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Quantum polar code[1]

Description

Entanglement-assisted CSS code utilized in a quantum polar coding scheme producing entangled pairs of qubits between sender and receiver. In such a scheme, the amplitude and phase information of a quantum state is handled in complementary fashion [2] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [3] and extended to arbitrary channels [4].

The scheme requires some a-priori quantum side information in the general case, making the associated code entanglement assisted [1]. The requirement of having quantum side information vanishes when the sum of the amplitude channel fidelity and the phase channel fidelity is not greater than 1. It is shown to vanish for the case of degradable noise channels [4]. A more complicated quantum polar-coding scheme that does not require pre-shared entanglement has also been derived [5].

Protection

Protects against Pauli noise and erasures.

Rate

The rate approaches the symmetric coherent information of arbitrary quantum channels [3].

Encoding

Encoding circuits can be viewed as branching-tree tensor networks [6].

Decoding

Quantum successive-cancellation list decoder (SCL-E) for quantum polar codes that do not need entanglement assistance [7].

Fault Tolerance

State preparation of a single logical qubit [8].

Cousins

  • Qubit CSS code— Quantum polar codes are CSS codes used in an entanglement generation scheme that generally requires entanglement assistance. They require assistance only to determine positions to store information which optimally protect against both bit and phase noise. Without this assistance, they are just CSS codes constructed out of polar codes. A variant of quantum polar codes exists that does not require entanglement assistance [5].
  • Polar code— Without entanglement assistance, quantum polar codes are CSS codes constructed out of polar codes.
  • Tensor-network code— Quantum polar encoding circuits can be viewed as branching-tree tensor networks [6].
  • Concatenated GKP code— Concatenation of GKP codes with quantum polar codes achieves a rate against the displacement channel [9].
  • Quantum Reed-Muller (RM) code— There are codes interpolating between quantum RM and quantum polar codes [10].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
EA qubit codeEAQECC Quantum
EA qubit stabilizer codeEAQECC Quantum
Parents
EA qubit stabilizer code
Quantum polar codes are CSS codes used in an entanglement generation scheme that generally requires entanglement assistance. They require assistance only to determine positions to store information which optimally protect against both bit and phase noise. Without this assistance, they are just CSS codes constructed out of polar codes. A variant of quantum polar codes exists that does not require entanglement assistance [5].
Quantum polar code
Children
\([[4,2,2]]\) Four-qubit code
\([[4,2,2]]\) code is a small quantum polar code [11].

References

[1]
J. M. Renes, F. Dupuis, and R. Renner, “Efficient Polar Coding of Quantum Information”, Physical Review Letters 109, (2012) arXiv:1109.3195 DOI
[2]
J. M. Renes and J.-C. Boileau, “Physical underpinnings of privacy”, Physical Review A 78, (2008) arXiv:0803.3096 DOI
[3]
M. M. Wilde and J. M. Renes, “Quantum polar codes for arbitrary channels”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1201.2906 DOI
[4]
M. M. Wilde and S. Guha, “Polar Codes for Degradable Quantum Channels”, IEEE Transactions on Information Theory 59, 4718 (2013) arXiv:1109.5346 DOI
[5]
J. M. Renes, D. Sutter, F. Dupuis, and R. Renner, “Efficient Quantum Polar Codes Requiring No Preshared Entanglement”, IEEE Transactions on Information Theory 61, 6395 (2015) arXiv:1307.1136 DOI
[6]
A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014) arXiv:1312.4578 DOI
[7]
A. Gong and J. M. Renes, “Improved Logical Error Rate via List Decoding of Quantum Polar Codes”, (2023) arXiv:2304.04743
[8]
A. Goswami, M. Mhalla, and V. Savin, “Fault-tolerant preparation of quantum polar codes encoding one logical qubit”, Physical Review A 108, (2023) arXiv:2209.06673 DOI
[9]
M. Subramanian, G. Zheng, and L. Jiang, “Achievable rates for concatenated square Gottesman-Kitaev-Preskill codes”, (2025) arXiv:2505.10499
[10]
K. Hidaka, D. Abdelhadi, and R. Urbanke, “Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes”, (2025) arXiv:2505.22142
[11]
Kyungjoo Noh, Leung code as quantum polar code, 2017.
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Zoo Code ID: quantum_polar

Cite as:
“Quantum polar code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_polar
BibTeX:
@incollection{eczoo_quantum_polar, title={Quantum polar code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_polar} }
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Permanent link:
https://errorcorrectionzoo.org/c/quantum_polar

Cite as:

“Quantum polar code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_polar

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/ea_stabilizer/quantum_polar.yml.