Using quantum switches to mitigate noise in Grover’s search algorithm

In this paper, we adopt an efficient data structure called the binary decision diagram (BDD) that exploits the structure displayed in quantum computing to simulate Grover's quantum search algorithm. First we adapt the original BDD and implement a series of algorithms to represent and manipulate matrices and vectors. Then instances of Grover's algorithm are simulated using our programme written in C++. In the last part we present an intensive analysis of the counter-intuitive experimental results in quantum mechanism, which may help understand the essence of quantum algorithms.

A fast natural algorithm for searching*1

The work presents a demonstration of functional understanding and practical implementation of quantum computing processes. The project considers both quantum operations (for example, superposition of qubits using Hadamard gates) and complex algorithms, such as Grover's quantum search algorithm. This algorithm is used to optimize the search for a specifically labeled element in an n-qubit space and is one of the main demonstrations of the quantum speedup that gives quantum computing an advantage over classical models. The presented codes are divided into independent, modular functions, which reflects the observance of functional programming principles in quantum algorithms. It is these systematic calculations that improve the transparency, accessibility, and reusability of the code, which is critically important in a quantum development environment where experimentation and multiple testing of algorithms are widely used. The code was simulated using Qiskit's "Aer", which provides multishot execution and statistical analysis of the results. In addition, the implementation of Grover's algorithm used oracle modeling.

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

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.

Quantum computing introduces an efficient way for unfolding complex problems on computing systems. It uses concept of superposition and entanglement, & hence supports intrinsic parallelism for obtaining the speedup in context of time over classical computing. Quantum algorithms can be efficiently implemented over quantum computing system. Most popular quantum algorithms like Grover's and Shor's provide a great foundation to specifically design the algorithms. These algorithms work on improving the lower runtime bounds of classical algorithms. Grover's algorithm exhibits a quadratic polynomial speedup while searching an element among ‘N’ elements unstructured database. The algorithmic strategy can be applied over various computer science related applications and some important are being considered in this article i.e. pattern matching, data mining and machine learning. This article explores quantum fundamentals; history based evolution followed by Grover's search algorithm in detail with cascading advantages, limitations & applications and also finally concludes with the significance of quantum computing & quantum algorithmic domains.

It is found that systematic errors in phase inversions and random errors in Hadmard-Walsh transformations are the dominant gate imperfection in Grover's quantum search algorithm. They lead to reductions in the maximum probability of the marked state and affect the efficiency of the algorithm. Given the degree of inaccuracy, we find that to guarantee a half-rate of success, the size of the database should be on the order of $O(1/{\ensuremath{\delta}}^{2}),$ where $\ensuremath{\delta}$ is the uncertainty.

Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|\phi>$ of $n$ qubits. It is shown that the optimal time to perform the measurement is independent of $| \phi>$, namely, it is identical to the optimal time in the original algorithm in which $| \phi > = | 0>$, with the same number of marked states, $r$. The probability of success $P_{\rm s}$ is obtained, in terms of the amplitudes of the state $| \phi>$, and is shown to be independent of $r$. A class of states, which includes fixed points and cycles of the Grover iteration operator is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.

A significant amount of study is being done to review the security promises made for the various ciphers now in use as a result of the development of quantum computing technology. A general attack against symmetric key cryptography primitives that can reduce search costs to the square root is Grover's search algorithm. To implement Grover's algorithm, it is necessary that the target cipher be implemented as a quantum circuit. Despite being relatively new, this area of study has received significant attention from the research community. The authors have estimated the cost of Grover's key search attack against the stream cipher Atom, for the first time, under circuit depth restrictions defined in National Institute of Standards and Technology (NIST) PQC standardisation process. The authors implement the quantum circuit of Atom in QISKIT, (open‐source software development kit for working with quantum computers running on IBM Quantum Experience). The results are also compared with other existing literature on LFSR‐based stream ciphers, such as Grain‐v1, Grain‐128‐AEAD, and Lizard. The authors also find that, to the best of their knowledge, in the existing literature on estimating the cost of Grover's attack on symmetric ciphers, Atom is the only 128‐bit key cipher that meets the threshold of ≈2 170 set by NIST for quantum security of 128‐bit key ciphers. The authors also analyse the security of Atom against quantum TMDTO attacks.

The development of programming languages for quantum computing has increased rapidly over the last few years, making it practical for a hands-on approach in teaching quantum computation. In this paper, I introduce the standard textbook example of searching for one item out of four using Grover's search algorithm and extend it by including a quantum database. In addition to explaining how to include a quantum database using a quantum circuit model, I give a complete program implementing the algorithm written in the quantum computing language Qiskit from IBM. If presented in the classroom, this extension of Grover's search algorithm alleviates some unsatisfying aspects of the standard textbook example. If given as an exercise to students, it allows them to extend the standard example and provides experience in developing quantum algorithms. I also outline searching for one (or more) items out of eight with a quantum database, which could be used for student projects.

Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of $N$ items laid out in $d$ spatial dimensions can be searched in time of order $\sqrt{N}$ for $d>2$, and in time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$ for $d=2$. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that $\sqrt{N}$ speedup can also be achieved on the hypercube. We show that full $\sqrt{N}$ speedup can be achieved on a $d$-dimensional periodic lattice for $d>4$. In $d=4$, the quantum walk search algorithm takes time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$, and in $d<4$, the algorithm does not provide substantial speedup.

In order to understand the bounds of utilization of the Grover's search algorithm for the large unstructured data in presence of the quantum computer noise, we undertake a series of simulations by inflicting various types of noise, modelled by the IBM QISKit. We apply three forms of Grover's algorithms: (1) the standard one, with 4-10 qubits, (2) recently published modified Grover's algorithm, set to reduce the circuit depth, and (3) the algorithms in (1) and (2) with multi-control Toffoli's modified by addition of an ancilla qubit. Based on these simulations, we find the upper bound of noise for these cases, establish its dependence on the quantum depth of the circuit and provide comparison among them. By extrapolation of the fitted thresholds, we predict what would be the typical gate error bounds when apply the Grover's algorithms for the search of a data in a data set as large as thirty two thousands.

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.

Quantum cooperative search algorithm for 3-SAT

We propose and study Th-QAOA (pronounced Threshold QAOA), a variation of the Quantum Alternating Operator Ansatz (QAOA) that replaces the standard phase separator operator, which encodes the objective function, with a threshold function that returns a value $1$ for solutions with an objective value above the threshold and a $0$ otherwise. We vary the threshold value to arrive at a quantum optimization algorithm. We focus on a combination with the Grover Mixer operator; the resulting GM-Th-QAOA can be viewed as a generalization of Grover's quantum search algorithm and its minimum/maximum finding cousin to approximate optimization. Our main findings include: (i) we provide intuitive arguments and show empirically that the optimum parameter values of GM-Th-QAOA (angles and threshold value) can be found with $O(\log(p) \times \log M)$ iterations of the classical outer loop, where $p$ is the number of QAOA rounds and $M$ is an upper bound on the solution value (often the number of vertices or edges in an input graph), thus eliminating the notorious outer-loop parameter finding issue of other QAOA algorithms; (ii) GM-Th-QAOA can be simulated classically with little effort up to 100 qubits through a set of tricks that cut down memory requirements; (iii) somewhat surprisingly, GM-Th-QAOA outperforms non-thresholded GM-QAOA in terms of approximation ratios achieved. This third result holds across a range of optimization problems (MaxCut, Max k-VertexCover, Max k-DensestSubgraph, MaxBisection) and various experimental design parameters, such as different input edge densities and constraint sizes.