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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2022
    • (edited Jun 16th 2022)

    We are finalizing an article:



    Abstract. While the classification of non-interacting crycrystalline topological insulator phases by equivariant K-theory has become widely accepted, its generalization to anyonic interacting phases – hence to phases with topologically ordered ground states supporting topological braid quantum gates – has remained wide open.

    On the contrary, the success of K-theory with classifying non-interacting topological phases seems to have tacitly been perceived as precluding a K-theoretic classification of interacting topological order; and instead a mix of other proposals has been explored. However, only K-theory connects closely to the actual physics of valence electrons; and self-consistency demands that any other proposal must connect to K-theory.

    Here we provide a detailed argument for the classification of symmetry protected/enhanced 𝔰𝔲(2)\mathfrak{su}(2)-anyonic topological order, specifically in interacting 2d semi-metals, by the twisted equivariant differential (TED) K-theory of configuration spaces of points in the complement of nodal points inside the crystal’s Brillouin torus orbi-orientifold.

    We argue, in particular, that:

    1. topological 2d semi-metal\;phases modulo global mass terms are classified by the flat differential twisted equivariant K-theory of the complement of the nodal points;

    2. nn-electron interacting phases are classified by the K-theory of configuration spaces of nn points in the Brillouin torus;

    3. the somewhat neglected twisting of equivariant K-theory by “inner local systems” reflects the effective “fictitious” gauge interaction of Chen, Wilczeck, Witten & Halperin (1989), which turns fermions into anyonic quanta;

    4. the induced 𝔰𝔲(2)\mathfrak{su}(2)-anyonic topological order is reflected in the twisted Chern classes of the interacting valence bundle over configuration space, constituting the hypergeometric integral construction of monodromy braid representations.

    A tight dictionary relates these arguments to those for classifying defect brane charges in string theory [[SS22-Any]], which we expect to be the images of momentum-space 𝔰𝔲(2)\mathfrak{su}(2)-anyons under a non-perturbative version of the AdS/CMT correspondence.


    Comments are welcome. If you do have a look, please grab the latest pdf version from behind the above link.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2022
    • (edited Jun 19th 2022)

    This goes along with the following brief note:



    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.


    Comments are welcome. If you do have a look, please grab our latest pdf version from behind the above link.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 19th 2022

    Looks intriguing.

    That talk you list in the brief note

    • [39] D. Licata, Homotopy theory in type theory(2013), []

    has several authors it seems,

    Guillaume Brunerie, Daniel R. Licata,and Peter LeFanu Lumsdaine, Joint work with Eric Finster, Kuen-Bang Hou (Favonia), Michael Shulman

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 19th 2022

    In fact, this subsumes dependent linear homotopy types…

    What precisely does ’this’ refer to here?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTime7 days ago
    • (edited 7 days ago)

    has several authors it seems,

    Right, not sure why I made that mistake – maybe I was misled by coming from the blog entries that seem to go with these slides. In any case, I have fixed this now also on the nLab, e.g. here.

    What precisely does ’this’ refer to here?

    It means to be continuing from the previous sentence, which was about “a language for linear homotopy types”. So now to highlight that these may be dependent, in general.

    For the moment I have left this as is, since any further character on this page will make the file break out of its 5 page limit :-). But I’ll think about what to do about it.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTime7 days ago

    How exactly is the chemical potential μ_F determined? Is it something that can be computed from the Hamiltonians H_k, or must be given separately?

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTime7 days ago

    This strong link between type theory, mathematics and computation reminds me of Paul Taylor’s Abstract Stone Duality, fwiw.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTime7 days ago
    • (edited 7 days ago)

    How exactly is the chemical potential μ F\mu_F determined? Is it something that can be computed from the Hamiltonians H kH_k, or must be given separately?

    The chemical potential of electrons in a crystalline material (also “electro-chemical potential” or “Fermi energy”) is determined by the electron’s Hamiltonian and the total number (number density) of electrons, which for the usual case of electrically neutral materials is equal to the number of protons in the crystal lattice.

    Namely, the chemical potential sets essentially the mean in the Fermi-Dirac distribution of electrons in a given potential, and is constrained such that the total number derived from the distribution equals the given total number.

