Notes on the connectivity of Cayley coset digraphs

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

Isospectral deformations of the Dirac operator

We give more details about an integrable system in which the Dirac operator D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) = d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure

Approximation by Quantum Circuits

In a recent preprint by Deutsch et al. [1995] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on $n$ qubits by 2-qubit unitary operations. We address that comment by proving strong lower bounds on the approximation capabilities of g-qubit unitary operations for fixed g. We consider approximation of unitary operations on subspaces as well as approximation of states and of density matrices by quantum circuits in several natural metrics. The ability of quantum circuits to probabilistically solve decision problem and guess checkable functions is discussed. We also address exact unitary representation by reducing the upper bound by a factor of n^2 and by formalizing the argument given by Barenco et al. [1995] for the lower bound. The overall conclusion is that almost all problems are hard to solve with quantum circuits.Comment: uuencoded, compressed postscript, LACES 68Q-95-2