize magnon-mediated unitary gates working in off-resonance conditions │Δ│≥κ. For example, θn=π/2 gate can be realized with Δ=18(2/19)½κ and T10=(19/2)½π/κ. While such off-resonance operations require longer interaction times, their advantage in comparison to resonance-type operations is that the magnonic and photonic subsystems remain only weakly coupled throughout the whole process. This may lead to weaker decoherence of the quantum photonic subsystem caused by spin-lattice relaxation processes in the magnonic resonator. In general, n-th unitary branch Tn(Δ) has n different detuning values Δ, for which a given exchange angle θn (up to multiplies of π) is achieved, which leaves a lot of space for optimization of practical magnon-mediated unitary gates.In summary, we proposed a simple model of magnon-mediated interaction in dynamically-tuned hybrid magnon-photon quantum systems. We derived analytically the conditions under which such interaction results in a unitary operation on the photonic subsystem and obtained expressions for the parameters of the control pulse necessary for achieving the desired operation. Our results can be used for estimation and optimization of performance parameters of practical hybrid magnon-photon systems. **

Because in IBM’s quantum computers, they only provide quantum gates of single quantum bit and two quantum bits, quantum gates of three quantum bits and many quantum bits must manually be decomposed into quantum gates of single quantum bit and two quantum bits. A good quantum algorithm of solving any given problem with the size of the input of n bits must have a constant successful probability of measuring its answer(s) that is close to one as soon as possible. In this chapter, we first illustrate how to decompose quantum gates of three quantum bits and many quantum bits into quantum gates of single quantum bit and two quantum bits. Next, we introduce how to write quantum programs with version 2.0 of Open QASM to implement decomposition among various kinds of quantum gates. A quantum search algorithm that is sometimes known as Grover’s algorithm to find an item in unsorted databases with 2n items that satisfies any given condition can give a quadratic speed-up and is the best one known. Hence, we then describe how to write quantum programs with version 2.0 of Open QASM to implement the quantum search algorithm in order to solve various applications.

Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points. A choice among these trajectories can then be made to obtain the best computational complexity and control over errors. As an explicit example, Grover's quantum search algorithm is described as a Hamiltonian evolution problem. It is shown that the computational complexity has a power-law dependence on error when a straightforward Lie-Trotter discretisation formula is used, and it becomes logarithmic in error when reflection operators are used. The exponential change in error control is striking, and can be used to improve many importance sampling methods. The key concept is to make the evolution steps as large as possible while obeying the constraints of the problem. In particular, we can understand why overrelaxation algorithms are superior to small step size algorithms.

By studying the diffusion operator in Grover’s quantum search algorithm, we find an analogy between quantum searching and classical, elastic two-body collisions. We exploit this mathematical analogy using a thought experiment involving multiple collisions between two bodies to help illuminate why Grover’s quantum search algorithm works. Related issues are discussed.

Quantum algorithms use the principles of quantum mechanics, such as, for example, quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimization, simulation, and solving large systems of linear equations. Here, we implement Grover's quantum algorithm, proposed to find an element in an unsorted list, using a single nuclear 3/2 spin carried by a Tb ion sitting in a single molecular magnet transistor. The coherent manipulation of this multilevel quantum system (qudit) is achieved by means of electric fields only. Grover's search algorithm is implemented by constructing a quantum database via a multilevel Hadamard gate. The Grover sequence then allows us to select each state. The presented method is of universal character and can be implemented in any multilevel quantum system with nonequal spaced energy levels, opening the way to novel quantum search algorithms.

The paper examines the roles played by superposition and entanglement in quantum computing. The analysis is illustrated by discussion of a 'classical' electronic implementation of Grover's quantum search algorithm. It is shown explicitly that the absence of multi-particle entanglement leads to exponentially rising resources for implementing such quantum algorithms.

Grover’s search algorithm is one of the most famous quantum algorithms, which enables search in unstructured dataspaces with a quadratic speedup over any known classical algorithm. This is one of the first algorithms to show quantum advantage over classical methods of computation. Besides its most anticipated application of brute force attack on symmetric key encryptions like AES, it is today being used in many modern applications as a subroutine. In this chapter, we describe the mathematical model of Grover’s quantum search algorithm, its behavior, and the theory of its circuit implementation. This chapter also discusses an exact variant of this algorithm and recent applications of Grover’s quantum search.

A modification of Tulsi's quantum search algorithm with intermediate measurements of the control qubit is presented. In order to analyze the effect of measurements in quantum searches, a different choice of the angular parameter is used. The study is performed for several values of time lapses between measurements, finding close relationships between probabilities and correlations (mutual information and cumulative correlation measure). The order of this modified algorithm is estimated, showing that for some time lapses the performance is improved, and becomes of order O(N) (classical brute-force search) when measurements are taken in every step. The results provide a possible way to analyze improvements to other quantum algorithms using one, or more, control qubits.

A proof-of-concept application of a quantum algorithm to multiloop Feynman integrals in the Loop-Tree Duality (LTD) framework is applied to a representative four-loop topology. Causality obtained through the LTD formalism, is a suitable problem to address with quantum computers given the straightforward possibility to encode the two on shell states of a propagator on the two states of a qubit. A modification of Grover's quantum search algorithm is developed for querying multiple solutions over the unstructured set of solutions generated after the application of the LTD. The quantum algorithm is successfully implemented on IBM Quantum and QUTE simulators. \n

We report the implementation of Grover's quantum search algorithm in the\nscalable system of trapped atomic ion quantum bits. Any one of four possible\nstates of a two-qubit memory is marked, and following a single query of the\nsearch space, the marked element is successfully recovered with an average\nprobability of 60(2)%. This exceeds the performance of any possible classical\nsearch algorithm, which can only succeed with a maximum average probability of\n50%.\n