Error correction and decoding for quantum stabilizer codes

In this paper, we study the complexity of Hamiltonians whose groundstate is a stabilizer code. We introduce various notions of $k$-locality of a stabilizer code, inherited from the associated stabilizer group. A choice of generators leads to a Hamiltonian with the code in its groundspace. We establish bounds on the locality of any other Hamiltonian whose groundspace contains such a code, whether or not its Pauli tensor summands commute. Our results provide insight into the cost of creating an energy gap for passive error correction and for adiabatic quantum computing. The results simplify in the cases of XZ-split codes such as Calderbank-Shor-Steane stabilizer codes and topologically-ordered stabilizer codes arising from surface cellulations.

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Pseudocodeword-based Decoding of Quantum Stabilizer Codes

2019 IEEE International Symposium on Information Theory (ISIT) ◽

10.1109/isit.2019.8849833 ◽

2019 ◽

Author(s):

July X. Li ◽

Pascal O. Vontobel

Keyword(s):

Stabilizer Codes ◽

Quantum Stabilizer Codes

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New construction of binary and nonbinary quantum stabilizer codes based on symmetric matrices

International Journal of Modern Physics B ◽

10.1142/s0217979219502746 ◽

2019 ◽

Vol 33 (24) ◽

pp. 1950274 ◽

Cited By ~ 2

Author(s):

Duc Manh Nguyen ◽

Sunghwan Kim

Keyword(s):

Inner Product ◽

Symmetric Matrices ◽

Parity Check ◽

Stabilizer Codes ◽

Construction Methods ◽

Singleton Bound ◽

New Construction ◽

Symplectic Inner Product ◽

Quantum Stabilizer Codes

In this paper, we propose two construction methods for binary and nonbinary quantum stabilizer codes based on symmetric matrices. In the first construction, we use the identity and symmetric matrices to generate parity-check matrices that satisfy the symplectic inner product (SIP) for the construction of quantum stabilizer codes. In the second construction, we modify the first construction to generate parity-check matrices based on the Calderbank–Shor–Stean structure for the construction of quantum stabilizer codes. The binary and nonbinary quantum stabilizer codes whose parameters achieve equality of the quantum singleton bound are investigated with the code lengths ranging from 4 to 12.

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An Explicit Construction of Quantum Stabilizer Codes From Quasi-Cyclic Codes

IEEE Communications Letters ◽

10.1109/lcomm.2020.2974731 ◽

2020 ◽

Vol 24 (5) ◽

pp. 1067-1071

Author(s):

Jingjie Lv ◽

Ruihu Li ◽

Junli Wang

Keyword(s):

Cyclic Codes ◽

Explicit Construction ◽

Stabilizer Codes ◽

Quantum Stabilizer Codes ◽

Quasi Cyclic Codes

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New Constructions of Quantum Stabilizer Codes Based on Difference Sets

Symmetry ◽

10.3390/sym10110655 ◽

2018 ◽

Vol 10 (11) ◽

pp. 655 ◽

Cited By ~ 6

Author(s):

Duc Nguyen ◽

Sunghwan Kim

Keyword(s):

Difference Sets ◽

Combinatorial Design ◽

Inner Product ◽

Code Construction ◽

Stabilizer Codes ◽

Stabilizer Code ◽

The Difference ◽

Symplectic Inner Product ◽

Quantum Stabilizer Code ◽

Quantum Stabilizer Codes

In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is first explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.

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