Abstract In this chapter, we shift our perspective from seeing the unitary evolution of a quantum state as a physical process that ‘just happens’ to instead seeing it as an executable operation in a programmable computer. To do this, we introduce the quantum circuit (or gate) model, which subsequently allows us to present and analyse quantum algorithms and quantum communication protocols. The universality of quantum computing – that every quantum mechanical system (composed of qubits) can be represented as such – is established, as is the closely related notion of universal gatesets. The chapter also briefly introduces some important topics in quantum compilation, in which some quantum circuit is rewritten as a functionally equivalent alternative that is more suitable (in some sense) for running on a particular quantum computer.
Abstract In this chapter, we shift our perspective from seeing the unitary evolution of a quantum state as a physical process that ‘just happens’ to instead seeing it as an executable operation in a programmable computer. To do this, we introduce the quantum circuit (or gate) model, which subsequently allows us to present and analyse quantum algorithms and quantum communication protocols. The universality of quantum computing – that every quantum mechanical system (composed of qubits) can be represented as such – is established, as is the closely related notion of universal gatesets. The chapter also briefly introduces some important topics in quantum compilation, in which some quantum circuit is rewritten as a functionally equivalent alternative that is more suitable (in some sense) for running on a particular quantum computer.
Abstract In this chapter, we describe the circuit model of quantum computation, focusing on the initialization of a set of qubits, their unitary evolution, and final projective measurement. We outline Di Vincenzo’s criteria for constructing a functional quantum computer. We describe single-qubit gates and measurements, and we present two-qubit gates. We prove that single-qubit gates and a particular two-qubit gate, the CNOT gate, are universal for quantum computation. In the last part of the chapter, we present the Clifford group of unitary transformations that stabilize the Pauli group, and the Gottesman-Knill theorem.
Quantum circuits If the state of a quantum system is a kind of information, then the dynamics of that system is a kind of information processing. This is the basic idea behind quantum computing , which seeks to exploit the physics of quantum systems to do useful information processing. In this chapter we will acquaint ourselves with a few of the ideas of quantum computing, using the idealized quantum circuit model. Then we will turn our attention to an actual quantum process, nuclear magnetic resonance , that can be understood as a realization of quantum information processing. In a quantum circuit, we have a set of n qubit systems whose dynamical evolution is completely under our control. We represent the evolution as a sequence of unitary operators, each of which acts on one or more of the qubits. Graphically, we represent the qubits as a set of horizontal lines, and the various stages of their evolution as a series of boxes or other symbols showing the structure of the sequence of operations. In such a diagram, time runs from left to right (see Fig. 18.1.). The n qubits form a kind of “computer,” and its overall evolution amounts to a “computation.” The key idea is that very complicated unitary operations on the n qubits can be built up step-by-step from many simple operations on one, two or a few qubits.
We are now ready to see how computing with qubits can be done. In this book, we will mainly use the circuit model for computation which was first introduced by Deutsch [25]. We will represent by quantum “wires,” the qubits upon which manipulations. The length of the wire is to be interpreted as the time axis. Manipulations on qubits can be done using basic unitary operators that are the equivalents of logic “gates.” An algorithm, or a complete set of steps for achieving a processing task, is a combination of wires and gates representing a quantum circuit. This circuit must be thought of as a time sequence of events with every wire a way of representing qubit states, and with gates representing processing of those states.
Abstract Having spent almost an entire book on working out that Hopfield nets and adiabatic quantum computing are closely related concepts, this chapter departs from our main story line of problem solving as energy minimization and concludes our study with an outlook to the possibilities of quantum gate computing. Indeed, the central ideas behind quantum gate computing differ from most of what we studied so far but our study still has us prepared for getting to grips with the basics of this paradigm. As this paradigm posits that quantum algorithms are sequences of unitary operators which each act on the joint state of a qubit system, we will look at the circuit model of quantum computing and see that quantum circuits consist of quantum gates which are interconnected by quantum wires where the latter represent the states of qubits on which the former perform unitary operations. As a practical example, we discuss how such circuits could compute univariate Boolean functions. This then seamlessly allows us to finally study Deutsch’s problem and algorithm and therefore to see an instance of a computational problem for which quantum computing algorithms can be exponentially faster than their classical counterparts.
Quantum computing stands as a pioneering field within computer science, leveraging the enigmatic principles of quantum mechanics to execute intricate calculations. At its heart lies the qubit, the elemental building block of quantum information, which resides in a state of superposition, representing both 0 and 1 concurrently. Quantum computers, constructed from assemblies of qubits, inhabit an expansive and intricate 2^n-dimensional complex Hilbert space, with their evolution governed by the Schrödinger equation. The process of quantum computation can be distilled into three key stages:1) Initialization of the input state: Initiating the quantum process; 2)Application of unitary transformations: Executing the desired transformations on the input state; 3) Measurement of the output state. Quantum computers serve as universal machines, capable of decomposing any unitary transformation into a sequence of fundamental gates. This chapter will reveal the importance of quantum mechanics, quantum computation and its applications.
In recent years, notable progress has been achieved in the theoretical investigation of quantum systems as computational tools. This has given rise to the development of quantum computing and quantum information, fields that delve into the feasibility of employing quantum systems for information processing objectives. Essential to the manipulation of qubits and the facilitation of quantum computations are quantum gates. Comparable to classical gates, these quantum counterparts are actions designed to alter the state of qubits. Among them are the Hadamard gate, CNOT gate, and Toffoli gate, each imbued with distinct functionalities that collectively enrich the repertoire of quantum computation tools. As we progress through this chapter, we embark on a journey that unveils the complexities of quantum communication. From the foundational concepts of quantum mechanics to the advanced realms of quantum teleportation, we have witnessed the potency of quantum entanglement to teleport quantum states. Furthermore, we have delved into the practical implementation of circuits using Qiskit, gaining a grasp of the art of orchestrating qubit operations, measurements, and corrections. Standing at the convergence of the quantum and classical realms, this chapter aims to provide a comprehensive perspective, exposing the intricate web of quantum communication and computing, while paving the way for a future in which quantum technologies redefine the boundaries of the achievable.