# Triangular color codes on trivalent graphs with flag qubits

@article{Chamberland2019TriangularCC, title={Triangular color codes on trivalent graphs with flag qubits}, author={Christopher Chamberland and Aleksander Kubica and Theodore J. Yoder and Guanyu Zhu}, journal={arXiv: Quantum Physics}, year={2019} }

The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the…

## Figures and Tables from this paper

## 44 Citations

Quantum error correction with the color-Gottesman-Kitaev-Preskill code

- Materials Science, PhysicsPhysical Review A
- 2021

Jiaxuan Zhang, 2 Jian Zhao, 2 Yu-Chun Wu, 2, 3, ∗ and Guo-Ping Guo 2, 3, 4 Key Laboratory of Quantum Information, Chinese Academy of Sciences, School of Physics, University of Science and Technology…

Finding fault-tolerant Clifford circuits using satisfiability modulo theories solvers and decoding merged color-surface codes

- Computer Science
- 2022

This work shows how satisﬁability modulo theories (SMT) solvers can be used to automate the construction of Clifford circuits with certain fault-tolerance properties and applies their techniques to a fault-Tolerant magic state preparation protocol.

Using redundant ancilla encoding and flags for very low overhead fault-tolerant magic state preparation

- Computer Science
- 2020

This work introduces a new concept which they call redundant ancilla encoding, and shows that for physical error rates near 10−4 and under a full circuit level noise model, this scheme can produce magic states using an order of magnitude fewer qubits compared to the most competitive magic state distillation schemes.

Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes

- Physics, Computer SciencePRX Quantum
- 2022

This work presents a comprehensive architectural analysis for a fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes, and proposes a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout.

Circuit-level protocol and analysis for twist-based lattice surgery

- Computer SciencePhysical Review Research
- 2022

The first circuit-level error correction simulations during twist-based lattice surgery using a biased depolarizing noise model are undertaken and a slight decrease in the threshold for timelike logical failures is indicated.

Decoder for the Triangular Color Code by Matching on a Möbius Strip

- Computer SciencePRX Quantum
- 2022

This work proposes a decoder for the planar color code on a triangular lattice where it is found that a modification of the decoder that manually compares inequivalent recovery operators can correct all errors of weight (d − 1)/2 for codes with distance d ≤ 15.

Fault-Tolerant Parity Readout on a Shuttling-Based Trapped-Ion Quantum Computer

- PhysicsPhysical Review X
- 2022

J. Hilder, D. Pijn, O. Onishchenko, A. Stahl, M. Orth, B. Lekitsch, A. Rodriguez-Blanco, M. Müller, 4 F. Schmidt-Kaler, and U. G. Poschinger ∗ Institut für Physik, Universität Mainz, Staudingerweg 7,…

Finding fault-tolerant Clifford circuits using SMT solvers

- Computer ScienceArXiv
- 2022

This work shows how satisfiability modulo theories (SMT) solvers can be used to automate the construction of Clifford circuits with certain fault-tolerance properties and applies their techniques to a fault-Tolerant magic state preparation protocol.

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

- PhysicsQuantum
- 2022

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological…

LILLIPUT: a lightweight low-latency lookup-table decoder for near-term Quantum error correction

- Computer Science
- 2022

To enable real-time decoding in near-term QEC, LILLIPUT– a Lightweight Low Latency Look-Up Table decoder is proposed, which tolerates an error on any operation in the quantum hardware, including gates and measurements, and the number of tolerated errors grows with the size of the code.

## References

SHOWING 1-10 OF 67 REFERENCES

Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits

- Computer Science, PhysicsPhysical Review X
- 2020

This work modify minimum weight perfect matching decoding to efficiently and scalably incorporate information from measurements of the flag qubits and correct up to the full code distance while respecting the limited connectivity.

Efficient color code decoders in $d\geq 2$ dimensions from toric code decoders

- Computer Science
- 2019

An efficient decoder of the color code in two and three dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation is introduced.

FLAG FAULT-TOLERANT ERROR CORRECTION WITH ARBITRARY DISTANCE CODES

- Computer Science
- 2017

In this disclosure, example fault-tolerant EC protocols are disclosed that use flag circuits, which signal when errors resulting from υ faults have weight greater than υ.

Almost-linear time decoding algorithm for topological codes

- Computer ScienceQuantum
- 2021

This work designs a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and Erasure and has a worst case complexity of O(nα(n), where n is the number of physical qubits and α is the inverse of Ackermann's function, which is very slowly growing.

Flag fault-tolerant error correction, measurement, and quantum computation for cyclic Calderbank-Shor-Steane codes

- Computer SciencePhysical Review A
- 2020

Some critical properties of CSS codes constructed from classical cyclic codes are proved that enable the construction of a flag fault-tolerant error correction scheme and a family of circuits, requiring only four ancilla qubits, applicable to cyclic CSS codes of distance 3.

Logical performance of 9 qubit compass codes in ion traps with crosstalk errors

- PhysicsQuantum Science and Technology
- 2020

We simulate four quantum error correcting codes under error models inspired by realistic noise sources in near-term ion trap quantum computers: $T_2$ dephasing, gate overrotation, and crosstalk. We…

2D Compass Codes

- Computer SciencePhysical Review X
- 2019

This work examines threshold behavior in this broad class of local codes by trading locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information, and analyzes these codes with asymmetric and spatially inhomogeneous Pauli noise in the code capacity and phenomenological models.

Advantages of versatile neural-network decoding for topological codes

- Computer SciencePhysical Review A
- 2019

Using neural networks simplifies the design of decoders and does not require prior knowledge of the underlying noise, and neural decoder threshold for the two-dimensional color code is very close to the toric code threshold.

Fault-tolerant magic state preparation with flag qubits

- Computer ScienceQuantum
- 2019

This paper presents a scheme to fault-tolerantly and directly prepare magic states using flag qubits and shows that the overhead can be improved by several orders of magnitude.

Fault-tolerant preparation of approximate GKP states

- Computer ScienceNew Journal of Physics
- 2019

This work provides rigorous definitions of fault-tolerance and introduces a fault-Tolerant phase estimation protocol for preparing Gottesman-Kitaev-Preskill states and derives analytic expressions which describe the leading order effects of the non-linear dispersive shift and Kerr non- linearity.