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arXiv:0904.2656

Diagrams of States in Quantum Information: an Illustrative Tutorial

Sara Felloni, Alberto Leporati, G. Strini

Published Apr 17, 2009• 3 citations
arXivHTMLPDF

Abstract

We present "Diagrams of States", a way to graphically represent and analyze how quantum information is elaborated during the execution of quantum circuits. This introductory tutorial illustrates the basics, providing useful examples of quantum computations: elementary operations in single-qubit, two-qubit and three-qubit systems, immersions of gates on higher dimensional spaces, generation of single and multi-qubit states, procedures to synthesize unitary, controlled and diagonal matrices. To perform the analysis of quantum processes, we directly derive diagrams of states from physical implementations of quantum circuits associated to the processes. Complete diagrams are then rearranged into simplified diagrams, to visualize the overall effects of computations. Conversely, diagrams of states help to conceive new quantum algorithms, by schematically describing desired manipulations of quantum information with intuitive diagrams and then by guessing the equivalent complete diagrams, from which the corresponding quantum circuit is obtained effortlessly. Related examples and analysis of complex algorithms will be provided in future works, for whose comprehension this first tutorial offers the necessary introduction.

DOI
10.48550/arxiv.0904.2656

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We present "Diagrams of States", a way to graphically represent and analyze how quantum information is elaborated during the execution of quantum circuits. This introductory tutorial illustrates the basics, providing useful examples of quantum computations: elementary operations in single-qubit, two-qubit and three-qubit systems, immersions of gates on higher dimensional spaces, generation of single and multi-qubit states, procedures to synthesize unitary, controlled and diagonal matrices. To perform the analysis of quantum processes, we directly derive diagrams of states from physical implementations of quantum circuits associated to the processes. Complete diagrams are then rearranged into simplified diagrams, to visualize the overall effects of computations. Conversely, diagrams of states help to conceive new quantum algorithms, by schematically describing desired manipulations of quantum information with intuitive diagrams and then by guessing the equivalent complete diagrams, from which the corresponding quantum circuit is obtained effortlessly. Related examples and analysis of complex algorithms will be provided in future works, for whose comprehension this first tutorial offers the necessary introduction.
We explore the main processes involved in the evolution of general quantum systems by means of Diagrams of States, a novel method to graphically represent and analyze how quantum information is elaborated during computations performed by quantum circuits. We present quantum diagrams of states for representations of quantum states by density matrices, partial trace operations, density matrix purification and time-evolution by Kraus operators. Following these representations, we describe by diagrams of states the most general transformations related to singlequbit decoherence and errors. Diagrams of states prove to be a useful approach to analyze quantum computations, by offering an intuitive graphic representation of the processing of quantum information. They also help in conceiving novel quantum computations, from describing the desired information processing to deriving the final implementation by quantum gate arrays.
We explore entanglement in composite quantum systems and how its peculiar properties are exploited in quantum information and communication protocols by means of Diagrams of States, a novel method to graphically represent and analyze how quantum information is elaborated during computations performed by quantum circuits. We present quantum diagrams of states for Bell states generation, measurements and projections, for dense coding and quantum teleportation, for probabilistic quantum machines designed to perform approximate quantum cloning and universal NOT and, finally, for quantum privacy amplification based on entanglement purification. Diagrams of states prove to be a useful approach to analyze quantum computations, by offering an intuitive graphic representation of the processing of quantum information. They also help in conceiving novel quantum computations, from describing the desired information processing to deriving the final implementation by quantum gate arrays.
The aim of this paper is to introduce a new graphic representation of quantum states by means of a specific application: the analysis of two models of quantum copying machines. The graphic representation by diagrams of states offers a clear and detailed visualization of quantum information's flow during the unitary evolution of not too complex systems. The diagrams of states are exponentially more complex in respect to the standard representation and this clearly illustrates the discrepancy of computational power between quantum and classical systems. After a brief introductive exposure of the general theory, we present a constructive procedure to illustrate the new representation by means of concrete examples. Elementary diagrams of states for single-qubit and two-qubit systems and a simple scheme to represent entangled states are presented. Quantum copying machines as imperfect cloners of quantum states are introduced and the quantum copying machines of Griffiths and Niu and of Buzek and Hillery are analyzed, determining quantum circuits of easier interpretation. The method has indeed shown itself to be extremely successful for the representation of the involved quantum operations and it has allowed to point out the characteristic aspects of the quantum computations examined.
Quantum Multiple-Valued Decision Diagrams have proven their utility in efficiently representing and manipulating reversible and quantum functions. This work, acting as a mini-review of the topic, aims to provide a comprehensive guide for implementing reversible and quantum circuit simulators using such diagrams by integrating existing methodologies into a single cohesive framework. The operation and purpose of all relevant algorithms and routines are explained, and pseudo-code schematics are also detailed for each. A proof-of-concept implementation is presented at the end, made openly available to facilitate comprehension and research in the field of decision-diagram-based simulation of Quantum Computing.
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 In quantum circuits, qubits and the quantum gates acting on them have traditionally been analysed using matrix algebra and Dirac notation. While powerful, these can be unintuitive for conceptual understanding and rapid problem solving. In this work, a new schematic representation method is developed that visualizes the effects of quantum gates on qubits without relying on complex mathematical operations. In the new notation, quantum bits (qubits) are represented using black (|0〉) and white (|1〉) circles. When a quantum gate is applied to a qubit, the circle representing the qubit is visually modified. For example, a Hadamard gate transforms a solid black or white circle into a half-black and half-white circle representing superposition. The work shows how this method simplifies the visualization of quantum algorithms, entanglement, and multi- qubit operations. The work proposes how this method can act as a visual aid to support the understanding of quantum algorithms, entanglement, and multi-qubit operations. Thus, the initial conceptual analysis of basic quantum circuits can be facilitated by simple schematic representations, reducing the immediate cognitive load of complex mathematical operations for beginners.