Discrete Wigner functions and the phase space representation of quantum computers

Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.

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.

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

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.

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.

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.

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.

A fast natural algorithm for searching*1

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 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.

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.

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.

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.

An analog of a quantum search method, known as Grover's algorithm, is modeled without entanglement on the macroscopic level, using numerical simulation of microwave devices and methods. An array of microstrip annular-ring resonators simulates a quantum bit array. An oracle that performs a test to determine which element in the array is the answer to the search algorithm is provided by a microwave-frequency plane wave, modulated by a Gaussian pulse. A single annular ring with a resonant frequency identical to the frequency of the incident pulse serves as the answer to the search. It is shown that the number of Gaussian pulses needed to identify the answer element is equal to the radicN iterations predicted by the Grover algorithm. The decay in a microwave-resonator element is used to show that the quantum dual - a spontaneous decay of the excited energy level of a quantum bit, also described as its decoherence - can be a serious obstacle to the development of large quantum-bit data arrays