The Landauer resistance and band spectra for the counting quantum turing machine

Abstract

In other work, the generalized counting quantum Turing machine (GCQTM) was studied. For any N this machine enumerates the first 2 N integers in succession as binary strings. The generalization consists of associating a potential with read 1 steps only. The Landauer Resistance (LR) and band spectra were determined for the tight binding Hamiltonians associated with the GCQTM for energies below the potential height. Here these calculations are extended to energies both above and below the barrier height. For parameters and potentials in the electron region, The LR fluctuates rapidly between very high and very low values as a function of momentum. The rapidity and extent of the fluctuations increases rapidly with increasing N . For N = 18, the largest value considered, the LR shows good transmission probability as a function of momentum with numerous holes of very high LR values present. This is true for energies both above and below the potential height. It is suggested that the main features of the LR can be explained by coherent superposition of the component waves reflected from or transmitted through or across the 2 N −1 potentials present in the distribution. If this explanation is correct, it provides a dramatic illustration of the effects of quantum nonlocality.

Models of Quantum Turing Machines

“…The case 0 ≤ γ l,s ≤ 1 was considered elsewhere [12]. A detailed analysis of a simple case was done for which γ l,s < 1 corresponded to the introduction of potential barriers in the computation paths [13,20]. For this example the distribution and widths of the potential barriers in the paths was quasiperiodic [21] and corresponded to a generalized substitution sequence [22][23][24].…”

“…The eigenstates and spectra of H are much more complex in that reflections and transmissions occur at the path locations of the potentials [11]. A specific example analyzed elsewhere [12,13,20] showed a complex band structure for the spectrum of H.…”