A graph-based formalism for surface codes and twists

Sarkar, Rahul; Yoder, Theodore J.

doi:10.48550/arxiv.2101.09349

Cited by 4 publications

(5 citation statements)

References 29 publications

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“…• The family of genon codes defined in this work overlaps and is inspired by the graphbased formalism of Sarkar-Yoder [33]. They were the first to identify the well-known [ [5,1,3]] as a member of a family of topological codes defined on a torus.…”

Section: Related Workmentioning

confidence: 99%

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A fault-tolerant non-Clifford gate for the surface code in two dimensions

Brown

2020

Sci. Adv.

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Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive development. This alleviates the need for distillation or higher-dimensional components to complete a universal gate set. The operation uses both local transversal gates and code deformations over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. These decoding algorithms allow us to draw upon the advantages of three-dimensional models using only a two-dimensional array of live qubits. Our gate is completed using parity checks of weight no greater than four. We therefore expect it to be amenable with near-future technology. As the gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation considerably.

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Section: Related Workmentioning

confidence: 99%

“…See [33], Theorem 3.9., where they define a family of codes called the cyclic toric codes when gcd(a, b) = 1.…”

Section: Genus Onementioning

confidence: 99%

A fault-tolerant non-Clifford gate for the surface code in two dimensions

Brown

2020

Sci. Adv.

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“…In this scenario, the graph G cannot be consistently two-colored and as a consequence, it can not be directly transformed to a graph G corresponding to a 2 chain complex. Fortunately, we can still use the algebraic tools by constructing the "doubled" graph G d ( L 1,d , L 2,d ) [31]: Without loss of generality, we assume L 1 is even while L 2 is odd in 1-norm. We then obtain the doubled graph G d by combining two copies of G together along L 1 .…”

Section: Deriving the Effective Code Distance For The Gtcsmentioning

confidence: 99%

“…As an example, we show how we construct the doubled graph for a GTC with the doubling was introduced and analyzed in pure graphbased formalism in Ref. [31]. Here we formalize the codes associated with the doubled graph from the perspective of algebraic topology.…”

Section: Deriving the Effective Code Distance For The Gtcsmentioning

confidence: 99%

Tailored XZZX codes for biased noise

Xu¹,

Nam²,

Seif³

et al. 2022

Preprint

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Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.

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“…Non-CSS stabilizers can be similarly defined by plaquettes [16], [17] but care needs to be taken for commutation relations. Some general constructions are provided in [23]. Herein, we consider the XZZX codes in [16], where each code has a rectangular layout rotated by a special angle.…”

Section: Non-css Toric Codesmentioning

confidence: 99%

Comparison of 2D topological codes and their decoding performances

Kuo¹,

Lai²

2022

Preprint

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Topological quantum codes are favored because they allow qubit layouts that are suitable for practical implementation. An N -qubit topological code can be decoded by minimum-weight perfect matching (MWPM) with complexity O(poly(N )) if it is of CSS-type. Recently it is shown that various quantum codes, including non-CSS codes, can be decoded by an adapted belief propagation with memory effects (denoted MBP) with complexity almost linear in N . In this paper, we show that various twodimensional topological codes, CSS or non-CSS, regardless of the layout, can be decoded by MBP, including color codes and twisted XZZX codes. We will comprehensively compare these codes in terms of code efficiency and decoding performance, assuming perfect error syndromes.

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A graph-based formalism for surface codes and twists

Cited by 4 publications

References 29 publications

A fault-tolerant non-Clifford gate for the surface code in two dimensions

A fault-tolerant non-Clifford gate for the surface code in two dimensions

Tailored XZZX codes for biased noise

Comparison of 2D topological codes and their decoding performances

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