Quantum state engineering with QW - code

Code and data used for the paper: L. Innocenti, H. Majury, T. Giordani, N. Spagnolo, F. Sciarrino, M. Paternostro, A. Ferraro, Quantum state engineering using one-dimensional discrete-time quantum walks (Phys. Rev. A 96, 062326 (2017), arXiv:1710.10518 [quant-ph]).

Requirements

Many functions used here rely on functionality provided by the Mathematica packages QM and QuantumWalks. Both of these packages can be installed by simply git cloneing the repositories into the directoy given by Mathematica evaluating FileNameJoin@{$UserBaseDirectory, "Applications"}. Furthermore, the MaTeX package is required for a few functionalities (but this should be automatically installed by QM).

Package files:

• reachabilitySolutions.m: Defines a set of function to easily handle the analytical reachability conditions to obtain target superpositions after projection over the + state. Mainly used in reachabilityConditions.nb.
• searchCoinParameters.m: Defines the function searchCoinParameters, which is used to run the numerical optimization procedure to find sets of coin parameters generating a target superposition. A number of utilities to handle and process the output of searchCoinParameters are also defined here.
• stabilityAnalysis.m: Defines the function plotFidelityVsParameters, which can be used to plot the fidelity as a function of variations of single parameters.

Notebooks:

• reachabilityConditions.nb: In this notebook we find and study the analytical solutions that give target superpositions after projection. For a few steps the equations can solved analytically, while for more steps (up to 5 here) they are solved numerically.
• numericalMaximization.nb: For more then a few steps the methods in reachabilityConditions.nb are not viable anymore. Here a new function, searchCoinParameters, is used to solve via optimization the equations for up to 20 steps (and possibly more). This is done writing the output fidelity between target state and obtained state as a function of the set of coin parameters of the walk. After this, NMaximize or FindMaximum can be used to find sets of solutions. To boost the efficiency the fidelity function is compiled in C.