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Quantum Computing by Simulations.

Michail A. Zak

Published Jun 1, 2014• 0 citations
OpenAlex

Abstract

Quantum computing by simulations is based upon similarity between mathematical formalism of quantum mechanics and phenomena to be computed. It exploits a dynamical convergence of several competing phenomena to an attractor which can represent an extrenum of a function, an image, a solution to a system of ODE, or a stochastic process. In this chapter, a quantum version of recurrent nets (QRN) as an analog computing device is discussed. This concept is introduced by incorporating classical feedback loops into conventional quantum networks. It is shown that the dynamical evolution of such networks, which interleave quantum evolution with measurement and reset operations, exhibit novel dynamical properties. Moreover, decoherence in quantum recurrent networks is less problematic than in conventional quantum network architectures due to the modest phase coherence times needed for network operation. It is proven that a hypothetical quantum computer can implement an exponentially larger number of the degrees of freedom within the same

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Quantum computing by simulations is based upon similarity between mathematical formalism of quantum mechanics and phenomena to be computed. It exploits a dynamical convergence of several competing phenomena to an attractor which can represent an extrenum of a function, an image, a solution to a system of ODE, or a stochastic process. In this chapter, a quantum version of recurrent nets (QRN) as an analog computing device is discussed. This concept is introduced by incorporating classical feedback loops into conventional quantum networks. It is shown that the dynamical evolution of such networks, which interleave quantum evolution with measurement and reset operations, exhibit novel dynamical properties. Moreover, decoherence in quantum recurrent networks is less problematic than in conventional quantum network architectures due to the modest phase coherence times needed for network operation. It is proven that a hypothetical quantum computer can implement an exponentially larger number of the degrees of freedom within the same
Abstract. We introduce the concept of quantum recurrent networks by incorporating classical feedback loops into conventional quantum networks. We show that the dynamical evolution of such networks, which interleave quantum evolution with measurement and reset operations, exhibit novel dynamical properties finding application in pattern recognition, optimization and simulation. Moreover, decoherence in quantum recurrent networks is less problematic than in conventional quantum network architectures due to the modest phase coherence times needed for network operation.
This chapter points out the limitations in simulating quantum systems in classical computers. It describes the various stages of quantum simulation using quantum computers and relevant quantum circuits. Quantum simulators are classified into two types: analog quantum simulators and digital quantum simulators. In an analog quantum simulator the physical properties of a targeted model are reproduced on a physical set-up under externally controlled conditions. Quantum simulation will be considered successful only if the initial state preparation, the implementation of the time evolution and the measurements of observables are realized using only polynomial resources. The chapter discusses the quantum circuits for measurement of the dynamical correlation functions and the spectrum of the Hermitian operators. Computing the time evolution operator is equivalent to the task of implementing a unitary matrix and any unitary transformation can be done in a quantum computer using a universal set of quantum gates.
Quantum simulation involves using a quantum system to study another quantum system. The proposal to simulate quantum physics using quantum mechanical devices, as outlined by Richard Feynman in 1982, stems from classical computers' challenges in simulating quantum systems. The pursuit of simulating complex quantum systems has led to the intersection of quantum mechanics and quantum computing. This offers new opportunities to understand and manipulate the quantum world. Central to describing quantum dynamics is the Hamiltonian operator H^\hat{H}H^, which governs the time evolution of physical systems and serves as the mathematical foundation for quantum simulations. This thesis explores the methodologies and advancements in simulating Hamiltonian operators on quantum computers, focusing on decomposing these operators into quantum gates and the efficiency gains over classical computational approaches. Initially, the thesis explains the fundamental theory and basics of quantum computing. Further it dives into the foundations of the Hamiltonian operator within quantum mechanics, highlighting its role in describing the dynamics of quantum systems. The complexity and high-dimensional nature of Hamiltonian matrices necessitate using quantum computers to simulate quantum phenomena accurately and efficiently. The process of simulating the Hamiltonian operator on a quantum computer involves a detailed examination of the strategies for decomposing the operator into a sequence of quantum gates. This decomposition is crucial for translating the abstract mathematical description of quantum systems into the physical operations of a quantum computer. By utilizing the principles of quantum mechanics, such as superposition and entanglement, quantum gates manipulate qubits to mimic the evolution of quantum states as the Hamiltonian operator dictates. Latter portions of the thesis are dedicated to comparing the efficiency of quantum simulations against traditional classical simulations and inspecting efficient quantum simulations. Quantum computers offer a polynomial or an exponential speedup for certain classes of problems, such as those involving highly entangled quantum states or requiring the simulation of complex quantum dynamics. This advantage is due to the quantum computer's ability to naturally represent and process quantum information - a task for which classical computers require exponential resources as the system size increases. The efficiency of simulating the Hamiltonian operator on a quantum computer is further enhanced through optimization techniques and algorithmic improvements. These include error mitigation strategies and the development of quantum algorithms tailored to specific types of Hamiltonian operators. The thesis presents a comprehensive analysis of these methods, demonstrating how they make quantum simulations more practical and accessible for scientific research and technological development. In conclusion this thesis is a literature review that examines the transformative potential of quantum simulations. By efficiently simulating the Hamiltonian operator on quantum computers researchers can explore quantum phenomena with precision, paving the way for breakthroughs in quantum physics, materials science, and beyond. The advancements in quantum computing technology and algorithmic approaches discussed here highlight the current capabilities and future explorations in the quantum realm.
Recently, there has been a rapid increase interest in quantum computers. Their work is based on the use of quantum-mechanical phenomena such as superposition and entanglement for computing to transform input data into outputs, which can actually provide effective performance 3-4 orders of magnitude higher than any modern computing devices. In the past few decades, there has been an acute problem of creating a quantum computer that uses quantum mechanical effects for computations. Currently, there are prototypes of devices of this class, but it is not yet possible to achieve an effective solution to the planned tasks with their help due to a number of difficulties in implementation. However, there is a mathematical apparatus that describes the behavior of quantum particles using a wave function. Hence, it becomes possible to simulate quantum computations on the classical architecture and check the reliability of algorithms belonging to the NP class, which are theoretically executed on a quantum computer in polynomial time and which cannot be solved on classical computers within an acceptable time frame. Effectively simulate the operation of a quantum system on devices with classical architecture is impossible, it remains to propose methods for acceleration of quantum computing on classical systems.
A quantum device simulating the human decision making process is introduced. It consists of quantum recurrent nets generating stochastic processes which represent the motor dynamics, and of classical neural nets describing the evolution of probabilities of these processes which represent the mental dynamics. The autonomy of the decision making process is achieved by a feedback from the mental to motor dynamics which changes the stochastic matrix based upon the probability distribution. This feedback replaces unavailable external information by an internal knowledge- base stored in the mental model in the form of probability distributions. As a result, the coupled motor-mental dynamics is described by a nonlinear version of Markov chains which can decrease entropy without an external source of information. Applications to common sense based decisions as well as to evolutionary games are discussed. An example exhibiting self-organization is computed using quantum computer simulation. Force on force and mutual aircraft engagements using the quantum decision maker dynamics are considered.