Depth-2 QAC circuits cannot simulate quantum parity

We show that the quantum parity gate on $n > 3$ qubits cannot be cleanly simulated by a quantum circuit with two layers of arbitrary C-SIGN gates of any arity and arbitrary 1-qubit unitary gates, regardless of the number of allowed ancilla qubits. This is the best known and first nontrivial separation between the parity gate and circuits of this form. The same bounds also apply to the quantum fanout gate. Our results are incomparable with those of Fang et al. [3], which apply to any constant depth but require a sublinear number of ancilla qubits on the simulating circuit.Comment: 21 pages, 2 figure

Weyl asymptotics: From closed to open systems

We present microwave experiments on the symmetry reduced 5-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to an non-integer value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.Comment: 9 pages, 7 figure

Resummation of projectile-target multiple scatterings and parton saturation

In the framework of a toy model which possesses the main features of QCD in the high energy limit, we conduct a numerical study of scattering amplitudes constructed from parton splittings and projectile-target multiple interactions, in a way that unitarizes the amplitudes without however explicit saturation in the wavefunction of the incoming states. This calculation is performed in two different ways. One of these formulations, the closest to field theory, involves the numerical resummation of a factorially divergent series, for which we develop appropriate numerical tools. We accurately compare the properties of the resulting amplitudes with what would be expected if saturation were explicitly included in the evolution of the states. We observe that the amplitudes have similar properties in a small but finite range of rapidity in the beginning of the evolution, as expected. Some of the features of reaction-diffusion processes are already present in that range, even when saturation is left out of the model.Comment: 14 pages, 16 figure

Vector continued fraction algorithms.

We consider the construction of rational approximations to given power series whose coefficients are vectors. The approximants are in the form of vector-valued continued fractions which may be used to obtain vector Padeapproximants using recurrence relations. Algorithms for the determination of the vector elements of these fractions have been established using Clifford algebras. We devise new algorithms based on these which involve operations on vectors and scalars only — a desirable characteristic for computations involving vectors of large dimension. As a consequence, we are able to form new expressions for the numerator and denominator polynomials of these approximants as products of vectors, thus retaining their Clifford nature

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials