arXiv subjects

Jens Niklas Eberhardt

Publications and source records attributed to Jens Niklas Eberhardt.

At least 19 recordsLinked to original sources

arXiv2026

Condensed configurations and valuative matroid invariants

Condensed configurations are compact incidence data obtained by grouping the cyclic flats of a matroid. We show that their inverse incidence matrices give explicit Schubert expansions and hence determine every valuative or covaluative matroid invariant. For the extended binary Golay matroid, this unexpectedly produces non-real-rooted Kazhdan-Lusztig and Z-polynomials. To our knowledge, the latter is the first counterexample in the literature.

math.CO↗

arXiv2026

Nearest-neighbour gates are all you need: High-rate quantum low-density parity-check codes on a planar grid

High-performance quantum low-density parity-check codes promise substantial reductions in the overhead of fault-tolerant quantum computation, but most constructions require long-range connectivity or qubit shuttling, both of which are difficult to realise in superconducting architectures. Here we introduce a family of quantum low-density parity-check codes that, for the first time, combines planar open-boundary layouts, finite-size advantages over surface codes, and syndrome extraction using only nearest-neighbour gates on a square grid of qubits. The key idea is to generate check-data connectivity dynamically: nearest-neighbour iSWAP walks both define the stabiliser supports and implement their measurement, avoiding the need for a long-range hardware graph. The resulting circuits achieve optimal constant-depth stabiliser measurement, independent of code size, and naturally remove leakage from the system by exchanging the role of check and data qubits at each syndrome extraction round. We find finite-size instances such as a [[323,14,15]] code, whose code-efficiency ratio is nearly an order of magnitude larger than that of rotated surface-code patches. At around 30 circuit qubits per logical qubit, the best directional tile-code layouts reduce the per-logical per-round logical error rate by up to a factor of 1000 relative to rotated surface-code memories. These results show that the advantages of quantum low-density parity-check codes can survive compilation into strictly planar nearest-neighbour circuits, bringing low-overhead fault-tolerant memories closer to near-term hardware.

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arXiv2025

The Parabolic K-motivic Hecke Category

We define and study the parabolic K-motivic Hecke category of a (possibly disconnected) Kac-Moody group. Our main result is a combinatorial description via singular K-theory Soergel bimodules which arise from the equivariant algebraic K-theory of parabolic Bott-Samelson resolutions. In the spherical affine case, the K-motivic Hecke category serves as one side of a conjectural quantum K-theoretic derived Satake equivalence, addressing a conjecture of Cautis-Kamnitzer.

math.RT↗

arXiv2025

Logical Operators and Derived Automorphisms of Tile Codes

The recently introduced tile codes are a promising alternative to surface codes, combining two-dimensional locality with higher encoding efficiency. While surface codes are well understood in terms of their logical operators and boundary behavior, much less is known about tile codes. In this work, we establish a natural and precise description of their logical operator space. We prove that, under mild assumptions, any tile code admits a canonical symplectic basis of logical operators supported along lattice boundaries, which can be generated efficiently by a simple cellular automaton with the number of update rules only depending on the non-locality of the tile code. Further, we develop algebraic and algebro-geometric frameworks for tile codes, by resolving them by translationally invariant Pauli stabilizer models and showing that they arise as derived sections of a Koszul complex on $\mathbb{P}^1 \times \mathbb{P}^1$. Finally, we introduce the concept of derived automorphisms for quantum codes. These are automorphism-like operations that can exist even for codes that do not have symmetries. We explain how derived automorphisms can be implemented for tile codes in a low-overhead and fault-tolerant manner by extending the lattice on one side and shrinking it on the other. While this operation is trivial for the surface code, it induces a product of logical CNOT gates on the encoded information. Our results provide new structural insights into tile codes and lay the groundwork for tile codes as building blocks for fault-tolerant quantum computation.

