Constructions of Quantum Convolutional Codes

We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum convolutional code by applying a product code construction to an arbitrary classical convolutional code and an arbitrary quantum block code. We show that the resulting codes have highly structured and efficient encoders. Furthermore, we show that the resulting quantum circuits have finite depth, independent of the lengths of the input stream, and show that this depth is polynomial in the degree and frame size of the code.Comment: 5 pages, to appear in the Proceedings of the 2007 IEEE International Symposium on Information Theor

Smith forms of circulant polynomial matrices

We obtain the Smith normal forms of a class of circulant polynomial matrices (λ-matrices) in terms of their “associated polynomials” when these polynomials do not have repeated roots. We apply this to the case when the associated polynomials are products of cyclotomic polynomials and show that the entries of the Smith normal form are products of cyclotomics

A local construction of the Smith normal form of a matrix polynomial

We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular row and column operations to the original matrix. The performance of the algorithm in exact arithmetic is reported for several test cases.Comment: 26 pages, 6 figures; introduction expanded, 10 references added, two additional tests performe

Bit Complexity of Jordan Normal Form and Spectral Factorization

We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An $\tilde{O}(n^{\omega+3}a+n^4a^2+n^\omega\log(1/\epsilon))$ time algorithm for finding an $\epsilon-$approximation to the Jordan Normal form of an integer matrix with $a-$bit entries, where $\omega$ is the exponent of matrix multiplication. (2) An $\tilde{O}(n^6d^6a+n^4d^4a^2+n^3d^3\log(1/\epsilon))$ time algorithm for $\epsilon$-approximately computing the spectral factorization $P(x)=Q^*(x)Q(x)$ of a given monic $n\times n$ rational matrix polynomial of degree $2d$ with rational $a-$bit coefficients having $a-$bit common denominators, which satisfies $P(x)\succeq 0$ for all real $x$. The first algorithm is used as a subroutine in the second one. Despite its being of central importance, polynomial complexity bounds were not previously known for spectral factorization, and for Jordan form the best previous best running time was an unspecified polynomial in $n$ of degree at least twelve \cite{cai1994computing}. Our algorithms are simple and judiciously combine techniques from numerical and symbolic computation, yielding significant advantages over either approach by itself.Comment: 19p