The accurate evaluation of diagonal unitary operators is often the most resourceintensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we give a simple way to construct efficient circuits for diagonal unitaries without ancillas, using a correspondence between Walsh functions and a basis for diagonal operators. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitaryê ( ) if x in the x basis, to that of finding the minimallength Walsh-series approximation to the function f(x). We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New J. Phys. 16 (2014) 033040 J Welch et al. The second equality shows that the functional form is the same whether x is continuous or discrete, the only difference being the number of bits in the expansion of x. This makes Walsh series useful for representing discretely sampled continuous functions. New J. Phys. 16 (2014) 033040 J Welch et al =Û e if that is diagonal in this basis is given in terms of its eigenvalues asˆ= f k f k k . New J. Phys. 16 (2014) 033040 J Welch et al 0 1 j j New J. Phys. 16 (2014) 033040 J Welch et al 5 New J. Phys. 16 (2014) 033040 J Welch et al 7 7, which means that 7 qubits are necessary to represent the series. New J. Phys. 16 (2014) 033040 J Welch et al

Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ò) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n, in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than  --error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.

Traffic flow management (TFM) in the U.S. is the process by which the Federal Aviation Administration (FAA), with the participation of airspace users, seeks to balance the capacity of airspace and airport resources with the demand for these resources. This is a difficult process, complicated by the presence of severe weather or unusually high demand. TFM in en-route airspace is concerned with managing airspace demand, specifically the number of flights handled by air traffic control (ATC) sectors; a sector is the volume of airspace managed by an air traffic controller or controller team. Therefore, effective decision-making requires accurate sector demand predictions. While it is commonly accepted that the sector demand predictions used by current and proposed TFM decision support systems contain significant uncertainty, this uncertainty is typically not quantified or taken into account in any meaningful way. The work described here is focused on measuring the uncertainty in sector demand predictions under current operational conditions, and on applying those measurements towards improving the performance and human factors of TFM decision support systems.

Reciprocal space measurements of spin diffusion in a single crystal of calcium fluoride (CaF2) have been extended to dipolar ordered states. The experimental results for the component of the spin diffusion rate parallel to the external field are D(parallel)(D)=29+/-3x10(-12) cm(2)/s for the [001] direction and D(parallel)(D)=33+/-4x10(-12) cm(2)/s for the [111] direction. The measured diffusion rates for dipolar order are faster than those for Zeeman order and are considerably faster than predicted by simple theoretical models. It is suggested that constructive interference in the transport of the two-spin states is responsible for this enhancement. As expected, the anisotropy in the diffusion rates is observed to be significantly less for dipolar order compared to the Zeeman case.