# An Introduction to Quantum Computing

## Phillip Kaye

Language: English

Pages: 288

ISBN: 019857049X

Format: PDF / Kindle (mobi) / ePub

This concise, accessible text provides a thorough introduction to quantum computing - an exciting emergent field at the interface of the computer, engineering, mathematical and physical sciences. Aimed at advanced undergraduate and beginning graduate students in these disciplines, the text is technically detailed and is clearly illustrated throughout with diagrams and exercises. Some prior knowledge of linear algebra is assumed, including vector spaces and inner products. However, prior familiarity with topics such as tensor products and spectral decomposition is not required, as the necessary material is reviewed in the text.

Algorithms in a Nutshell

Physically Based Rendering: From Theory to Implementation

A Bug Hunter's Diary: A Guided Tour Through the Wilds of Software Security

Compiler Design: Virtual Machines

However, in this text we will restrict use of the word ‘gate’ to refer to operations that do not output such classical information. Thus the most general kind of operation implementable by a gate is a superoperator. Furthermore, for convenience, we will restrict attention to unitary gates. 61 TEAM LinG 62 A QUANTUM MODEL OF COMPUTATION Fig. 4.1 A quantum circuit. The 4-qubit state | 0 | 0 | 0 | 0 enters the circuit on the left. The boxes labelled U 1 , U 2 , U 3 , U 4 represent quantum

between a control qubit in the state α 0 | 0 + α 1 | 1 and a target qubit in the state γ 0 | 0 + γ 1 | 1 . TEAM LinG AN APPLICATION OF QUANTUM TELEPORTATION 83 Bell Bell Fig. 5.3 Teleportation circuits to teleport the states α 0 | 0 + α 1 | 1 and γ 0 | 0 + γ 1 | 1 . Note that the state of the two logical qubits is unaffected by this circuit, and so can be viewed as an implementation of the identity operation. Fig. 5.4 A cnot gate between the pair of teleported states. The overall effect

probability that the computation ends up in state 3 after the second step. This is calculated by first determining the proba- bility associated with each computation ‘path’ that could end up at the state 3, and then by adding the probabilities for all such paths. There are four compu- tation paths that can leave the computation in state 3 after the first step. The computation can proceed from state 0 to state j and then from state j to state 3, for any of the four j ∈ { 0 , 1 , 2 , 3 }. The

MAJORITY is the THRESHOLD M function, defined as THRESHOLD M (X) = 1 if X has at least M ones, and 0 otherwise. It can be shown that deg(THRESHOLD M ) ∈ Θ( M ( N − M + 1)). Note that this means that the exact quantum counting algorithm described in Section 8.3 makes an optimal number of queries (up to a constant factor). The PARITY function is defined as PARITY(X) = 1 if X has an odd number of ones, and 0 if X has an even number of ones. The degree of the PARITY function is deg(PARITY) = N

could 2 2 modify the three-bit code accordingly by encoding 0 as 111 and 1 as 000). If p = 1 then the error channel completely randomizes the information, and there 2 is no hope of error correction helping. The three-bit code is an example of a repetition code, since codewords are formed by simply repeating the value of each logical bit a specified number of times. We will see later that simple repetition codes do not exist for quantum error correction, but that the idea can be modified