Parameterized stable homotopy theory organizes local systems of spectra over homotopy types, governed by a "yoga" of six functors. To provide semantics for the recently developed Linear Homotopy Type Theory (LHoTT), good model categories of these spectra are required, preferably monoidal with respect to the external smash product. We focus on the case of parameterized $H\mathbb{K}$ -module spectra ( $\infty$ -local systems), motivated by recent applications of parameterized homotopy to topological quantum computing. While traditionally treated via dg-categories, we leverage combinatorial model structures on simplicial chain complexes to construct the first dedicated global model structure for $\mathbb{K}$ -linear $\infty$ -local systems, which offers better control than existing models for general parameterized spectra. In particular, when restricted to base 1-types, our model structure is monoidal with respect to the external tensor product, making it a candidate target semantics for the multiplicative fragment of LHoTT.Parameterized stable homotopy theory organizes local systems of spectra over homotopy types, governed by a "yoga" of six functors. To provide semantics for the recently developed Linear Homotopy Type Theory (LHoTT), good model categories of these spectra are required, preferably monoidal with respect to the external smash product. We focus on the case of parameterized $H\mathbb{K}$-module spectra ($\infty$-local systems), motivated by recent applications of parameterized homotopy to topological quantum computing. While traditionally treated via dg-categories, we leverage combinatorial model structures on simplicial chain complexes to construct the first dedicated global model structure for $\mathbb{K}$-linear $\infty$-local systems, which offers better control than existing models for general parameterized spectra. In particular, when restricted to base 1-types, our model structure is monoidal with respect to the external tensor product, making it a candidate target semantics for the multiplicative fragment of LHoTT.

math.ATCross-listedarXiv:2604.05671Apr 8, 2026

Topological Quantum Computation

Jiannis K. Pachos

Lecture notes in computer scienceJan 1, 20137 citations

Topological Quantum Computation

R. W. Ogburn, John P. Preskill

Lecture notes in computer scienceJan 1, 199980 citations

Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory

Juan Ospina

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.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEMay 21, 20150 citations

Cups and Gates II: Operads, twists, and addressable intra-code gates

Nikolas P. Breuckmann, Margarita Davydova, Jens Niklas Eberhardt +2 more

In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of $C^{k-1}Z$ logical gates that we called copy-cup on $k$ copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from $\mathbb{Z}/2$ to $\mathbb{Z}/2^k$ , enabling all diagonal logical gates such as single-qubit rotations $\operatorname{R}(2π/2^k)$ . Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable $T$ gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from $E_\infty$ -structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of $C^{k-1}Z$ logical gates that we called copy-cup on $k$ copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from $\mathbb{Z}/2$ to $\mathbb{Z}/2^k$, enabling all diagonal logical gates such as single-qubit rotations $\operatorname{R}(2π/2^k)$. Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable $T$ gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from $E_\infty$-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.

quant-pharXiv:2610.05214Oct 6, 20260 citations

A Global Model Structure for

\mathbb{K}

-Linear

\infty

-Local Systems

H\mathbb{K}

-module spectra (

\infty

-local systems), motivated by recent applications of parameterized homotopy to topological quantum computing. While traditionally treated via dg-categories, we leverage combinatorial model structures on simplicial chain complexes to construct the first dedicated global model structure for

\mathbb{K}

-linear

\infty

-local systems, which offers better control than existing models for general parameterized spectra. In particular, when restricted to base 1-types, our model structure is monoidal with respect to the external tensor product, making it a candidate target semantics for the multiplicative fragment of LHoTT.

C^{k-1}Z

logical gates that we called copy-cup on

k

copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from

\mathbb{Z}/2

\mathbb{Z}/2^k

, enabling all diagonal logical gates such as single-qubit rotations

\operatorname{R}(2π/2^k)

. Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable

T

gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from

E_\infty

-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.