Grover Iterations Research Articles

A METHOD FOR SOLVING A SYSTEM OF EQUATIONS BASED ON THE PRSCIIPLE OF TRAINING GENERATIVE-ADVERSARIAL NEURAL NETWORKS (GAN) USING A MODIFIED GROVER ALGORITHM

The article proposes a method for solving a system of equations based on a quantum Grover search algorithm. Finding a solution to a system of equations is a computationally complex process and can be considered as an algorithmic primitive for solving various problems. The computational complexity of finding a solution to a system of equations has led to attempts to implement this problem using quantum computing. So, the concept of Quantum Linear System Problem (QLSP) is well known - the solution of systems of linear equations using a quantum computer. The method proposed in the article is considered within the framework of solving a system of algebraic equations. A feature of this method is a modification of the Grover algorithm, which consists in placing the condition of each equation in a separate Grover iteration, which differs from the usual use of Grover iterations - repeats of the oracle and the diffusion operator, in which the oracle does not change. Thus, the construction of its own oracle function of the Grover algorithm for each equation of the system is implemented within the framework of the implementation of the general scheme. A feature of the proposed method is the approximation of the problem of solving a system of equations to a problem resembling the pRSCIiple of training generative-adversarial neural networks (GAN) using Grover’s algorithm, since Grover’s algorithm allows analyzing all possible values of variables. Thanks to the use of the modified Grover algorithm, the proposed method is not limited by the mandatory condition that the number of equations is equal to the number of unknowns, since solutions to incomplete systems of equations can be found within the limits imposed by the size of the allocated quantum registers. A quantum circuit optimization method is also proposed, which consists in implementing some calculations directly in the body of the Grover algorithm. The claimed efficiency of the proposed method is O(2n/m). The method proposed in the article allows us to obtain a quantum primitive for solving a wide range of practical problems.

Tsallis relative α entropy of coherence dynamics in Grover′s search algorithm

Quantum coherence plays a central role in Grover’s search algorithm. We study the Tsallis relative entropy of coherence dynamics of the evolved state in Grover’s search algorithm. We prove that the Tsallis relative entropy of coherence decreases with the increase of the success probability, and derive the complementarity relations between the coherence and the success probability. We show that the operator coherence of the first relies on the size of the database the success probability and the target states. Moreover, we illustrate the relationships between coherence and entanglement of the superposition state of targets, as well as the production and deletion of coherence in Grover iterations.

Conditions for advantageous quantum Bitcoin mining

Our aim is to determine the conditions for quantum computing technology to give rise to the security risks associated with quantum Bitcoin mining. Specifically, we determine the speed and energy efficiency a quantum computer needs to offer an advantage over classical mining. We analyze the setting in which the Bitcoin network is entirely classical except for a single quantum miner with a small hash rate compared to the network. We develop a closed-form approximation for the probability that the quantum miner successfully mines a block, with this probability dependent on the number of Grover iterations the quantum miner applies before making a measurement. Next, we show that for a quantum miner that is “peaceful”, this success probability is maximized if the quantum miner applies Grover iterations for 16 ​min before measuring, which is surprising, as the network mines blocks every 10 ​min on average. Using this optimal mining procedure, we show that the quantum miner outperforms a classical computer in efficiency (cost per block) if the condition Q ​< ​Crb is satisfied, where Q is the cost of a Grover iteration, C is the cost of a classical hash, r is the quantum miner's speed in Grover iterations per second, and b is a factor that attains its maximum if the quantum miner uses our optimal mining procedure. This condition lays the foundation for determining when quantum mining and the known security risks associated with it will arise.

Quantum search degeneration under amplitude noise in queries to the oracle

We examine how amplitude noise in queries to the oracle degrades a performance of quantum search algorithm. The Grover search and similar techniques are widely used in various quantum algorithms, including cases where rival parties are fighting over confidential data. Hence, the oracle-box wires become the subject of competing activity with an alteration of their normal functioning. Of course, many kinds of errors could arise in this way. Possible influence of dephasing on quantum search was already addressed in the literature. Amplitude damping is another type of errors that should be analyzed first. To study the problem, we introduce a simple model of collective distortions with the use of amplitude damping channel. All the quantities of interest should be considered as functions of the number of Grover's iterations. In particular, we investigate the success probability with respect to variations the parameter that characterizes the level of amplitude errors. The success probability degrades significantly even if the error amount is not essential. Namely, this probability soon enough reduces to a value around one half. We also study trade-off relations between quantum coherence and the success probability in the presence of amplitude noise.

Quantum-accessible reinforcement learning beyond strictly epochal environments

In recent years, quantum-enhanced machine learning has emerged as a particularly fruitful application of quantum algorithms, covering aspects of supervised, unsupervised and reinforcement learning. Reinforcement learning offers numerous options of how quantum theory can be applied, and is arguably the least explored, from a quantum perspective. Here, an agent explores an environment and tries to find a behavior optimizing some figure of merit. Some of the first approaches investigated settings where this exploration can be sped-up, by considering quantum analogs of classical environments, which can then be queried in superposition. If the environments have a strict periodic structure in time (i.e. are strictly episodic), such environments can be effectively converted to conventional oracles encountered in quantum information. However, in general environments, we obtain scenarios that generalize standard oracle tasks. In this work, we consider one such generalization, where the environment is not strictly episodic, which is mapped to an oracle identification setting with a changing oracle. We analyze this case and show that standard amplitude-amplification techniques can, with minor modifications, still be applied to achieve quadratic speed-ups. In addition, we prove that an algorithm based on Grover iterations is optimal for oracle identification even if the oracle changes over time in a way that the “rewarded space” is monotonically increasing. This result constitutes one of the first generalizations of quantum-accessible reinforcement learning.