    This is easy to say in the case “at absolute zero” where the temperature T=0T=0. In this limit the Fermi-Dirac distribution is a step function and the chemical potential equals the eigenvalue of the electron Hamiltonian which is reached by filling the available number of electrons into eigenstates, starting at the lowest energy and occupying each state with a single electron.

    In Kittel 1953 this is discussed around p. 136. The account of this topic is pretty homegeneous across textbooks, but see also in Li 2006 around Fig. 2.3: On the following p. 53 the statement concerning T=0T=0 is fully explicit.

    For positive temperature the chemical potential increases, but slowly, so that it’s value at T=0T=0 typically gives a good approximation.

    In fact, the K-theory classification of valence bundles tacitly assumes the case T=0T=0: The valence bundle whose K-theory class one considers is by definition the sub-bundle of the full Bloch state bundle spanned by the lowest few Bloch eigenstates up to the number enforced by charge neutrality.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTime1 day ago

    In reaction to a question, we have added here a few more words on the meaning and relevance of “topological-hardware awareness”:

    • On topological-hardware awareness via TED-K

      From a broad perspective, all quantum gates are linear maps/linear operators (semantically), hence are functions between linear types (syntactically). On a finer level however, some such (families of) linear operators/function are much more readily implemented on given physical hardware than others.

      Existing quantum programming languages (QPLs) tend to have little reflection of this more detailed information, they are not “hardware aware”. In contrast, in TED-K implemented in cohesive HOTT one would have dependent linear types much as, say, Quipper does, but in addition infrastructure for naturally speaking about that particular class of these which matches expected TQC hardware functionality.

      Concretely, the elementary quantum gates that are expected to be realized by a topological quantum computer are of a very specific and peculiar kind: They are families of linear maps which are parameterized by elements of a braid group and which constitute a specific class of braid representations, namely “monodromy braid representations on 𝔰𝔲(2)\mathfrak{su}(2)-conformal blocks”.

      A traditional QPL like Quipper has no information about this. If one were to run (in the future) a Quipper program on a topological quantum computer, there would have to be a compiler involved which provides this missing information: The compiler would have to read in functions between linear types specified in Quipper, and would try to approximate them as composites of linear operators appearing in a monodromy braid representations on 𝔰𝔲(2)\mathfrak{su}(2)-conformal blocks.

      As a result, the generic Quipper program would tend to have an inefficient compilation to any given TQC machine. It would not be “aware” that this is the hardware that it is running on.

      In contrast, the TED-K language scheme that we are describing would be adding to the expressiveness of a language like Quipper the information about the native quantum gates that the TQC hardware would provide. The programmer would be provided with one very particular dependent linear homotopy type (the fiberwise TED-K type of fibrations of shapes of configuration spaces) and would be guaranteed that the linear maps (functions between linear types) which are obtained by type transport in/on this specific dependent linear type are those which the TQC hardware has an efficient implementation of.

      Of course, one could imagine adding this hardware-information by brute force to a QPL Quipper: One could write a Quipper library which encodes by hand the tensor functions which arise in 𝔰𝔲(2)\mathfrak{su}(2)-monodromy braid representations. The programmer could then decide to prefer composites of these pre-defined linear maps to build their quantum circuits, much like a contemporary high-level language programmer might choose to insert fine-tuned “assembler” commands for guarantee of verbatim efficient implementation on the underlying hardware.

      However, in both cases this would be a hack: the assembler code is alien to the ambient high-level language that calls it, just as Quipper would not provide any language handle on what it is that a would-be TQC library is providing.

      Our observation is that in homotopy type theory supporting a minimum of cohesion, this “TQC assembler code library”, if you wish, would automatically and natively be available, constructed “simply” as the 0-truncation of a certain dependent function type whose semantics is that of certain TED K-theory groups.

      This seems like a remarkable confluence of language and quantum physics for TQC. In the companion note Anyonic topological order in TED-K we explain that these TED-K types are a remarkably accurate reflection not just of topological quantum gates as such, but generally of the “topological phases of quantum materials” which are expected to serve as hardware implementing such gates. With a language for TED-K in cohesive HoTT, the distinction between a quantum programming language and an actual simulation of the underlying topological quantum hardware at just the right “universal” level of resolution would disappear.