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Paper 1

Climbing the Clifford Hierarchy

Luca Bastioni, Samuel Glandon, Tefjol Pllaha, Madison Stewart, Phillip Waitkevich

Year

2026

Journal

arXiv preprint

DOI

arXiv:2603.12088

arXiv

2603.12088

The Clifford Hierarchy has been a central topic in quantum computation due to its strong connections with fault-tolerant quantum computation, magic state distillation, and more. Nevertheless, only sections of the hierarchy are fully understood, such as diagonal gates and third level gates. The diagonal part of the hierarchy can be climbed by taking square roots and adding controls. Similarly, square roots of Pauli gates (first level) are Clifford gates (climb to the second level). Based on this theme, we study gates whose square roots climb to the next level. In particular, we fully characterize Clifford gates whose square roots climb to the third level.

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Paper 2

Non-Hermitian spectral flows and Berry-Chern monopoles

Lucien Jezequel, Pierre Delplace

Year

2022

Journal

arXiv preprint

DOI

arXiv:2209.03876

arXiv

2209.03876

We propose a non-Hermitian generalization of the correspondence between the spectral flow and the topological charges of band crossing points (Berry-Chern monopoles). A class of non-Hermitian Hamiltonians that display a complex-valued spectral flow is built by deforming an Hermitian model while preserving its analytical index. We relate those spectral flows to a generalized Chern number that we show to be equal to that of the Hermitian case, provided a line gap exists. We demonstrate the homotopic invariance of both the non-Hermitian Chern number and the spectral flow index, making explicit their topological nature. In the absence of a line gap, our system still displays a spectral flow whose topology can be captured by exploiting an emergent pseudo-Hermitian symmetry.

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