A Term-Rewriting Semantics for Pure Quantum States

In 2017, Terry Rudolph introduced an elementary rewriting system that relies on a representation of quantum states as misty states to accurately describe the basics of quantum circuits and quantum computation to high-school and middle-school students. The accessibility and effectiveness of the system are remarkable: every calculation can be done to good-enough accuracy, and perhaps with a small overhead, using just a tiny, universal set of gates chosen to take advantage of a remarkable mathematical result by Yaoyun Shi, leveraging another powerful result by A. Y. Kitaev. The misty formalism greatly simplifies calculations and makes them accessible to first-time learners using only simple arithmetic, and without sacrificing accuracy; it, too, is universal, inasmuch as you can use it to do any quantum calculation with maybe just a small overhead. We don't advocate that we should recast all of quantum theory into this formalism. The misty state picture is a good way of getting people to the heart of some nontrivial quantum theory without having to first absorb a huge amount of (what might initially seem largely) irrelevant math. Our argument is that the misty formalism can effectively be used to facilitate a transition to the full, conventional quantum-mathematical apparatus. To this end, we start by reviewing the original proposal, consider its strengths and limitations, and show it in action via entanglement swapping. We then extend the formalism through a new category of (irreducible) misty states acting as fixed points, and present the GHZ game in this new, general setting and representational semantics.