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arXiv:2206.05294 · PRX Quantum

Self-Correction from Higher-Form Symmetry Protection on a Boundary

Charles Stahl

Published Sep 29, 2023• 3 citations
arXivHTMLPDF

Abstract

Recent work has shown that a self-correcting quantum memory can exist in three spatial dimensions, provided that it is protected by a 1-form symmetry.Requiring that the dynamics of a system obey this type of symmetry is equivalent to enforcing a macroscopic number of symmetry terms throughout the bulk.In this paper, we show how to replace the explicit 1-form symmetry with an emergent 1-form symmetry in the bulk and an explicit 1-form symmetry on the boundary.To do so, we use the extended excitations of a three-dimensional (3D) toric code to confine anyons in a two-dimensional (2D) toric code on the boundary.The boundary anyons are bound to the bulk excitations by the explicit 1-form symmetry.Although the symmetry still has to be explicitly enforced on the boundary, this could conceivably be a more attainable constraint due to the accessibility of the boundary qubits.Furthermore, this only requires O(L 2 ) terms for a system of linear size L, instead of O(L 3 ) terms.

DOI
10.1103/prxquantum.4.030341

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Recent work has shown that a self-correcting quantum memory can exist in three spatial dimensions, provided that it is protected by a 1-form symmetry.Requiring that the dynamics of a system obey this type of symmetry is equivalent to enforcing a macroscopic number of symmetry terms throughout the bulk.In this paper, we show how to replace the explicit 1-form symmetry with an emergent 1-form symmetry in the bulk and an explicit 1-form symmetry on the boundary.To do so, we use the extended excitations of a three-dimensional (3D) toric code to confine anyons in a two-dimensional (2D) toric code on the boundary.The boundary anyons are bound to the bulk excitations by the explicit 1-form symmetry.Although the symmetry still has to be explicitly enforced on the boundary, this could conceivably be a more attainable constraint due to the accessibility of the boundary qubits.Furthermore, this only requires O(L 2 ) terms for a system of linear size L, instead of O(L 3 ) terms.
Whether self-correcting quantum memories can exist at nonzero temperature in a physically reasonable setting remains a great open problem. It has recently been argued [S. Roberts and S. D. Bartlett, Phys. Rev. X 10, 031041 (2020)] that symmetry-protected topological (SPT) systems in three space dimensions subject to a strong constraint---that the quantum dynamics respect a 1-form symmetry---realize such a quantum memory. We illustrate how this works in Walker-Wang codes, which provide a specific realization of these desiderata. In this setting we show that it is sufficient for the 1-form symmetry to be enforced on a subvolume of the system. This strongly suggests that the SPT character of the state is not essential. We confirm this by constructing an explicit example with a trivial (paramagnetic) bulk that realizes a self-correcting quantum memory. We therefore show that the enforcement of a 1-form symmetry on a measure-zero subvolume of a three-dimensional system can be sufficient to stabilize a self-correcting quantum memory at nonzero temperature.
We study the two-dimensional toric-code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow an arbitrary increase in the lifetime of the stored quantum information by making LLL, the linear size of the memory, larger [Chesi et al., Phys. Rev. A 82, 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough LLL. This influence can be either beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. [Phys. Rev. B 83, 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in L2{L}^{2}L2 for both periodic and open boundary conditions, although the exponent in the latter case is found to be less favorable. However, LLL is upper bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.
The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (``self-correction''), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.
Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high-energy scales while other topological excitations have low energies. The low-energy properties of topological orders in this limit, with the emergent higher symmetries, may be described by higher symmetry-protected topological order. This motivates us, as a simplest example, to study a lattice model of Zn{\mathbb{Z}}_{n}Zn​-1 symmetry-protected topological (1-SPT) states in 3+13+13+1 dimensions for even nnn. We write an exactly solvable lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with nontrivial self-statistics. For the n=2n=2n=2 case, where the bulk classification is given by an integer mmm mod 4, we show that the boundary can be gapped with double-semion topological order for m=1m=1m=1 and toric code for m=2m=2m=2. The bulk ground-state wave-function amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.
This work explores a deformation of Kitaev's toric code that induces a phase transition out of the topologically ordered phase. By placing the model on a cylinder, the bulk global 1-form symmetries separate into distinct boundary operators, allowing us to show that the transition is accompanied by the condensation of all anyons in a given boundary. Using a lower dimensional (1 + 1)-dimensional boundary Hamiltonian, we extract an effective central charge and find a pronounced suppression near βcβ_cβc​, followed by its restoration at strong coupling.