Quantum computations have so far proved to be more powerful than classical computations, but quantum computers still have not been put into practical use due to several technical issues. One of the most serious problems for realizing quantum computers is decoherence that occurs inevitably since our apparatus are surrounded with environment and open systems. In this paper, we give some surveys on a variety of effects of decoherence in quantum algorithms such as Grover's database search and quantum walks, and we show how quantum algorithms work under decoherence, how sensitive they are against decoherence, and how to implement a robust quantum circuit.
Quantum computations have so far proved to be more powerful than classical computations, but quantum computers still have not been put into practical use due to several technical issues. One of the most serious problems for realizing quantum computers is decoherence that occurs inevitably since our apparatus are surrounded with environment and open systems. In this paper, we give some surveys on a variety of effects of decoherence in quantum algorithms such as Grover's database search and quantum walks, and we show how quantum algorithms work under decoherence, how sensitive they are against decoherence, and how to implement a robust quantum circuit.
Recently several quantum search algorithms based on quantum walks were proposed. Those algorithms differ from Grover's algorithm in many aspects. The goal is to find a marked vertex in a graph faster than classical algorithms. Since the implementation of those new algorithms in quantum computers or in other quantum devices is error-prone, it is important to analyze their robustness under decoherence. In this work we analyze the impact of decoherence on quantum search algorithms implemented on two-dimensional grids and on hypercubes.
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics.
The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of N items requires Ω(N) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in O{\sqrt{N}} queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alternate derivation of Grover's famous result. We also subject the Grover-executing computer to a generic external influence without needing to know the specifics of the Hamiltonian insofar as the histories decohere. Depending on the amount of decoherence, which is captured in our model by a single parameter related to the amount of information obtained by the environment, the search time can range from quantum to classical. Thus, we identify a key effect induced by the environment that can adversely affect a quantum computer's performance and demonstrate exactly how classical computing can emerge from quantum laws.
Abstract. Grover’s database search algorithm, although discovered in the context of quantum computation, can be implemented using any physical system that allows superposition of states. A physical realization of this algorithm is described using coupled simple harmonic oscillators, which can be exactly solved in both classical and quantum domains. Classical wave algorithms are far more stable against decoherence compared to their quantum counterparts. In addition to providing convenient demonstration models, they may have a role in practical situations, such as catalysis. Keywords: Grover’s database search algorithm, Catalysis, Resonance
Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed. The main blocks of quantum logic, schemes of implementation of quantum calculations, as well as some known today effective quantum algorithms, called to realize advantages of quantum calculations upon classical, are concerned. Among them special place is taken by Shor's algorithm of number factorization, Grover's algorithm of unsorted database search and, finally, the most promising in application methods of quantum phenomena simulation, particularly quantum chaos. The most perspective methods of experimental realization of quantum computer, namely nuclear-magnetic resonance and trapped ions realizations, are discussed. Phenomena of decoherence, its influence on quantum computer stability and methods of quantum error correction are described.
Grover’s database search algorithm, although discovered in the context of quantum computation, can be implemented using any physical system that allows superposition of states. A physical realization of this algorithm is described using coupled simple harmonic oscillators, which can be exactly solved in both classical and quantum domains. Classical wave algorithms are far more stable against decoherence compared to their quantum counterparts. In addition to providing convenient demonstration models, they may have a role in practical situations, such as catalysis.