Optimal thresholds for fracton codes and random spin models with subsystem symmetry

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Fracton models provide examples of novel gapped quantum phases of matter that host intrinsically immobile excitations and therefore lie beyond the conventional notion of topological order. Here, we calculate optimal error thresholds for quantum error correcting codes based on fracton models. By mapping the error-correction process for bit-flip and phase-flip noises into novel statistical models with Ising variables and random multibody couplings, we obtain models that exhibit an unconventional subsystem symmetry instead of a more usual global symmetry. We perform large-scale parallel tempering Monte Carlo simulations to obtain disorder-temperature phase diagrams, which are then used to predict optimal error thresholds for the corresponding fracton code. Remarkably, we found that the X-cube fracton code displays a minimum error threshold (7.5%) that is much higher than 3D topological codes such as the toric code (3.3%), or the color code (1.9%). This result, together with the predicted absence of glass order at the Nishimori line, shows great potential for fracton phases to be used as quantum memory platforms.
© 2022 American Physical Society H. S. and M. A. M.-D. acknowledge support from the Spanish MINECO grants MINECO/FEDER Projects FIS2017-91460-EXP, PGC2018-099169-B-I00 FIS-2018, and with O. V. from CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM). H. S. has also been supported by the Natural Sciences and Engineering Research Council of Canada and by the National Natural Science Foundation of China (Grant No. 12047503). M. A. M.-D. has also been supported by MCIN with funding from European Union Next-GenerationEU (PRTR-C17.I1) and the Ministry of Economic Affairs Quantum ENIA project, and partially by the U.S. Army Research Office through Grant No. W911NF-14-1-0103. J. S.-K., K. L., and L. P. acknowledge support from FP7/ERC Consolidator Grant QSIMCORR, No. 771891, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2111-390814868. The project and research is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the High-tech Agenda Bayern Plus. Our simulations make use of the ALPSCore library [66] and the TKSVM library [67,68].
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