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arXiv:1207.0479 · Physical Review A

Topological subsystem codes from graphs and hypergraphs

Pradeep Kiran Sarvepalli, Kenneth R. Brown

Published Oct 31, 2012• 15 citations
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Abstract

Topological subsystem codes were proposed by Bombin based on 3-face-colorable cubic graphs. Suchara, Bravyi, and Terhal generalized this construction and proposed a method to construct topological subsystem codes using 3-valent hypergraphs that satisfy certain constraints. Finding such hypergraphs and computing their parameters, however, is a nontrivial task. We propose families of topological subsystem codes that were previously not known. In particular, our constructions give codes which cannot be derived from Bombin's construction. We also study the error recovery schemes for the proposed subsystem codes and give detailed schedules for the syndrome measurement that take advantage of the 2-locality of the gauge group. The study also leads to a new and general construction for color codes.

DOI
10.1103/physreva.86.042336

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Topological subsystem codes were proposed by Bombin based on 3-face-colorable cubic graphs. Suchara, Bravyi, and Terhal generalized this construction and proposed a method to construct topological subsystem codes using 3-valent hypergraphs that satisfy certain constraints. Finding such hypergraphs and computing their parameters, however, is a nontrivial task. We propose families of topological subsystem codes that were previously not known. In particular, our constructions give codes which cannot be derived from Bombin's construction. We also study the error recovery schemes for the proposed subsystem codes and give detailed schedules for the syndrome measurement that take advantage of the 2-locality of the gauge group. The study also leads to a new and general construction for color codes.
Topological subsystem codes can combine the advantages of both topological codes and subsystem codes. Suchara et al. proposed a framework based on hypergraphs for construction of such codes. They also studied the performance of some subsystem codes. Later Bravyi et al. proposed a subsystem surface code. Building upon these works, we propose efficient decoding algorithms for large classes of subsystem codes on hypergraphs and surfaces. We also propose a construction of the subsystem surface codes that includes the code proposed by Bravyi et al. Our simulations for the subsystem code on the square octagon lattice resulted in a noise threshold of 1.75%. This is comparable to previous result of 2% by Bombin et al. who used a different algorithm.
Topological subsystem codes can combine the advantages of both topological codes and subsystem codes. In this paper we study two classes of topological subsystem codes with an emphasis on decoding them. First, we generalize the two step decoding algorithm of Suchara et al. to all topological subsystem color codes. Then, we propose a construction of subsystem surface codes and develop decoders for them generalizing the results of Bravyi et al. Our simulations for the topological subsystem color code derived from the square octagon lattice resulted in a noise threshold of 1.75%. This is comparable to the previous result of 1.95% by Bombin et al., who used a different algorithm.
Topological subsystem codes proposed recently by Bombin are quantum error correcting codes defined on a two-dimensional grid of qubits that permit reliable quantum information storage with a constant error threshold. These codes require only the measurement of two-qubit nearest-neighbor operators for error correction. In this paper we demonstrate that topological subsystem codes (TSCs) can be viewed as generalizations of Kitaev's honeycomb model to trivalent hypergraphs. This new connection provides a systematic way of constructing TSCs and analyzing their properties. We also derive a necessary and sufficient condition under which a syndrome measurement in a subsystem code can be reduced to measurements of the gauge group generators. Furthermore, we propose and implement some candidate decoding algorithms for one particular TSC assuming perfect error correction. Our Monte Carlo simulations indicate that this code, which we call the five-squares code, has a threshold against depolarizing noise of at least 2%.
A promising approach to overcome decoherence in quantum computing schemes is to perform active quantum error correction using topology. Topological subsystem codes incorporate both the benefits of topological and subsystem codes, allowing for error syndrome recovery with only 2-local measurements in a two-dimensional array of qubits. We study the error threshold for topological subsystem color codes under very general external noise conditions. By transforming the problem into a classical disordered spin model, we estimate using Monte Carlo simulations that topological subsystem codes have an optimal error tolerance of 5.5(2)5.5(2)%5.5(2). This means there is ample space for improvement in existing error-correcting algorithms that typically find a threshold of approximately 22%2.
We introduce a family of two-dimensional (2D) topological subsystem quantum error-correcting codes. The gauge group is generated by two-local Pauli operators, so that two-local measurements are enough to recover the error syndrome. We study the computational power of code deformation in these codes and show that boundaries cannot be introduced in the usual way. In addition, we give a general mapping connecting suitable classical statistical mechanical models to optimal error correction in subsystem stabilizer codes that suffer from depolarizing noise.
Topological subsystem color codes (TSCCs) are an important class of topological subsystem codes that allow for syndrome measurement with only 2-body measurements. It is expected that such low complexity measurements can help in fault tolerance. While TSCCs have been studied over depolarizing noise model, their performance over the erasure channel has not been studied as much. Recently, we proposed erasure decoders for TSCCs and reported a threshold of 9.7%. In this paper, we continue our study of TSCCS over the erasure channel. We propose two erasure decoders for topological subsystem color codes. These decoders employ a mapping of the TSCCs to topological color codes (TCCs). In addition, these decoders use the technique of gauge fixing, where some of the gauge operators of the subsystem code are promoted to stabilizers. We perform gauge fixing using 4-body and 8-body gauge operators. With partial gauge fixing, we obtained a threshold of 17.7% on a TSCC derived from the square octagon lattice. Using an order maximal gauge fixing decoder we were able to improve the threshold to 44%. The performance of the order maximal gauge fixing decoder can be further improved to close to 50% in conjunction with an optimal erasure decoder for topolological color codes. We also study the correctability of erasures on the subsystem codes.