A Quantum Abacus for Teaching Quantum Algorithms.

At the time of this writing more than 60 (sixty) companies in the world are building quantum computers. These computers, based on quantum physics principles, are radically different from those that operate according to the more familiar principles of classical physics. A quantum algorithm takes a number of classical bits as its input, manipulates them so as to create a superposition of all their possible states, further manipulates this exponentially large superposition to obtain the final quantum result, and then measures the result to get (with the appropriate probability distribution) the same number of output bits as in its input. For the middle phase, there are elementary operations which count as one step and yet manipulate all the exponentially many amplitudes of the superposition. The natural language of these quantum gates is that of linear algebra in a complex (Hilbert) vector space. Since 2017 it is known that it is possible to replace the linear algebra with some string-rewriting rules which are 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) and Sofia Economou and Ed Barnes (Virginia Tech) as well as several other educators at the high-school level. In this paper 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, probability, entanglement, phase, interference and unitary state evolution, as they occur in well-known quantum algorithms. We give 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 paper. The system we present allows a novice learner to actually trace a quantum algorithm as if it were a classical computation, which is a rare (and, frankly, borderline incredible) luxury in the area of quantum computation, where traditional debugging is impossible. Examples include the phase kickback phenomenon and the famous Deutsch-Josza algorithm. We end with a discussion (and more examples) of 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.