Transmission and Spectral Aspects of Tight Binding Hamiltonians for the Counting Quantum Turing Machine

TL;DR

This paper analyzes the spectral and transmission properties of a generalized quantum Turing machine with potentials, revealing how these potentials influence quantum computation paths, reflection, and transmission characteristics.

Contribution

It extends previous work by determining the spectral and transmission properties of a specific GQTM with potential distributions linked to quasiperiodic sequences.

Findings

Landauer Resistance fluctuates rapidly with momentum.

Good transmission occurs over certain momentum regions.

Energy band spectra are computed for energies below the barrier height.

Abstract

It was recently shown that a generalization of quantum Turing machines (QTMs), in which potentials are associated with elementary steps or transitions of the computation, generates potential distributions along computation paths of states in some basis B. The distributions are computable and are thus periodic or have deterministic disorder. These generalized machines (GQTMs) can be used to investigate the effect of potentials in causing reflections and reducing the completion probability of computations. This work is extended here by determination of the spectral and transmission properties of an example GQTM which enumerates the integers 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 quasiperiodic distribution that corresponds to a substitution sequence. The energy band spectra and Landauer Resistance (LR) are calculated for energies below the barrier height by use of transfer matrices. The LR fluctuates rapidly with momentum with minima close to or at band-gap edges. For several values of the parameters, there is good transmission over some momentum regions.