Schreiber Topological Quantum Programming in TED-K

Accepted contribution to PlanQC 2022 (for further details see below):

on a scheme for topological quantum programming in terms of twisted equivariant differential K-theory implemented in cohesive homotopy type theory (compatible with the quantum programming scheme QS).

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:

  1. our recent result [[SS22-Any, SS22-Ord]] that wavefunctions of realistic and technologically viable anyon species – namely of 𝔰𝔲 ( 2 ) \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;

  2. 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 as Agda-\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.

Futher details:

The above file is just a brief announcement note. Here are some further details:

  • (1) on the conceptual role of “topological hardware-awareness”

  • (2) technical details of syntax/semantics for topological braid quantum gates

Background results:

Related articles:

Related presentations:

Last revised on May 25, 2024 at 10:03:50. See the history of this page for a list of all contributions to it.