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arXiv:2308.00138 · Physical review. B./Physical review. B

Boundaries and defects in the cubic code

Cory T. Aitchison, Daniel Bulmash, Arpit Dua, Andrew C. Doherty, Dominic J. Williamson

Published May 9, 2024• 7 citations
arXivHTMLPDF

Abstract

Haah's cubic code is the prototypical type-II fracton topological order. It instantiates the no stringlike operator property that underlies the favorable scaling of its code distance and logical energy barrier. Previously, the cubic code was only explored in translation-invariant systems on infinite and periodic lattices. In these settings, the code distance scales superlinearly with the linear system size, while the number of logical qubits within the degenerate ground space exhibits a complicated functional dependence that undergoes large fluctuations within a linear envelope. Here, we extend the cubic code to systems with open boundary conditions and crystal lattice defects. We characterize the condensation of topological excitations in the vicinity of these boundaries and defects, finding that their inclusion can introduce local stringlike operators and enhance the mobility of otherwise fractonic excitations. Despite this, we use these boundaries and defects to define new encodings where the number of logical qubits scales linearly without fluctuations, and the code distance scales superlinearly, with the linear system size. These include a subsystem encoding with open boundary conditions and a subspace encoding using lattice defects.

DOI
10.1103/physrevb.109.205125

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Haah's cubic code is the prototypical type-II fracton topological order. It instantiates the no stringlike operator property that underlies the favorable scaling of its code distance and logical energy barrier. Previously, the cubic code was only explored in translation-invariant systems on infinite and periodic lattices. In these settings, the code distance scales superlinearly with the linear system size, while the number of logical qubits within the degenerate ground space exhibits a complicated functional dependence that undergoes large fluctuations within a linear envelope. Here, we extend the cubic code to systems with open boundary conditions and crystal lattice defects. We characterize the condensation of topological excitations in the vicinity of these boundaries and defects, finding that their inclusion can introduce local stringlike operators and enhance the mobility of otherwise fractonic excitations. Despite this, we use these boundaries and defects to define new encodings where the number of logical qubits scales linearly without fluctuations, and the code distance scales superlinearly, with the linear system size. These include a subsystem encoding with open boundary conditions and a subspace encoding using lattice defects.
Quantum error-correcting codes (QECs) are essential components of quantum computation and have deep connections to quantum phases of matter. A key obstruction to passive self-correcting QECs is the presence of string logical operators, which can generate logical errors through constant-energy-barrier processes. Haah's Codes (fracton codes) showed that three-dimensional stabilizer codes can forbid such string logical operators, but their translation-invariant structure supports self-similar fractal logical operators with a logarithmic energy barrier. We introduce the qutrit random cubic codes, a family of local qutrit Calderbank-Shor-Steane stabilizer Hamiltonians with similar cube-check structure as Haah's Code 1 but built from spatially varying stabilizers. We prove that these models retain the no-string property and numerically observe that they have properties distinct from translation-invariant fracton codes: the smallest ground-state degeneracy exponent is k=2k=2k=2 for odd LLL and k=4k=4k=4 for even LLL; noncontractible plane-logical operators span the entire logical space; and charge-push diagnostics show that the self-similar fractal operators are absent. These results demonstrate that constrained randomness can fundamentally change the nature of stabilizer codes and improve their self-correction properties. They further point to broader families of quantum error-correcting codes and quantum phases beyond canonical topological and fracton orders.
ZN\mathbb{Z}_NZN​ generalizations of three-dimensional stabilizer codes
In this work, we generalize several three-dimensional Z2 stabilizer models--including the X-cube model, the three-dimensional toric code, and Haah's code--to their ZN counterparts. Under periodic boundary conditions, we analyze their ground state degeneracies and topological excitations, and uncover behaviors that strongly depend on system size. For the X-cube model, we identify excitations with mobility restricted under local operations but relaxed under nonlocal ones derived from global topology. These excitations, previously confined to open boundaries in the Z2 model, now appear even under periodic boundaries. In the toric code, we observe nontrivial braiding between string and point excitations despite the absence of ground state degeneracy, indicating long-range entanglement independent of topological degeneracy. Again, this effect extends from open to periodic boundaries in the generalized models. For Haah's code, we find new excitations--fracton tripoles and monopoles--that remain globally constrained, along with a relaxation of immobility giving rise to lineons and planons. These results reveal new forms of topological order and suggest a broader framework for understanding fracton phases beyond the conventional Z2 setting.
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.
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The authors generalize here three-dimensional \ensuremath{\mathbb{Z}}2{}_{2}2​ stabilizer models---the X-cube, toric code, and Haah's code---to their \ensuremath{\mathbb{Z}}N{}_{N}N​ counterparts, revealing size-dependent ground-state degeneracy and novel excitations. They demonstrate that the generalized X-cube hosts quasi-fractons that remain immobile under local operations but become mobile via global ones. In the toric code, both closed-loop and open-string excitations emerge, while Haah's code features new fracton tripole and monopole excitations. These findings uncover unconventional topological orders and broaden the understanding of fracton phases beyond the conventional \ensuremath{\mathbb{Z}}2{}_{2}2​ framework.
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.