Analysis of the possibilities and peculiarities of programming cryptology problems on a quantum computer

This paper is devoted to detailing the possibilities and features of quantum computer use for cryptological problems, their demonstration, justification of approaches to the possibilities analysis and studying features of cryptanalysis problems programming on quantum computers. The possibilities and availability of hardware for solving cryptanalysis problems through quantum methods are analyzed and existing restrictions of their use are determined. The quantum computer and quantum computer programming features are considered. The possibilities of quantum computer use for cryptanalysis are also considered with the Grover’s method example. The essence of Grover’s method and features of its application for cryptanalysis are given. An example of its application to the search space which is represented by a quantum register of 56 qubits is given as well. The quantum computer application of Grover’s method on quantum computer accessible through a loud service is considered. Schemes for conducting a search by Grover's method for quantum computer application are developed, containing a different number of Grover iterations to study the need for executing a full cycle, the possibility to stop and evaluate search results at a certain stage. The developed circuits are tested on quantum computers with different architectures and on a quantum simulator provided for the analysis of circuits intended to run on a quantum computer. The comparison of the expected and obtained results of the Grover’s method application at different search stages on quantum computer is given.

On the insecurity of quantum Bitcoin mining

Grover's algorithm confers on quantum computers a quadratic advantage over classical computers for searching in an arbitrary data set, a scenario that describes Bitcoin mining. It has previously been argued that the only side-effect of quantum mining would be an increased difficulty. In this work, we argue that a crucial argument in the analysis of Bitcoin security breaks down when quantum mining is performed. Classically, a Bitcoin fork occurs rarely, i.e., when two miners find a block almost simultaneously, due to propagation time effects. The situation differs dramatically when quantum miners use Grover's algorithm, which repeatedly applies a procedure called a Grover iteration. The chances of finding a block grow quadratically with the number of Grover iterations applied. Crucially, a miner does not have to choose how many iterations to apply in advance. Suppose Alice receives Bob's new block. To maximize her revenue, she should stop and measure her state immediately in the hopes that her block (rather than Bob's) will become part of the longest chain. The strong correlation between the miners' actions and the fact that they all measure their states at the same time may lead to more forks -- which is known to be a security risk for Bitcoin. We propose a mechanism that, we conjecture, will prevent this form of quantum mining, thereby circumventing the high rate of forks.

Unity-Efficiency Parametric Down-Conversion via Amplitude Amplification.

We propose an optical scheme, employing optical parametric down-converters interlaced with nonlinear sign gates (NSGs), that completely converts an n-photon Fock-state pump to n signal-idler photon pairs when the down-converters' crystal lengths are chosen appropriately. The proof of this assertion relies on amplitude amplification, analogous to that employed in Grover search, applied to the full quantum dynamics of single-mode parametric down-conversion. When we require that all Grover iterations use the same crystal, and account for potential experimental limitations on crystal-length precision, our optimized conversion efficiencies reach unity for 1≤n≤5, after which they decrease monotonically for n values up to 50, which is the upper limit of our numerical dynamics evaluations. Nevertheless, our conversion efficiencies remain higher than those for a conventional (no NSGs) down-converter.

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).

Adaptive Phase Matching in Grover’s Algorithm

When the Grover’s algorithm is applied to search an unordered database, the successful probability usually decreases with the increase of marked items. In order to solve this problem, an adaptive phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater , the successful probability is equal to 1 with at most two Grover iterations. The validity of the new phase matching is verified by a search example.

Exact quantum algorithm to distinguish Boolean functions of different weights

In this work, we exploit the Grover operator for the weight analysis of a Boolean function, specifically to solve the weight-decision problem. The weight w is the fraction of all possible inputs for which the output is 1. The goal of the weight-decision problem is to find the exact weight w from the given two weights w1 and w2 satisfying a general weight condition as w1 + w2 = 1 and 0 < w1 < w2 < 1. First, we propose a limited weight-decision algorithm where the function has another constraint: a weight is in for integer k. Second, by changing the phases in the last two Grover iterations, we propose a general weight-decision algorithm which is free from the above constraint. Finally, we show that when our algorithm requires O(k) queries to find w with a unit success probability, any classical algorithm requires at least Ω(k2) queries for a unit success probability. In addition, we show that our algorithm requires fewer queries to solve this problem compared with the quantum counting algorithm.

Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure

The Groverian entanglement measure $G(\ensuremath{\psi})$ is applied to characterize pure quantum states $\ensuremath{\mid}\ensuremath{\psi}⟩$ of multiple qubits. This is an operational measure of entanglement in the sense that it quantifies the utility of the state $\ensuremath{\mid}\ensuremath{\psi}⟩$ as an initial state for the search algorithm. A convenient parametrization is presented, which allows us to calculate the Groverian measure analytically for certain states of high symmetry. A numerical procedure is used in order to calculate it for arbitrary pure states of multiple qubits. Using the Groverian measure to evaluate the entanglement produced by quantum algorithms may provide useful insight into the role of entanglement in making quantum algorithms powerful. Here we calculate $G(\ensuremath{\psi})$ for the intermediate states generated during the evolution of Grover's algorithm for various initial states and for different sets of marked states. It is shown that Grover's iterations generate highly entangled states in intermediate stages of the quantum search process, even if the initial state and the target state are product states.