An article that we are finalizing at CQTS:
David Jaz Myers, $\;$ Hisham Sati $\,$ and $\,$ Urs Schreiber:
Topological Quantum Gates in Homotopy Type Theory
download:
Abstract. 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 (elusive) physical realization as well as their quantum information-theoretic nature. Building on recent results on defect branes in string/M-theory [SS23a] and on their holographically dual anyonic defects in condensed matter theory [SS23b], here we explain (as announced in [SS22]) 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 the 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 (announced earlier) we explain how further passage to "dependent linear" homotopy types naturally extends this scheme to a full-blown quantum programming/certification language in which our topological quantum gates may be compiled into verified quantum circuits with quantum measurement gates and classical control.
Companion article:
Related articles:
Topological Quantum Programming in TED-K
PlanQC 2022 33 (2022)
[arXiv:2209.08331, video presentation: YT]
Anyonic defect branes in TED-K-theory
Rev. Math. Phys. (2023)
Anyonic topological order in TED K-theory
Rev. Math. Phys. (2023)
Expository talks:
Topological Quantum Gates in HoTT
talk at QFT and Cobordism 2023,
NYU Abu Dhabi, March 2023
video: YT
Related talks:
Towards verified topological hardware-aware quantum programming
slides: pdf (view on full screen)
talk at CQTS & TII Workshop 2023
NYU Abu Dhabi, 24 Feb 2023
TED K-theory of Cohomotopy moduli spaces and Anyonic Topological Order
talk at Higher Structures and Field Theory,
ESI Vienna, 25 Aug 2022
Topological Quantum Gates from M-Theory
talk at M-Theory and Mathematics 2023
NYU Abu Dhabi, Jan 2023
Last revised on September 6, 2023 at 15:09:04. See the history of this page for a list of all contributions to it.