The recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the so-called “open-closed stringy topological quantum computer” which is a theoretical topological quantum computer that employs a system of open-closed strings whose worldsheets are open-closed cobordisms. The open-closed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible open-closed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an open-closed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMT-solver Z3 with the SMT-LIB language. Some examples of application of ATPs and SMT-solvers in the mathematical setup of an open-closed stringy topological quantum computer will be provided.
The recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the so-called “open-closed stringy topological quantum computer” which is a theoretical topological quantum computer that employs a system of open-closed strings whose worldsheets are open-closed cobordisms. The open-closed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible open-closed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an open-closed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMT-solver Z3 with the SMT-LIB language. Some examples of application of ATPs and SMT-solvers in the mathematical setup of an open-closed stringy topological quantum computer will be provided.
Recently the celebrated Khovanov Homology was introduced as a target for Topological Quantum Computation given that the Khovanov Homology provides a generalization of the Jones polynomal and then it is possible to think about of a generalization of the Aharonov.-Jones-Landau algorithm. Recently, Lipshitz and Sarkar introduced a space-level refinement of Khovanov homology. which is called Khovanov Homotopy. This refinement induces a Steenrod square operation Sq2 on Khovanov homology which they describe explicitly and then some computations of Sq2 were presented. Particularly, examples of links with identical integral Khovanov homology but with distinct Khovanov homotopy types were showed. In the presente work we will introduce possible quantum algorithms for the Lipshitz- Sarkar-Steenrod square for Khovanov Homolog and their possible simulations using computer algebra.
Possible quantum algorithms for generalized Khovanov homology and the Rasmussen's invariant are proposed. Such algorithms are resulting from adaptations of the recently proposed Kauffman's algorithm for the standard Khovanov homology. The method that was applied consists in to write the relevant quantum invariant as the trace of a certain unitary operator and then to compute the trace using the Hadamard test. We apply such method to the quantum computation of the Jones polynomial, HOMFLY polynomial, Chromatic polynomial, Tutte polynomial and Bollobàs- Riordan polynomial. These polynomials are quantum topological invariants for knots, links, graphs and ribbon graphs respectively. The Jones polynomial is categorified by the standard Khovanov homology and the others polynomials are categorified by generalized Khovanov homologies, such as the Khovanov-Rozansky homology and the graph homologies. The algorithm for the Rasmussen's invariant is obtained using the gauge theory; and the recently introduced program of homotopyfication is linked with the super-symmetric quantum mechanics. It is claimed that a new program of analytification could be development from the homotopyfication using the celebrated Atiyah-Singer theorem and its super-symmetric interpretations. It is hoped that the super-symmetric quantum mechanics provides the hardware for the implementation of the proposed quantum algorithms.
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 that we are developing, for typed functional (hence verifiable) topological quantum programming which is 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 that wavefunctions of realistic and technologically viable anyon species -- namely of 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 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. 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.
A model of a D-Brane Topological Quantum Computer (DBTQC) is presented and sustained. The model is based on four-dimensional TQFTs of the Donaldson-Witten and Seiber-Witten kinds. It is argued that the DBTQC is able to compute Khovanov homology for knots, links and graphs. The DBTQC physically incorporates the mathematical process of categorification according to which the invariant polynomials for knots, links and graphs such as Jones, HOMFLY, Tutte and Bollobás-Riordan polynomials can be computed as the Euler characteristics corresponding to special homology complexes associated with knots, links and graphs. The DBTQC is conjectured as a powerful universal quantum computer in the sense that the DBTQC computes Khovanov homology which is considered like powerful that the Jones polynomial.
We present a comprehensive quantum algorithm for computing Khovanov cohomology, a categorification of the Jones polynomial, together with a clear narrative that connects its motivation, construction, and implementation. By simulating Laplacian operators derived from the Khovanov cochain complex, we estimate cohomological dimensions using quantum phase estimation, thermal sampling, and Quantum Singular Value Transformation (QSVT). The framework explicitly exploits the Frobenius algebra structure and cobordism maps, showing how these topological tools translate naturally into unitary quantum operations. Worked examples for the trefoil knot and Hopf link illustrate each step, from encoding the cube of resolutions to constructing differentials and Laplacians. This narrative approach not only refines and operationalizes the Laplacian method introduced by previous works but also provides a practical roadmap for extending categorified invariants into the realm of quantum algorithms suitable for near-term devices.
An emergent trend in quantum computation is the topological quantum computation (TQC). Briefly, TQC results from the application of quantum computation with the aim to solve the problems of quantum topology such as topological invariants for knots and links (Jones polynomials, HOMFLY polynomials, Khovanov polynomials); topological invariants for graphs (Tutte polynomial and Bollobás-Riordan polynomial); topological invariants for 3-manifolds (Reshetiskin-Turaev, Turaev-Viro and Turaer-Viro-Ocneanu invariants) and topological invariants for 4-manifolds( Crane-Yetter invariants). In a few words, TQC is concerned with the formulation of quantum algorithms for the computation of these topological invariants in quantum topology. Given that one of the fundamental achievements of quantum topology was the discovery of strong connections between monoidal categories and 3-dimensional manifolds, in TQC is possible and necessary to exploit such connections with the purpose to formulate universal quantum algorithms for topological invariants of 3-manifolds. In the present work we make an exploration of such possibilities. Specifically we search for universal quantum algorithms for generalized Turaev-Viro invariants of 3-manifolds such as the Turaev-Viro-Ocneanu invariants, the Kashaev-Baseilhac-Benedetti invariants of 3-manifolds with links and the Geer-Kashaev-Turaev invariants of 3-manifolds with a link and a principal bundle. We also look for physical systems (three dimensional topological insulators and three-dimensional gravity) over which implement the resulting universal topological quantum algorithms.