SashaLattice: a link-based simulator of quantum walks on the simple cubic lattice with 14 directions per cell, programmable boundary conditions on the links, and the Weyl automaton of D'Ariano and Perinotti reproduced to 1e-12

SashaLattice is a classical simulator of discrete-time quantum walks, that is of the one-particle sector of quantum cellular automata, on the simple cubic lattice. Each cell is connected to its 8 vertex neighbours and to its 6 face neighbours by 14 directed links; the 12 edge neighbours are not connected. The internal degree of freedom of the walker is the link, not the cell: the state is a 14-component vector per cell, one amplitude per incoming link, and the one-step unitary is U = S C, with C a block-diagonal coin acting cell by cell and S the shift, a pure permutation of link amplitudes. Each end of a link (input and output of a cell) carries a boundary condition in one of three states, reflecting, interacting or transparent. Chains of transparent cells are compiled once and crossed within a single step at no cost; a cell with three or more transparent ends is a splitter that costs one step; the number of steps needed by an evolution is the diameter of the compiled graph. As a benchmark we rebuilt the Weyl automaton of D'Ariano and Perinotti [1] from its eight transition matrices A_h on the eight vertex links and measured the Bloch matrix U(k) on the plane waves of three tori: the two non-zero eigenvalue branches agree with the dispersion relation of eq. 29 of [1] within 1e-12 at every wave-vector, for both automata A±. The measurement confirms the pairing of eq. 25 between the sign of zeta and the two automata. One result we had not expected: lifted to the eight links, the Weyl coin has rank two, and it is an isometry only on the subspace spanned by a two-component cell spinor. Thirteen invariants are kept as executable requirements and verified by a bench of 66 tests. The Italian version of the note is included as a second file.