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arXiv:2009.01252 · Quantum

Fractalizing quantum codes

Trithep Devakul, Dominic J. Williamson

Published Apr 22, 2021• 14 citations
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Abstract

We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms of well understood lower-dimensional spin models. Fractalization is also useful for deriving new spin models and quantum codes from known ones. We construct higher dimensional generalizations of fracton models that host extended fractal excitations. Finally, by applying fractalization to a 2D subsystem code, we produce a family of locally generated 3D subsystem codes that are conjectured to saturate a quantum information storage tradeoff bound.

DOI
10.22331/q-2021-04-22-438

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We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms of well understood lower-dimensional spin models. Fractalization is also useful for deriving new spin models and quantum codes from known ones. We construct higher dimensional generalizations of fracton models that host extended fractal excitations. Finally, by applying fractalization to a 2D subsystem code, we produce a family of locally generated 3D subsystem codes that are conjectured to saturate a quantum information storage tradeoff bound.
We generalize fractalization, a procedure for the construction of fracton models, from space to spacetime. We apply spacetime fractalization to construct fracton floquet codes with syndrome excitations that have limited mobility in space and time. This extends the notion of fracton order to intrinsically dynamical quantum phases of matter that are inequivalent to static fracton phases. We find spacetime type-II fracton floquet codes which have no topological excitations that are mobile in space or time. These codes exhibit an extreme form of quantum discrete time crystal order with response periods that scale exponentially in their linear system sizes. In this context, the no-strings rule that characterizes type-II fractons leads to a superlinear scaling of the floquet code fault-distance with time, potentially lowering the time overhead required for quantum error correction.
Constructing new quantum codes and understanding their error resilience are central challenges in the development of robust quantum memories. Topological codes are particularly promising due to their favorable error-correcting properties and their connections to phases of matter in many-body physics. In this thesis, we explore the interplay between classical Ising models and quantum error correction through subsystem symmetries, fracton topological order, and Kramers-Wannier-type duality. We study two three-dimensional classical self-dual Ising models with subsystem symmetries, the Tetrahedral Ising model and the Fractal Ising model, investigating their thermal behavior, their relation to fracton phases through subsystem-symmetry gauging, and the properties of the resulting fracton codes. Using a statistical-mechanical mapping, we determine the optimal code-capacity threshold of the Checkerboard code to be 0.107(3)0.107(3)0.107(3), which saturates the theoretical limit for CSS codes and represents the highest optimal error threshold among known three-dimensional codes. We relate this saturation to a generalized entropy relation for classical spin models satisfying a Kramers-Wannier-type duality, and argue how this prediction extends to CSS codes with zero encoding rate whose XXX- and ZZZ-noise models map to classically dual spin models. These findings establish fracton codes as highly resilient candidates for quantum memories and demonstrate the power of the statistical-mechanical framework, together with its duality predictions, in analyzing and constructing robust quantum error-correcting codes.
We study spin systems which exhibit symmetries that act on a fractal subset of sites, with fractal structures generated by linear cellular automata. In addition to the trivial symmetric paramagnet and spontaneously symmetry broken phases, we construct additional fractal symmetry protected topological (FSPT) phases via a decorated defect approach. Such phases have edges along which fractal symmetries are realized projectively, leading to a symmetry protected degeneracy along the edge. Isolated excitations above the ground state are symmetry protected fractons, which cannot be moved without breaking the symmetry. In 3D, our construction leads additionally to FSPT phases protected by higher form fractal symmetries and fracton topologically ordered phases enriched by the additional fractal symmetries.
Fracton codes host unconventional topological states of matter and are promising for fault-tolerant quantum computation due to their large coding space and strong resilience against decoherence and noise. In this paper, we investigate the ground-state properties and phase transitions of two prototypical self-dual fracton spin models—the tetrahedral Ising model and the fractal Ising model—which correspond to error-correction procedures for the representative fracton codes of type I and type II, the checkerboard code and the Haah's code, respectively, in the error-free limit. They are endowed with exotic symmetry-breaking properties that contrast sharply with the spontaneous breaking of global symmetries and deconfinement transition of gauge theories. To show these unconventional behaviors, which are associated with subdimensional symmetries, we construct and analyze the order parameters, correlators, and symmetry generators for both models. Notably, the tetrahedral Ising model acquires an extended semilocal ordering moment, while the fractal Ising model fits into a polynomial ring representation and leads to a fractal order parameter. Numerical studies combined with analytical tools show that both models experience a strong first-order phase transition with an anomalous L−(D−1) scaling, despite the fractal symmetry of the latter. Our paper provides a unique understanding of subdimensional symmetry breaking and makes an important step for studying quantum-error-correction properties of the checkerboard and Haah's codes. Published by the American Physical Society 2024