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arXiv2025

Universal Koszul Duality for Kac-Moody Groups

We prove a monoidal equivalence, called universal Koszul duality, between genuine equivariant K-motives on a Kac-Moody flag variety and constructible monodromic sheaves on its Langlands dual. The equivalence is obtained by a Soergel-theoretic description of both sides which extends results for finite-dimensional flag varieties by Taylor and the first author. Universal Koszul duality bundles together a whole family of equivalences for each point of a maximal torus. At the identity, it recovers an ungraded version of Beilinson-Ginzburg-Soergel's and Bezrukavnikov-Yun's Koszul duality for equivariant and unipotently monodromic sheaves. It also generalizes Soergel-theoretic descriptions for monodromic categories on finite-dimensional flag varieties by Lusztig-Yun, Gouttard and the second author. For affine Kac-Moody groups, our work sheds new light on the conjectured quantum Satake equivalences by Cautis-Kamnitzer and Gaitsgory. On our way, we establish foundations on six functors for reduced K-motives and introduce a formalism of constructible monodromic sheaves.

math.RT↗

arXiv2025

Planar quantum low-density parity-check codes with open boundaries

Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions $6 \leq k\leq13$, e.g., $[[78, 6, 6]]$, $[[107, 7, 7]]$, $[[268, 8, 12]]$, $[[405, 9, 15]]$, $[[348, 10, 13]]$, $[[450, 11, 15]]$, $[[386, 12, 12]]$, $[[362, 13, 11]]$, all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a $[[282,12,14]]$ code, exhibiting an efficiency metric ($kd^2/n$) an order of magnitude higher than the surface code. The construction combines boundary anyon condensation with the ``lattice grafting'' optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators whose distance scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.

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arXiv2025

Tile Codes: High-Efficiency Quantum Codes on a Lattice with Boundary

We introduce tile codes, a simple yet powerful way of constructing quantum codes that are local on a planar 2D-lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer weight. Our construction does not compromise on the fact that the codes are local on a lattice with open boundary conditions. Despite its simplicity, we use our construction to find codes with parameters $[[288, 8, 12]]$ using weight-6 stabilizers and $[[288, 8, 14]]$ using weight-8 stabilizers, outperforming all previously known constructions in this direction. Allowing for a slightly higher non-locality, we find a $[[512, 18, 19]]$ code using weight-8 stabilizers, which outperforms the rotated surface code by a factor of more than 12. Our approach provides a unified framework for understanding the structure of codes that are local on a 2D planar lattice and offers a systematic way to explore the space of possible code parameters. In particular, due to its simplicity, the construction naturally accommodates various types of boundary conditions and stabilizer configurations, making it a versatile tool for quantum error correction code design.

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arXiv2024

Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions

Quantum low-density parity-check codes are promising candidates for quantum error correcting codes as they might offer more resource-efficient alternatives to surface code architectures. In particular, bivariate bicycle codes have recently gained attention due to their 2D-local structure, high encoding rate, and promising performance under simulation. In this work, we will explore how one can transform bivariate bicycle codes defined on lattices with periodic boundary conditions to codes with the same locality properties on a 2D lattice with open boundary conditions. For this, we introduce the concept of pruning quantum codes. We explain how pruning bivariate bicycle codes is always possible when the codes are hypergraph products of two classical cyclic codes. We also indicate that this might be possible for more general bivariate bicycle codes by constructing explicit examples. Finally, we investigate fault-tolerant quantum computation using the constructed pruned codes by describing fold-transversal gates.

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arXiv2024

Two proofs of a Jantzen Conjecture for Whittaker Modules

We define a filtration of a standard Whittaker module over a complex semisimple Lie algebra and and establish its fundamental properties. Our filtration specialises to the Jantzen filtration of a Verma module for a certain choice of parameter. We prove that embeddings of standard Whittaker modules are strict with respect to our filtration, and that the filtration layers are semisimple. This provides a generalisation of the Jantzen conjectures to Whittaker modules. We prove these statements in two ways. First, we give an algebraic proof which compares Whittaker modules to Verma modules using a functor introduced by Backelin. Second, we give a geometric proof using mixed twistor $\mathcal{D}$-modules.

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arXiv2024

An Intersection Matrix for Affine Hyperplane Arrangements

For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of Schechtman-Varchenko, we show that there is a closed formula for its determinant that only depends on the combinatorics of the underlying matroid. We conjecture an analogous formula for its $q$-deformation. Our work also applies more generally in the setting of affine oriented matroids. Additionally, we give a representation-theoretic interpretation of our $q$-intersection matrix using Braden-Licata-Proudfoot-Websters's hypertoric category $\mathcal{O}$ (or more generally Kowalenko-Mautner's category $\mathcal{O}$ for oriented matroid programs). This paper is part of a broader program to categorify matroidal Schur algebras defined by Braden-Mautner.

