Transmission and spectral aspects of tight-binding Hamiltonians for the counting quantum Turing machine

One-dimensional (1D) systems with deterministic disorder, such as those with quasiperiodic or substitutional sequence potential distributions, have been extensively studied. It was recently shown that a generalization of quantum Turing machines (QTM's), in which potentials are associated with elementary steps or transitions of the computation, generates potential distributions along computation paths of stales in some basis B, which are computable and are thus periodic or have deterministic disorder. These generalized machines (GQTM's) can be used to investigate the effect of potentials in causing reflections and reducing the completion probability of computations. This paper expands on this work by determining the spectral and transmission properties of an example GQTM, which enumerates the integers in succession as binary strings. A potential is associated with just one type of step. For many computation paths the potential distributions are initial segments of a distribution that is quasiperiodic and corresponds to a substitution sequence. Thus the methods developed in the study of 1D systems can be used to calculate the energy band spectra and Landauer resistance (LR). For energies below the barrier height, the LR fluctuates rapidly with momentum with minima close to or at band-gap edges. Also for several values of the parameters involved there is good transmission over some momentum regions.