arXiv subjects

Sebastian Schönnenbeck

Publications and source records attributed to Sebastian Schönnenbeck.

7 recordsLinked to original sources

arXiv2024

Stabilizer operators and Barnes-Wall lattices

We give a simple description of rectangular matrices that can be implemented by a post-selected stabilizer circuit. Given a matrix with entries in dyadic cyclotomic number fields $\mathbb{Q}(\exp(i\frac{2π}{2^m}))$, we show that it can be implemented by a post-selected stabilizer circuit if it has entries in $\mathbb{Z}[\exp(i\frac{2π}{2^m})]$ when expressed in a certain non-orthogonal basis. This basis is related to Barnes-Wall lattices. Our result is a generalization to a well-known connection between Clifford groups and Barnes-Wall lattices. We also show that minimal vectors of Barnes-Wall lattices are stabilizer states, which may be of independent interest. Finally, we provide a few examples of generalizations beyond standard Clifford groups.

quant-ph↗

arXiv2024

Multi-qubit circuit synthesis and Hermitian lattices

We present new optimal and heuristic algorithms for exact synthesis of multi-qubit unitaries and isometries. For example, our algorithms find Clifford and T circuits for unitaries with entries in $\mathbb{Z}[i,1/\sqrt{2}]$. The optimal algorithms are the A* search instantiated with a new data structure for graph vertices and new consistent heuristic functions. We also prove that for some gate sets, best-first search synthesis relying on the same heuristic is efficient. For example, for two-qubit Clifford and T circuits, our best-first search runtime is proportional to the T-count of the unitary. Our algorithms rely on Hermite and Smith Normal Forms of matrices with entries in a ring of integers of a number field, and we leverage the theory of and algorithms for Hermitian lattices over number fields to prove efficiency. These new techniques are of independent interest for future work on multi-qubit exact circuit synthesis and related questions.

quant-ph↗

arXiv2019

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis of the equivariant spectral sequence combined with torsion subcomplex reduction.\_\_\_\_\_\_\_\_\_\_With an appendix by Bui Anh Tuan and Sebastian Sch{ö}nnenbeck

math.KT↗

arXiv2019

Automorphic Forms on Feit's Hermitian Lattices

We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.

math.NT↗

arXiv2017

Computing $S$-unit groups of orders

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of $S$-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

math.NT↗

arXiv2016

Simultaneous computation of Hecke operators

We present a method to compute two Hecke operators acting on a space of algebraic modular forms simultaneously based on an idea of Eichler's. We show that in certain cases this method can be used to obtain the action of the full Hecke algebra with respect to a hyperspecial subgroup and use it to compute Hecke eigenforms for compact forms of symplectic groups.

math.NT↗

arXiv2016

Resolutions for unit groups of orders

We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic forms and Opgenorth's theory of dual cones. The information from the complex is then used together with Wall's perturbation lemma to obtain the resolution.

math.KT↗