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An brief note that we are finalizing:
Hisham Sati, $\;$ Urs Schreiber:
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Topological Quantum Computation in TED-K
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on a scheme for topological quantum programming in terms of twisted equivariant differential K-theory implemented in cohesive homotopy type theory.
Abstract. While the realization of scalable quantum computation will arguably require topological stabilization and, with it, topological-hardware-aware quantum programming and topological-quantum circuit verification, the proper combination of these strategies into dedicated topological quantum programming languages has not yet received attention.
Here we describe a fundamental and natural scheme for typed functional (hence verifiable) topological quantum programming which is fully topological-hardware aware – in that it natively reflects the universal fine technical detail of topological q-bits, namely of symmetry-protected (or enhanced) topologically ordered Laughlin-type anyon ground states in topological phases of quantum materials.
What makes this work is:
our recent result $[$SS22-Any, SS22-Ord$]$ that wavefunctions of realistic and technologically viable anyon species – namely of $\mathfrak{su}(2)$-anyons such as the popular Majorana/Ising anyons but also of computationally universal Fibonacci anyons – are reflected in the twisted equivariant differential (TED) K-cohomology of configuration spaces of codimension=2 nodal defects in the host material’s crystallographic orbifold;
combined with our earlier observation $[$SS20-EPB, SS20-Orb, Sc14$]$ that such TED generalized cohomology theories on orbifolds interpret intuitionistically-dependent linear data types in cohesive homotopy type theory (HoTT), supporting a powerful modern form of modal quantum logic.
Not only should this emulation of anyonic topological hardware functionality via `TED-K` implemented in cohesive HoTT make advanced formal software verification tools available for hardware-aware topological quantum programming, but the constructive nature of type checking a
TED-K
quantum program in cohesive HoTT on a classical computer using existing software (such asAgda
-$\flat$), should amount at once to classically simulating the intended quantum computation at the deep level of physical topological q-bits.This would make `TED-K` in cohesive HoTT an ideal software laboratory for topological quantum computation on technologically viable types of topological q-bits, complete with ready compilation to topological quantum circuits as soon as the hardware becomes available.
In this short note we give an exposition of the basic ideas, a quick review of the underlying results and a brief indication of the basic language constructs for anyon braiding via `TED-K` in cohesive HoTT. The language system is under development at the Center for Quantum and Topological Systems at the Research Institute of NYU Abu Dhabi.
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Revision on June 21, 2022 at 02:02:40 by Urs Schreiber. See the history of this page for a list of all contributions to it.