Abstract
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.
Highlights
Any architecture proposed for information storage must be equipped with an error correction strategy to avoid the corruption of the data encoded, whether the information is classical or quantum in nature [8, 25, 30]
The subspace in which the quantum information is stored is the joint eigenspace of pairwise commuting operators, called stabilizers
We investigate a topological order that is entirely new to the context of quantum error correction, but is still in principle realizable with a qubit architecture
Summary
Any architecture proposed for information storage must be equipped with an error correction strategy to avoid the corruption of the data encoded, whether the information is classical or quantum in nature [8, 25, 30]. Other generalisations involving non-commuting stabiliser sets [27] have demonstrated the ability to produce gate sets which, while not universal, have enhanced computation power Taken together, these findings strongly motivate the quest for new topological quantum error correction codes with stabilizers outside the Pauli group. We modify existing lattice models for topological orders – twisted quantum double models – so that they give rise to Non-Pauli stabilizers In their original form, the local terms of these Hamiltonians do not commute in a particular excited subspace of the Hilbert space, which makes them – on first glance – unsuitable for stabilizer error correction. In most cases – namely, for Abelian twisted quantum doubles – these obstructions can be lifted completely by carefully modifying the offending terms in the Hamiltonian, yielding a true stabilizer code, consisting of commuting non-Pauli operators.
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