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arXiv2024

Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes

Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates without overhead. As examples, we construct $[[98,6,12]]$ and $[[162, 8, 12]]$ BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which might be of independent interest.

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arXiv2024

K-motives, Springer Theory and the Local Langlands Correspondence

We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive $p$-adic groups via a $K$-motivic Springer theory. We suggest a connection to the coherent Springer theory of Ben-Zvi, Chen, Helm, and Nadler through a categorical Chern character and outline results and conjectures on $K$-motives within the Langlands program. To achieve our results, we introduce a six functor formalism for reduced $K$-motives applicable to linearly reductive stacks and establish formality for categories of Springer $K$-motives. We work within a broader framework of Hecke algebras derived from Springer data. This makes the results applicable, for example, to the ($K$-theoretic) quiver Hecke and Schur algebra. Moreover, we relate our constructions to prior geometric realizations for graded Hecke algebras.

math.RT↗

arXiv2023

Standard Extension Algebras I: Perverse Sheaves and Fukaya Calculus

In this first of a series of articles on standard extension algebras we study standard perverse sheaves on varieties with $\mathbb{G}_m$-actions. Based on Braden's hyperbolic localisation, we describe their extension algebra geometrically via a convolution structure on the intersections of attracting with repelling cells. We introduce a multiplicative structure on open Richardson varieties which provides a practical way to compose these extensions in case of flag varieties. For open Richardson varieties in Grassmannians we construct two explicit cell decompositions, of Gauss- and of Deodhar-type. It is shown that the latter is a stratification with the same combinatorics as Deodhar's decomposition. We introduce a calculus of Fukaya diagrams to encode the geometry of the decompositions. It provides a model for the cohomology of open Richardson varieties and thus for standard extensions. The calculus is motivated by the Mak-Smith Fukaya-Seidel category of a natural Lefschetz fibration and should allow to compute morphism spaces in there. We finally discuss the relation of our work to extensions of (parabolic) Verma modules in category O as well as to the computation of R- and R'-polynomials.

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arXiv2022

K-theory Soergel Bimodules

We initiate the study of K-theory Soergel bimodules-a K-theory analog of classical Soergel bimodules. Classical Soergel bimodules can be seen as a completed and infinitesimal version of their new K-theoretic analog. We show that morphisms of K-theory Soergel bimodules can be described geometrically in terms of equivariant K-theoretic correspondences between Bott-Samelson varieties. We thereby obtain a natural categorification of K-theory Soergel bimodules in terms of equivariant coherent sheaves. We introduce a formalism of stratified equivariant K-motives on varieties with an affine stratification, which is a K-theoretric analog of the equivariant derived category of Bernstein-Lunts. We show that Bruhat-stratified torus-equivariant K-motives on flag varieties can be described in terms of chain complexes of K-theory Soergel bimodules. Moreover, we propose conjectures regarding an equivariant/monodromic Koszul duality for flag varieties and the quantum K-theoretic Satake.

math.RT↗

arXiv2022

K-Motives and Koszul Duality

We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of $K$-theory. For this, we introduce and study categories $\operatorname{DK}_{\mathcal{S}}(X)$ of $\mathcal{S}$-constructible $K$-motivic sheaves on varieties $X$ with an affine stratification $\mathcal{S}$. We show that there is a natural and geometric functor, called Beilinson realisation, from $\mathcal{S}$-constructible mixed sheaves $\operatorname{D}^{mix}_{\mathcal{S}}(X)$ to $\operatorname{DK}_{\mathcal{S}}(X)$. We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible $K$-motivic sheaves on Langlands dual flag varieties.

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arXiv2022

Integral Motivic Sheaves And Geometric Representation Theory

With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of graded $\ell$-adic sheaves.

math.AG↗

arXiv2021

Motivic Springer Theory

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures. As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.

math.RT↗

arXiv2021

Quantum Low-Density Parity-Check Codes

Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called low-density parity-check (LDPC) quantum codes. The codes we discuss are alternatives to the surface code, which is the currently leading candidate to implement quantum fault-tolerance. We introduce the zoo of LDPC quantum codes and discuss their potential for making quantum computers robust against noise. In particular, we explain recent advances in the theory of LDPC quantum codes related to certain product constructions and discuss open problems in the field.

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