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.

Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a ``quantum attack'' of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation of the elliptic curve group operation. Such an implementation requires we be able to compute inverses in the underlying field. In \cite{PZ03}, Proos and Zalka show how to implement the extended Euclidean algorithm to compute inverses in the prime field $\GF(p)$. They employ a number of optimizations to achieve a running time of $O(n^2)$, and a space-requirement of $O(n)$ qubits, where $n$ is the number of bits in the binary representation of $p$ (there are some trade-offs that they make, sacrificing a few extra qubits to reduce running-time). In practice, elliptic curve cryptosystems often use curves over the binary field $\GF(2^m)$. In this paper, I show how to implement the extended Euclidean algorithm for polynomials to compute inverses in $\GF(2^m)$. Working under the assumption that qubits will be an `expensive' resource in realistic implementations, I optimize specifically to reduce the qubit space requirement, while keeping the running-time polynomial. The implementation here differs from that in $\cite{PZ03}$ for $\GF(p)$, and we are able to take advantage of some properties of the binary field $\GF(2^m)$. I also optimize the overall qubit space requirement for computing the group operation for elliptic curves over $\GF(2^m)$ by decomposing the group operation to make it ``piecewise reversible'' (similar to what is done in \cite{PZ03} for curves over $\GF(p)$).

Quantum protocols often require the generation of specific quantum states. We describe a quantum algorithm for generating any prescribed quantum state. For an important subclass of states, including pure symmetric states, this algorithm is efficient.

If two parties, Alice and Bob, share some number, n, of partially entangled pairs of qubits, then it is possible for them to concentrate these pairs into some smaller number of maximally entangled states. We present a simplified version of the algorithm for such entanglement concentration, and we describe efficient networks for implementing these operations.

Algorithmic cooling is a potentially important technique for making scalable NMR quantum computation feasible in practice. Given the constraints imposed by this approach to quantum computing, the most likely cooling algorithms to be practicable are those based on simple reversible polarization compression (RPC) operations acting locally on small numbers of bits. Several different algorithms using 2-and 3-bit RPC operations have appeared in the literature, and these are the algorithms I consider in this note. Specifically, I show that the RPC operation used in all these algorithms is essentially a majority vote of 3 bits, and prove the optimality of the best such algorithm. I go on to derive some theoretical bounds on the performance of these algorithms under some specific assumptions about errors.KEY WORDS: algorithmic cooling; NMR quantum computation; reversible polarization compression; error analysis of cooling algorithms; cooling algorithms based on the 3-bit majority.