Abstract
A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D "cluster-state" model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally.
Highlights
Quantum-error-correcting codes can be used to protect information in a noisy quantum computer
A self-correcting quantum memory (SCQM) in d spatial dimensions is a quantum-manybody spin system with the following four properties: (i) The Hilbert space consists of a finite density of finite-dimensional spins in d spatial dimensions; (ii) the Hamiltonian H has local terms with bounded strength and range, such that each spin is in the support of only a constant number of terms; (iii) the ground space of H is degenerate such that a qubit can be encoded in the ground space and that this ground space is perturbatively stable; (iv) the lifetime of the stored information after coupling the system to a thermal bath must grow without bound in the system size
We show that spin-lattice models corresponding to 2D symmetry-enriched topological (SET)-ordered boundaries of thermally stable 3D SPTordered phases protected by a suitable 1-form symmetry can be self-correcting quantum memories
Summary
Quantum-error-correcting codes can be used to protect information in a noisy quantum computer. The four-dimensional generalization of the toric code [1] provides a canonical example of a self-correcting quantum memory. Symmetry can provide new directions in the search for self-correcting quantum memories, as the landscape of ordered spin-lattice models becomes even richer when one considers the interplay of symmetry and topology. We show that a self-correcting quantum memory can be encoded in a 2D SET boundary of this 3D model, and it is protected by the thermally stable SPT ordering of the bulk. We discuss the possibility of such 1-form symmetries being emergent in 3D topological models in Sec. V based around the gauge color code.
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