The formulation of the quantum description of the rotation angle of the plane rotator has been beset by many of the long-standing problems associated with harmonic-oscillator phases. We apply methods recently developed for oscillator phases to the problem of describing a rotation angle by a Hermitian operator. These methods involve use of a finite, but arbitrarily large, state space of dimension 2l+1 that is used to calculate physically measurable quantum properties, such as expectation values, as a function l. Physical results are then recovered in the limit as l tends to infinity. This approach removes the indeterminacies caused by working directly with an infinite-dimensional state space. Our results show that the classical rotation angle observable does have a corresponding Hermitian operator with well-determined and reasonable properties. The existence of this operator provides deeper insight into the quantum-mechanical nature of rotating systems.
The quantum approximate optimization algorithm (QAOA) and quantum annealing are two of the most popular quantum optimization heuristics. While QAOA is known to be able to approximate quantum annealing, the approximation requires QAOA angles to vanish with the problem size
n, whereas optimized QAOA angles are observed to be size-independent for small
n and constant in the infinite-size limit. This fact led to a folklore belief that QAOA has a mechanism that is fundamentally different from quantum annealing. In this work, we provide evidence against this by analytically showing that QAOA energy approximates that of quantum annealing under two conditions, namely that angles vary smoothly from one layer to the next and that the sum is bounded by a constant. These conditions are known to hold for near-optimal QAOA angles empirically. Our proof relies on a series expansion of QAOA energy in sum of angles, which we show converges to quantum annealing limit as QAOA depth grows for constant sum of angles even if angles do not vanish with problem size
n. A corollary of our results is a quadratic improvement for the bound on depth required to compile Trotterized quantum annealing of the SK model in the average case.
Lorentz transformations in time and two space dimensions consist of boosts and rotations, and combinations thereof. In general, the combination of two boosts is not another boost: It is a boost followed by a rotation. The rotation angle is called the Wigner angle. Although it is straightforward to determine the energy and direction of the combined boost, it is difficult to determine the Wigner angle. In this article, the vector, matrix and spinor derivations of formulas for the Wigner angle are reviewed, and the underlying mathematics and physics are discussed briefly. Although the derivations are different, the results they produce are equivalent, as they should be. Like many physics problems, if one looks at the problem in the right way, it is not difficult to solve.