  Despite the evident necessity of topological protection for realizing
scalable quantum computers, the conceptual underpinnings of topological quantum
logic gates had arguably remained shaky, both regarding their physical
realization as well as their information-theoretic nature.
  Building on recent results on defect branes in string/M-theory and on their
holographically dual anyonic defects in condensed matter theory, here we
explain how the specification of realistic topological quantum gates, operating
by anyon defect braiding in topologically ordered quantum materials, has a
surprisingly slick formulation in parameterized point-set topology, which is so
fundamental that it lends itself to certification in modern homotopically typed
programming languages, such as cubical Agda.
  We propose that this remarkable confluence of concepts may jointly kickstart
the development of topological quantum programming proper as well as of
real-world application of homotopy type theory, both of which have arguably
been falling behind their high expectations; in any case, it provides a
powerful paradigm for simulating and verifying topological quantum computing
architectures with high-level certification languages aware of the actual
physical principles of realistic topological quantum hardware.
  In a companion article, we will explain how further passage to "dependent
linear" homotopy data types naturally extends this scheme to a full-blown
quantum programming/certification language in which our topological quantum
gates may be compiled to verified quantum circuits, complete with quantum
measurement gates and classical control.
