The Quantum Computing Abacus

What is it about?

If you have tried teaching yourself (or others) the basics of Quantum Computing (or Quantum Information Science) using one of the many already available wonderful textbooks only to feel overwhelmed by the mathematical apparatus (or just its syntax) know that there is an alternative pedagogical approach that acts as a bridge to the standard quantum computation curriculum but in which the mathematics starts to feel supportive, organic and helpful, instead of oppressive. We should stress that, in our view, there is nothing wrong with mathematics; mathematics by itself is not oppressive. We use mathematics to help describe things going on in the physical world around us. To a physicist, the math is an inextricable part of our understanding. Unfortunately, not everyone is good at math, and most have little, if any, training in physics. A system devised by Terry Rudolph based on a rewriting system (based on “misty states”) can effectively be used to guide our students to the place where we would all like them to be, no less, but going through a stage where they feel that they “really understand” what our mathematics “means” in terms of stuff that goes on in the physical world

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Photo by Sigmund on Unsplash

Photo by Sigmund on Unsplash

Why is it important?

In 2017 Terry Rudolph proposed a method of teaching quantum mechanics and quantum computing using only the simple rules of arithmetic to students as early as sixth grade. The method is incredibly effective and in a series of papers we showed how we use it to introduce superposition, phase, interference and entanglement with virtually no mathematical overhead. Furthermore we showed that a complete eight week introductory course (for computer science sophomores as well as HS and middle school students and teachers) has been built around this approach with the following milestones: quantum gates and circuits, phase kickback, the Deutsch-Josza algorithm, Grover search, Bernstein-Vazirani and the extended Church-Turing thesis, superdense coding, the GHZ game, quantum teleportation, entanglement swapping. There is general consensus that the actual mathematics behind quantum computation is an inevitable and desirable destination for our students. But for those students that lack an adequate mathematical background (HS and younger students) one can reliably use Terry’s method (i.e., computing with "misty states") to communicate a visual and entirely operational understanding of key quantum computing concepts without resorting to complex numbers or matrix multiplication. In past conferences, workshops and tutorials we presented concrete evidence that the approach can create a genuine bridge to the actual mathematics behind quantum computation: we first identified an elementary break-even point when creating a three-qubit W-entangled state. Terry’s (misty state) formalism is based on a paper by Shi that Toffoli plus Hadamard gates are universal. When trying to create the W-entangled state we need to accommodate rotations and we must use controlled-Hadamard gates. And this is what allows for a breakeven point: a Hadamard gate controlled by the output of another Hadamard gate breaks the ubiquitous symmetry in Terry’s system, and from then on one has to carry around (i.e., specify) the actual probability amplitudes in misty states. This means that students can proceed to developing, in parallel, with (extended) misty states and Dirac notation. And after crossing that bridge we have an entirely conventional Quantum Computation course, but the intuition we acquired while computing with misty states remains with us.

Perspectives