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arXiv:2605.23119

Construction of EAQECCs with imperfect ebits

Guanmin Guo, Ruihu Li

quant-phcs.ITSubmitted May 22, 2026• Announced May 25, 2026• 0 citations
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Abstract

We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general $q$-ary case, where $q$ is a prime power. By leveraging the structure of the generalized Pauli group over $\mathbb{F}_q$ and symplectic geometry over $\mathbb{F}_q^{2n}$, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over $\mathbb{F}_q$ and additive codes over $\mathbb{F}_q^{2n}$ are derived. We further construct several families of $q$-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.

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We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general qqq-ary case, where qqq is a prime power. By leveraging the structure of the generalized Pauli group over Fq\mathbb{F}_qFq​ and symplectic geometry over Fq2n\mathbb{F}_q^{2n}Fq2n​, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over Fq\mathbb{F}_qFq​ and additive codes over Fq2n\mathbb{F}_q^{2n}Fq2n​ are derived. We further construct several families of qqq-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.
We introduce a stabilizer formalism for EAQECCs with noise ebits, using special subgroups of product groups of two Pauli groups. This formalism includes the two coding schemes,given by Lai and Brun (C.Y. Lai and T. A. Brun, PHYSICAL REVIEW A 86, 032319 (2012)), for EAQECCs with imperfect ebits as special cases. Then two equivalent formalisms of the formalism are derived in nomenclature of sympletic geometry and additive codes. We apply this theory to construct some EAQECCs with noise ebits, and analyze their performance.
Entanglement-assisted quantum error correction codes (EAQECCs) play an important role in quantum communications with noise. Such a scheme can use arbitrary classical linear code to transmit qubits over noisy quantum channels by consuming some ebits between the sender (Alice) and the receiver (Bob). It is usually assumed that the preshared ebits of Bob are error free. However, noise on these ebits is unavoidable in many cases. In this work, we evaluate the performance of EAQECCs with noisy ebits over asymmetric quantum channels and quantum channels with memory by computing the exact entanglement fidelity of several EAQECCs. We consider asymmetric errors in both qubits and ebits and show that the performance of EAQECCs in entanglement fidelity gets improved for qubits and ebits over asymmetric channels. In quantum memory channels, we compute the entanglement fidelity of several EAQECCs over Markovian quantum memory channels and show that the performance of EAQECCs is lowered down by the channel memory. Furthermore, we show that the performance of EAQECCs is diverse when the error probabilities of qubits and ebits are different. In both asymmetric and memory quantum channels, we show that the performance of EAQECCs is improved largely when the error probability of ebits is reasonably smaller than that of qubits.
In this paper, we introduce a unified framework to construct entanglement-assisted quantum error-correcting codes, including additive and nonadditive codes, based on the codeword stabilized framework on subsystems. The codeword stabilized (CWS) framework is a scheme to construct quantum error-correcting codes (QECCs) including both additive and nonadditive codes, and gives a method to construct a QECC from a classical error-correcting code in standard form. Entangled pairs of qubits (ebits) can be used to improve capacity of quantum error correction. In addition, it gives a method to overcome the dual-containing constraint. Operator quantum error correction (OQEC) gives a general framework to construct quantum error-correcting codes. We construct OQEC codes with ebits based on the CWS framework. This new scheme, entanglement-assisted operator codeword stabilized (EAOCWS) quantum codes, is the most general framework we know of to construct both additive and nonadditive codes from classical error-correcting codes. We describe the formalism of our scheme, demonstrate the construction with examples, and give several EAOCWS codes.
The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.
The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing preshared entanglement between the sender and receiver. In this paper, a new family of EAQECCs is constructed from negacyclic codes of length n=q2+1an=\frac{q^2+1}{a}n=aq2+1​, where qqq is an odd prime power, a=m2+12a=\frac{m^2+1}{2}a=2m2+1​ and mmm is an odd integer. Some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes are obtained in the sense that their parameters are not covered by the previously known ones.
Entanglement-assisted concatenated quantum codes (EACQCs), constructed by concatenating two quantum codes, are proposed. These EACQCs show significant advantages over standard concatenated quantum codes (CQCs). First, we prove that, unlike standard CQCs, EACQCs can beat the nondegenerate Hamming bound for entanglement-assisted quantum error-correction codes (EAQECCs). Second, we construct families of EACQCs with parameters better than the best-known standard quantum error-correction codes (QECCs) and EAQECCs. Moreover, these EACQCs require very few Einstein-Podolsky-Rosen (EPR) pairs to begin with. Finally, it is shown that EACQCs make entanglement-assisted quantum communication possible, even if the ebits are noisy. Furthermore, EACQCs can outperform CQCs in entanglement fidelity over depolarizing channels if the ebits are less noisy than the qubits. We show that the error-probability threshold of EACQCs is larger than that of CQCs when the error rate of ebits is sufficiently lower than that of qubits. Specifically, we derive a high threshold of 47% when the error probability of the preshared entanglement is 1% to that of qubits.