Quantum Computing
NOTES ON QUANTUM COMPUTING AND RELATED TOPICS
Abstract
These notes are intended as a simple introduction to the new field of quantum computing, quantum information theory and quantum cryptography. Undergraduate level quantum mechanics and mathematics is required for an understanding of these lectures. After an introduction to qubits and quantum registers, we introduce the key topics of entangled states and quantum logic gates.
For two qubit states, we introduce the four Bell states as a change of basis.
The essentials of quantum cryptography are then described, although this is just a straightforward application of quantum mechanics. The characters of
Alice, Bob and Eve are first introduced here. Two qubit Bell states are used to demonstrate a novel ’dense coding’ technique. Finally, in these communication applications, quantum teleportation is explained in detail, again making use of entangled Bell states. The technique of magnetic spin resonance is used as a familiar example to illustrate how qubit operations could in principle be realised. This leads on to the specification of quantum devices that can encode functions. All this is preparatory to a detailed discussion of two of the most significant quantum algorithms discovered to date, namely, Peter Shor’s factorization algorithm and Lov Grover’s quantum database search algorithm.
1. INTRODUCTION
The basic unit of a classical computer is a bit. This is a device that can be in one of two states. Usually this is a wire which is in the state j1 > if the wire carries a voltage and j0 > if it does not (more precisely the two states are distinguished by the electrode having a high or low voltage respectively). Thus such a bit can carry one binary digit, the two states representing the numbers 0 and 1. By assembling L such bits one can store numbers from 0 to 2L − 1. The memory of a modern computer contains of the order of 109 bits and the disk storage contains of the order of 1011 bits.
Abstract
These notes are intended as a simple introduction to the new field of quantum computing, quantum information theory and quantum cryptography. Undergraduate level quantum mechanics and mathematics is required for an understanding of these lectures. After an introduction to qubits and quantum registers, we introduce the key topics of entangled states and quantum logic gates.
For two qubit states, we introduce the four Bell states as a change of basis.
The essentials of quantum cryptography are then described, although this is just a straightforward application of quantum mechanics. The characters of
Alice, Bob and Eve are first introduced here. Two qubit Bell states are used to demonstrate a novel ’dense coding’ technique. Finally, in these communication applications, quantum teleportation is explained in detail, again making use of entangled Bell states. The technique of magnetic spin resonance is used as a familiar example to illustrate how qubit operations could in principle be realised. This leads on to the specification of quantum devices that can encode functions. All this is preparatory to a detailed discussion of two of the most significant quantum algorithms discovered to date, namely, Peter Shor’s factorization algorithm and Lov Grover’s quantum database search algorithm.
1. INTRODUCTION
The basic unit of a classical computer is a bit. This is a device that can be in one of two states. Usually this is a wire which is in the state j1 > if the wire carries a voltage and j0 > if it does not (more precisely the two states are distinguished by the electrode having a high or low voltage respectively). Thus such a bit can carry one binary digit, the two states representing the numbers 0 and 1. By assembling L such bits one can store numbers from 0 to 2L − 1. The memory of a modern computer contains of the order of 109 bits and the disk storage contains of the order of 1011 bits.