English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the topological stabilizer code and Floquet code defined on anon-orientable surface, which can be considered as families of codes extendingShor's 9-qubit code. We investigate the fault-tolerant logical gates of the$\mathbb{Z}_2$ toric code in this setup, which corresponds to $e\leftrightarrowm$ exchanging symmetry of the underlying $\mathbb{Z}_2$ gauge theory. We findthat non-orientable geometry provides a new way the emergent symmetry acts onthe code space, and discover the new realization of the fault-tolerant Hadamardgate of 2d $\mathbb{Z}_2$ toric code on a surface with a single cross-cap,dubbed an $\mathbb{RP}^2$ code. This Hadamard gate can be realized by aconstant-depth local unitary circuit modulo non-locality caused by a cross-cap,thus reduces the error propagation and eliminates the problem of thefactor-of-two distance reduction compared with the previously known realizationon a surface code. Via folding, the $\mathbb{RP}^2$ code can be turned into abilayer local quantum code, where the folded cross-cap is equivalent to abi-layer twist terminated on a gapped boundary and the logical Hadamard onlycontains local gates with intra-layer couplings. We further obtain the completelogical Clifford gate set for a stack of $\mathbb{RP}^2$ codes. We thenconstruct the honeycomb Floquet code in the presence of a single cross-cap, andfind that the period of the sequential Pauli measurements acts as a $HZ$logical gate on the single logical qubit, where the cross-cap enriches thedynamics compared with the orientable case. We find that the dynamics of thehoneycomb Floquet code is precisely described by a condensation operator of the$\mathbb{Z}_2$ gauge theory, and illustrate the exotic dynamics of our code interms of a condensation operator supported at a non-orientable surface.

Fault-tolerant logical gates via constant depth circuits and emergent symmetries on non-orientable topological stabilizer and Floquet codes