The module of vector-valued modular forms is Cohen-Macaulay

Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote the collection of modular forms on $H$. Then $M(\rho)$ is a $\textbf{Z}$-graded $M(H)$-module. It has been proven that $M(\rho)$ may not be projective as a $M(H)$-module. We prove that $M(\rho)$ is Cohen-Macaulay as a $M(H)$-module. We also explain how to apply this result to prove that if $M(H)$ is a polynomial ring then $M(\rho)$ is a free $M(H)$-module of rank $\textrm{dim } \rho.$Comment: Six page

An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.Comment: 46 pages, with large margins. Includes quant-ph/0004072 plus 30 pages of new material, mostly on fault-toleranc

The Heisenberg Representation of Quantum Computers

Since Shor's discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers - the difficulty of describing them on classical computers - also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can be described by their stabilizer, a group of tensor products of Pauli matrices. Even this simple group structure is sufficient to allow a rich range of quantum effects, although it falls short of the full power of quantum computation.Comment: 20 pages, LaTeX. Expanded version of a plenary speech at the 1998 International Conference on Group Theoretic Methods in Physic

Adrift on the Sea of Indeterminacy

Today\u27s conflicts scholars no doubt consider themselves a diverse bunch, with widely differing views about how law should be chosen in multistate disputes. But from the trenches, most of them look alike. Each waxes eloquent about the search for the perfect solution-the most intellectually and morally satisfying choice of law for each dispute-and each ends the theorizing by embracing some proposition that will prove wholly indeterminate in practice