Quantum Computing

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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. In early computers a memory device to store a bit consisted of a small toroid of ferromagnetic material with an electric coil wrapped around it. If the bit was “set” (i.e. in the state representing the number 1) then a current passed through the coil and the toroid produced a magnetic field. For the state representing 0 there was no current and consequently no magnetic field. Clearly the total number of such bits was limited by constraints of both size and cost and computer with more than 106 bits were rare. Since then we have seen the revolution in semiconductor technology and a great deal of effort has been put into reducing the size and costs of these binary bits. Nowadays a flat microchip with a surface area of order 1 cm2 can hold of the order of 108 bits. The small size of these memory chips has also had the effect of speeding up the rate at which computers can run; essentially this is because the electromagnetic signal has less distance to travel between components. The original motivation for imagining a “quantum computer” was based on pushing these improvements in technology to their physical limit. The smallest device one can imagine, that can exist in two states, is a single electron which has the property of spin whose component in a given direction (usually taken to be the z−direction) can take one of two values, 1 2h. We could take these two states

to represent the two states of a binary bit. The spin of an electron can be flipped by the application of an oscillating magnetic field with the correct (resonant) frequency, and can in principle be measured by applying a constant magnetic field in the z direction and observing the energy change. If this were a single outer electron of a molecule that represented one lattice point on the surface of a crystal, a surface area of order 1 cm2 could hold of order 1016 such bits. The difficulty, of course, is that to store and read different numbers we would need to be able to apply or measure magnetic fields that differentiated...
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