Local decoders for the 2D and 4D toric code

Breuckmann, Nikolas P.; Duivenvoorden, Kasper; Michels, D.; Terhal, Barbara M.

doi:10.26421/qic17.3-4-1

Cited by 31 publications

(36 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In 4D, a natural choice is to put qubits on 2-cells, so that one associates a Z-check with each 3-cell and an X-check with each edge. Beyond the 4D toric code which corresponds to a filling of flat 4D space, namely the honeycomb {4, 3, 3, 4} [10,11], generalizations of the hyperbolic surface codes to 4D are known to exist as well [12,13]. These codes have a number of logical qubits k which scales linearly with the number of physical qubits n, just like the hyperbolic surface codes.…”

Section: (B) Some Three-and Four-dimensional Examples Based On Regular Tessellationsmentioning

confidence: 99%

“…With the above considerations, we now describe a decoding protocol which satisfies the faulttolerant criteria outlined in Condition 5.1. Given the size of the small stellated dodecahedron code, it is possible to decode X and Z errors separately using full lookup tables (since each contains only 2 11 = 2048 syndromes). For a given syndrome s, the look-up table chooses the lowest weight error E that corresponds to the measured syndrome.…”

Section: Fault-tolerant Circuits For the Small Stellated Dodecahedron Codementioning

confidence: 99%

“…The interest in the small stellated dodecahedron code is that it can pack logical qubits very densely while, like the [ [9,1,3]] surface code, still allowing for plain fault-tolerant parity check measurements in combination with a look-up table decoder. Even denser packings of logical qubits in block stabilizer codes are certainly feasible: there are non-CSS codes such as [ [8,3,3]], [ [10,4,3]], [ [11,5,3]], [ [13,7,3]] and [ [14,8,3]] codes listed in [6]. However, one may expect that the construction of fault-tolerant parity check circuits for such codes requires resource-intense methods such as Steane, Shor or Knill error correction (EC), or flag-fault-tolerance methods [7,8].…”

Section: Introductionmentioning

confidence: 99%

“…Z (0,8) Z (0,9) Z (6,8) Z (6,9)X8 = X (0,6) X (0,10) X (1,3) X (1,11) X (3,5) X(5,11) X(6,8) X (8,10) . .…”

mentioning

confidence: 99%

See 3 more Smart Citations

The small stellated dodecahedron code and friends

Conrad

Chamberland

Breuckmann

et al. 2018

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges respectively vertices of a small stellated dodecahedron, making this code suitable for three-dimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold.This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

show abstract

Section: (B) Some Three-and Four-dimensional Examples Based On Regular Tessellationsmentioning

confidence: 99%

Section: Fault-tolerant Circuits For the Small Stellated Dodecahedron Codementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Z (0,8) Z (0,9) Z (6,8) Z (6,9)X8 = X (0,6) X (0,10) X (1,3) X (1,11) X (3,5) X(5,11) X(6,8) X (8,10) . .…”

mentioning

confidence: 99%

See 2 more Smart Citations

The small stellated dodecahedron code and friends

Conrad

Chamberland

Breuckmann

et al. 2018

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…For n = 3, we obtain the toric codes with parameters [[3m 3 , 3, m]], see [11], and for n = 4, we obtain the toric codes proposed by Breuckman et al [12] with parameters [[6m 4 , 6, m 2 ]].…”

Section: Introductionmentioning

confidence: 99%

Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

Carvalho

Soares

Silva

2021

Entropy

View full text Add to dashboard Cite

In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

show abstract

Single-shot quantum error correction with the three-dimensional subsystem toric code

Kubica

Vasmer

2022

Nat Commun

View full text Add to dashboard Cite

Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to be pSTC ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.

show abstract

Local decoders for the 2D and 4D toric code

Cited by 31 publications

References 29 publications

The small stellated dodecahedron code and friends

The small stellated dodecahedron code and friends

Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

Single-shot quantum error correction with the three-dimensional subsystem toric code

Contact Info

Product

Resources

About