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Paper 1

Qutrit codes within representations of SU(3)

Xzavier Herbert, Jonathan Gross, Michael Newman

Year

2023

Journal

arXiv preprint

DOI

arXiv:2312.00162

arXiv

2312.00162

We describe a quantum error-detecting and error-correcting code embedded within irreducible representations of SU(3). These logical qutrits inherit the He(3) symmetries induced by the representation, while protecting against small SU(3) displacements. We explore the general methodology for finding codes from structure-inducing representations of groups, together with symmetries inherited from finite subgroups, extending the case of spin representations of SU(2).

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Paper 2

Critical properties of the Anderson transition in random graphs: two-parameter scaling theory, Kosterlitz-Thouless type flow and many-body localization

Ignacio García-Mata, John Martin, Olivier Giraud, Bertrand Georgeot, Rémy Dubertrand, Gabriel Lemarié

Year

2022

Journal

arXiv preprint

DOI

arXiv:2209.04337

arXiv

2209.04337

The Anderson transition in random graphs has raised great interest, partly because of its analogy with the many-body localization (MBL) transition. Unlike the latter, many results for random graphs are now well established, in particular the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. Here we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length $ξ_\parallel$ than the one in the transverse direction, $ξ_\perp$. Importantly, these two lengths have different critical behaviors: $ξ_\parallel$ diverges with a critical exponent $ν_\parallel=1$, while $ξ_\perp$ reaches a finite universal value ${ξ_\perp^c}$ at the transition point $W_c$. Indeed, $ξ_\perp^{-1} \approx {ξ_\perp^c}^{-1} + ξ^{-1}$, with $ξ\sim (W-W_c)^{-ν_\perp}$ associated with a new critical exponent $ν_\perp = 1/2$, where $\exp( ξ)$ controls finite-size effects. The delocalized phase inherits the strongly non-ergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $ν=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $ξ_\perp$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a smallworld complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

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