Entangled qubits can increase the capacity of quantum error-correcting codes based on stabilizer codes. In addition, by using entanglement quantum stabilizer codes can be construct from classical linear codes that do not satisfy the dual-containing constraint. We show that it is possible to construct both additive and nonadditive quantum codes using the codeword stabilized quantum code framework. Nonadditive codes may offer improved performance over the more common stabilizer codes. Like other entanglement-assisted codes, the encoding procedure acts only on the qubits on Alice's side, and only these qubits are assumed to pass through the channel. However, errors in the codeword stabilized quantum code framework give rise to effective $Z$ errors on Bob's side. We use this scheme to construct entanglement-assisted nonadditive quantum codes, in particular, $($ (5,16,2;1) $)$ and $($ (7,4,5;4) $)$ codes.Entangled qubits can increase the capacity of quantum error-correcting codes based on stabilizer codes. In addition, by using entanglement quantum stabilizer codes can be construct from classical linear codes that do not satisfy the dual-containing constraint. We show that it is possible to construct both additive and nonadditive quantum codes using the codeword stabilized quantum code framework. Nonadditive codes may offer improved performance over the more common stabilizer codes. Like other entanglement-assisted codes, the encoding procedure acts only on the qubits on Alice's side, and only these qubits are assumed to pass through the channel. However, errors in the codeword stabilized quantum code framework give rise to effective $Z$ errors on Bob's side. We use this scheme to construct entanglement-assisted nonadditive quantum codes, in particular, $($(5,16,2;1)$)$ and $($(7,4,5;4)$)$ codes.

arXiv:1109.3358Physical Review ADec 22, 201125 citations

Family of fast quantum stabilizer codes constructions

Yuanli Cai, Xingxiang Liu

Optical ReviewMar 1, 20100 citations

Quantum stabilizer codes construction from Hermitian self-orthogonal codes over GF(4)

Duc Manh Nguyen, Sunghwan Kim

In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.

Journal of Communications and NetworksJun 1, 201818 citations

A Construction of Optimal Nonbinary Pure Quantum Stabilizer Codes

Sujuan Huang, Zhonghua Sun, Shixin Zhu

International Journal of Theoretical PhysicsApr 1, 20220 citations

A New Description of the Additive Quantum Codes

唐黎明, 刘焕平, 吕学琴

东北数学：英文版Jan 1, 20060 citations

No-Go Theorem on Fault Tolerant Gadgets for Multiple Logical Qubits

Aranya Chakraborty, Daniel Gottesman

Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. Motivated by this goal, we study several classes of fault-tolerant gadget constructions consisting of Clifford gates acting on the physical qubits, including transversal gadgets, code automorphisms, and fold-transversal gadgets. While stabilizer codes encoding a single logical qubit, most notably the [[7,1,3]] Steane code, are known to admit transversal implementations of the full logical Clifford group, no analogous examples are known for codes encoding multiple logical qubits. In this work, we prove a no-go theorem establishing that no stabilizer code admits a fully transversal implementation of the Clifford group on more than one logical qubit. We further strengthen this result by showing that fold-transversal implementations of the full logical Clifford group are impossible for stabilizer codes encoding more than two logical qubits. More generally, we introduce the notion of k-fold transversal gadgets and prove that implementing the full Clifford group on k logical qubits requires at least k-fold transversal gadgets at the physical level. In addition, we analyze code-automorphism based constructions and demonstrate that they also fail to realize the full Clifford group on multiple logical qubits for any stabilizer code. Together, these results place fundamental constraints on fault-tolerant Clifford gadget design and show that stabilizer codes supporting the full logical Clifford group on multiple logical qubits via these architectures do not exist. Since the Clifford group is a core component of universal gate sets, our findings imply that quantum computing with codes encoding multiple logical qubits within a single code block necessarily entails more complex constructions for fault tolerance.Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. Motivated by this goal, we study several classes of fault-tolerant gadget constructions consisting of Clifford gates acting on the physical qubits, including transversal gadgets, code automorphisms, and fold-transversal gadgets. While stabilizer codes encoding a single logical qubit, most notably the [[7,1,3]] Steane code, are known to admit transversal implementations of the full logical Clifford group, no analogous examples are known for codes encoding multiple logical qubits. In this work, we prove a no-go theorem establishing that no stabilizer code admits a fully transversal implementation of the Clifford group on more than one logical qubit. We further strengthen this result by showing that fold-transversal implementations of the full logical Clifford group are impossible for stabilizer codes encoding more than two logical qubits. More generally, we introduce the notion of k-fold transversal gadgets and prove that implementing the full Clifford group on k logical qubits requires at least k-fold transversal gadgets at the physical level. In addition, we analyze code-automorphism based constructions and demonstrate that they also fail to realize the full Clifford group on multiple logical qubits for any stabilizer code. Together, these results place fundamental constraints on fault-tolerant Clifford gadget design and show that stabilizer codes supporting the full logical Clifford group on multiple logical qubits via these architectures do not exist. Since the Clifford group is a core component of universal gate sets, our findings imply that quantum computing with codes encoding multiple logical qubits within a single code block necessarily entails more complex constructions for fault tolerance.

quant-pharXiv:2602.13395PRX QuantumFeb 17, 20260 citations

Z

errors on Bob's side. We use this scheme to construct entanglement-assisted nonadditive quantum codes, in particular,

(

(5,16,2;1)

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and

(

(7,4,5;4)

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codes.