We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path-integral representation. We calculate the asymptotics using a method that is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that works over the full range of positions, n, including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of this system have run into difficulties in the transitional range, because the approximations on which they were based break down here. The fact that there are two different kinds of approach to this problem (path integral versus Schrodinger wave mechanics) is ultimately a manifestation of the equivalence between the path-integral formulation of quantum mechanics and the original formulation developed in the 1920s. We also discuss how and why our approach is related to the two methods that have already been used to analyse these systems.
We now very briefly describe the methods previous authors have used. They have so far followed two approaches to determine the limiting behaviour of the @math -functions as @math . The translational invariance of this problem means that it has a simple description in momentum space, and the Schr " o dinger approach relies on that fact. We will describe this in more detail in the section below. Beginning with the recursion relation , Nayak and Vishwanath @cite_4 showed that where and @math is the matrix .
@cite_4
2649051464
It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math .