Universal set

Abstract: We use quantum process tomography to characterize a full universal set of all-microwave gates on two superconducting single-frequency single-junction transmon qubits. All extracted gate fidelities, including those for Clifford group generators, single-qubit $\ensuremath{\pi}/4$ and $\ensuremath{\pi}/8$ rotations, and a two-qubit controlled-not, exceed $95%$ ($98%$), without (with) subtracting state preparation and measurement errors. Furthermore, we introduce a process map representation in the Pauli basis which is visually efficient and informative. This high-fidelity gate set serves as a critical building block towards scalable architectures of superconducting qubits for error correction schemes and pushes up on the known limits of quantum gate characterization.

Abstract: A trisecting-and-acting model explains three-way decisions (3WD) in terms of two basic tasks. One task is to divide a universal set into three pair-wise disjoint regions called a trisection or a tri-partition of the universal set. The other task is to act upon objects in one or more regions by developing appropriate strategies. 3WD are a class of effective ways and heuristics commonly used in human problem solving and information processing. We argue that 3WD are built on solid cognitive foundations and offer cognitive advantages and benefits. We demonstrate the flexibility and general applicability of 3WD by using examples from across many fields and disciplines.

Abstract: Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory.

Abstract: We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.