A Quantum Abacus for Teaching Quantum Algorithms

Today, more than sixty companies (in the world) are building quantum computers. The natural language of their quantum gates is that of linear algebra in a complex (Hilbert) vector space. Since 2017 it is known that one can replace the linear algebra with some string-rewriting rules no more complicated than the basic rules of arithmetic. The original system was introduced by Terry Rudolph and has been promoted and disseminated in large-scale outreach projects (among others) by Diana Franklin (University of Chicago), Sofia Economou and Ed Barnes (Virginia Tech) and other educators at the high-school level. In this workshop we show how a slightly modified (though still very elementary) system can be used to communicate a visual and entirely operational understanding of key quantum computation concepts such as: superposition, entanglement, phase, interference and unitary state evolution, as they occur in quantum algorithms. Examples include the phase kickback phenomenon, teleportation, and the famous Deutsch-Josza, Bernstein-Vazirani and Grover algorithms along with the GHZ game. We work out concrete examples of proving properties for quantum gates and quantum circuits without resorting at all to complex numbers or matrix multiplication; only simple, abacus-like operations are used, hence the title of the tutorial. We show how this approach can create a genuine bridge to the mathematics of quantum computation, that is, of vector and tensor algebras in complex spaces for students who may have little or no proper mathematical background.