Hadamard Eigenvectors with Misty States

The misty states formalism is a set of rewriting rules (with strict syntax and acting on a small set of carefully selected quantum gates) with operations no more complicated than the rules of basic arithmetic. Introduced in 2017 by Terry Rudolph, this formalism (i.e., syntactic rules plus gates) is universal, the reason being a remarkable mathematical result due to Yaoyun Shi, leveraging another powerful result from Alexei Kitaev. Here we show that the formalism is powerful (robust) enough to handle correctly quantum gates and states for which (although well-formed) it has not been specifically designed.