On The Distribution of Partial Sums of Randomly Weighted Powers of Uniform Spacings

Objectives: To study the asymptotic theory of the randomly wieghted partial sum process of powers of k-spacings from the uniform distribution. Methods: Earlier results on the distribution of the uniform incremental randomly weighted sums. Methods: Based on theorems on weak and strong approximations of partial sum processes. Results and conculsions: Our main contribution is to prove the weak convergence of weighted sum of powers of uniform spacings.

Near-linear constructions of exact unitary 2-designs

A unitary 2-design can be viewed as a quantum analogue of a 2-universal hash function: it is indistinguishable from a truly random unitary by any procedure that queries it twice. We show that exact unitary 2-designs on n qubits can be implemented by quantum circuits consisting of Oe(n) elementary gates in logarithmic depth. This is essentially a quadratic improvement in size (and in width times depth) over all previous implementations that are exact or approximate (for sufficiently strong approximations).