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Long-range enhanced surface code (LRESC)[1]

Description

Code constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code. This family interpolates between surface codes and hypergraph codes since the hypergraph product of two repetition codes yields the planar surface code. The construction uses small \([3,2,2]\) and \([6,2,4]\) LDPC codes concatenated with \([4,1,4]\) and \([2,1,2]\) repetition codes, respectively. An example using a \([5,2,3]\) code is also presented.

Gates

Patch-transversal gates for suitable seed codes [1].

Realizations

Preparation of GHZ state of four logical qubits with beyond break-even fidelity in a \([[25,4,3]]\) LRESC [2].

Cousins

  • La-cross code— La-cross codes yield LRESCs for \(k=2\). La-cross codes have a number of long-range stabilizers that scales linearly with code size, while the number of LRESC long-range stabilizers can be tuned to scale between the square-root of the size and linearly in the size.
  • Low-density parity-check (LDPC) code— LRESCs are constructed constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code.
  • Repetition code— LRESCs are constructed constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code.
  • Concatenated code— LRESCs are constructed constructed using a hypergraph product of two copies of a concatenated LDPC-repetition seed code.
  • Kitaev surface code— LRESCs reduce to planar surface codes when a trivial LDPC code is used in the hypergraph product.

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Fiber-bundle codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Homological product code
Hypergraph product (HGP) codeCSS Generalized homological-product Lattice stabilizer QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Hypergraph product (HGP) code
LRESCs are constructed using a hypergraph product a concatenated LDPC-repetition code with itself.
Long-range enhanced surface code (LRESC)

References

[1]
Y. Hong, M. Marinelli, A. M. Kaufman, and A. Lucas, “Long-range-enhanced surface codes”, Physical Review A 110, (2024) arXiv:2309.11719 DOI
[2]
Y. Hong, E. Durso-Sabina, D. Hayes, and A. Lucas, “Entangling Four Logical Qubits beyond Break-Even in a Nonlocal Code”, Physical Review Letters 133, (2024) arXiv:2406.02666 DOI
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Zoo Code ID: lresc

Cite as:
“Long-range enhanced surface code (LRESC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/lresc
BibTeX:
@incollection{eczoo_lresc, title={Long-range enhanced surface code (LRESC)}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/lresc} }
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Permanent link:
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Cite as:

“Long-range enhanced surface code (LRESC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/lresc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/concatenated/lresc.yml.