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Physics Communication

Quantum Spectral Algorithms for Khovanov Cohomology via Laplacian Formalism and Frobenius Cobordisms

Listiyanto, Celfien Septian Widya Pratama, Siti Wahyuni

Published Aug 27, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.15294/pc.v9i2.30159

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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.
Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in 4D4D4D supersymmetric Yang--Mills theory. Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown. To address this challenge, this work initiates the study of efficient quantum algorithms for Khovanov homology. We provide simple proofs that increasingly accurate additive approximations to the ranks of Khovanov homology are DQC1-hard, BQP-hard, and #P-hard, respectively. For the first two approximation regimes, we propose a novel quantum algorithm. Our algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap, for which we give numerical and analytical evidence. Our approach introduces a pre-thermalization procedure that allows our quantum algorithm to succeed even if the Betti numbers of Khovanov homology are much smaller than the dimensions of the corresponding chain spaces, overcoming a limitation of prior quantum homology algorithms. We introduce novel connections between Khovanov homology and graph theory to derive analytic lower bounds on the spectral gap.
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.
In this paper, we give a quantum statistical interpretation of the bracket polynomial state sum K , the Jones polynomial V K (t) and virtual knot theory versions of the Jones polynomial, including the arrow polynomial.We use these quantum mechanical interpretations to give new quantum algorithms for these Jones polynomials.In those cases where the Khovanov homology is defined, the Hilbert space C(K ) of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers.There is a natural unitary transformation U: C(K ) → C(K ) such that K = Trace(U ), where K denotes the evaluation of the state sum model for the corresponding polynomial.We show that for the Khovanov boundary operator ∂: C(K ) → C(K ), we have the relationship ∂U + U ∂ = 0. Consequently, the operator U acts on the Khovanov homology, and we obtain a direct relationship between the Khovanov homology and this quantum algorithm for the Jones polynomial.
This talk will explain the structure of Khovanov homology for knots and links and will present a quantum algorithm for its computation that is based on the use of combinatorial Hodge theory. By using combinatorial Hodge theory we can formulate the problem of computing the homology in terms of the determination of the dimension of the eigenvalue one subspace of a unitary operator that is associated with the boundary and coboundary operators of the Khovanov complex. We then use the quantum phase estimation algorithm to find the dimension of this subspace. This is a first foray into the problems of producing a quantum algorithm for the significant knot and link invariant Khovanov homology. There are other deeper problems related to the structure of the chain complex and we shall discuss these difficulties. This paper represents joint work with Sam Lomonaco and Nadya Shirakova.
Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations. A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.
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.