Efficient fault-tolerant quantum computing

Fault tolerant quantum computing methods which work with efficient quantum error correcting codes are discussed. Several new techniques are introduced to restrict accumulation of errors before or during the recovery. Classes of eligible quantum codes are obtained, and good candidates exhibited. This permits a new analysis of the permissible error rates and minimum overheads for robust quantum computing. It is found that, under the standard noise model of ubiquitous stochastic, uncorrelated errors, a quantum computer need be only an order of magnitude larger than the logical machine contained within it in order to be reliable. For example, a scale-up by a factor of 22, with gate error rate of order $10^{-5}$, is sufficient to permit large quantum algorithms such as factorization of thousand-digit numbers.Comment: 21 pages plus 5 figures. Replaced with figures in new format to avoid problem

The Landauer Resistance and Band Spectra for the Counting Quantum Turing Machine

The generalized counting quantum Turing machine (GCQTM) is a machine which, for any N, 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 both above and below the potential 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 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 the $2^{N-1}$ potentials in the distribution. If this explanation is correct, it provides a dramatic illustration of the effects of quantum nonlocality.Comment: 19 pages Latex, elsart.sty file included, 12 postscript figures, Submitted to PhysComp96 for publication in Physica-

Tight Binding Hamiltonians and Quantum Turing Machines

This paper extends work done to date on quantum computation by associating potentials with different types of computation steps. Quantum Turing machine Hamiltonians, generalized to include potentials, correspond to sums over tight binding Hamiltonians each with a different potential distribution. Which distribution applies is determined by the initial state. An example, which enumerates the integers in succession as binary strings, is analyzed. It is seen that for some initial states the potential distributions have quasicrystalline properties and are similar to a substitution sequence.Comment: 4 pages Latex, 2 postscript figures, submitted to Phys Rev Letter

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

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.Comment: 22 pages Latex, 13 postscript figures, Submitted to Phys. Rev.