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arXiv:2603.19597

Stabilizer Formalism for EAQECCs with Noise ebits

Ruihu Li, Guanmin Guo, Yang Liu, Hao Song

quant-phSubmitted Mar 20, 2026• Announced Mar 23, 2026• 0 citations
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Abstract

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.

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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 $q$-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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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.

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Stabilizer Formalism: The Algebraic Core of Quantum Error Correction — E8 Intelligence Research

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Entanglement-assisted codeword-stabilized quantum codes with imperfect ebits

Byungkyu Ahn, Jeonghwan Shin, Jun Heo

In quantum communication systems, quantum error-correcting codes (QECCs) are known to exhibit improved performance with the use of error-free entanglement bits (ebits). In practical situations, ebits inevitably suffer from errors, and as a result, the error-correcting capability of the code is diminished. Previous studies have proposed two different schemes as a solution. One study uses only one QECC to correct errors on the receiver side (i.e., Bob) and sender side (i.e., Alice). The other uses different QECCs on each side. In this paper, we present a method to correct errors on both sides by using single nonadditive entanglement-assisted codeword stabilized quantum error-correcting code (EACWS QECC). We use the property that the number of effective error patterns decreases as much as the number of ebits. This property results in a greater number of logical codewords using the same number of physical qubits.

arXiv:1604.06191•ICT Express•Jun 1, 2016•0 citations
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.
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.
The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes assumes that the ebits of the receiver are error free. In practical situations, errors on these ebits are unavoidable, which diminishes the error-correcting ability of these codes. We consider two different versions of this problem. We first show that any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code that can correct errors on the qubits of both sender and receiver. These EAQEC codes are equivalent to standard stabilizer codes, and hence the decoding techniques of standard stabilizer codes can be applied. Several EAQEC codes of this type are found to be optimal. In a second scheme, the receiver uses a standard stabilizer code to protect the ebits, which we call a ``combination code.'' The performances of different quantum codes are compared in terms of the channel fidelity over the depolarizing channel. We give a formula for the channel fidelity over the depolarizing channel (or any Pauli error channel), and show that it can be efficiently approximated by a Monte Carlo calculation. Finally, we discuss the tradeoff between performing extra entanglement distillation and applying an EAQEC code with imperfect ebits.
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.
FINDING: Quantum error correction (QEC) is the mathematical scaffold for fault-tolerant quantum computation, with the stabilizer formalism as its core algebraic structure. | MATH: Stabilizer codes are defined by an abelian subgroup S \mathcal{S} S of the Pauli group Pn \mathcal{P}_n Pn​; codewords are simultaneous +1+1+1 eigenstates of all S∈S S \in \mathcal{S} S∈S. The code space dimension is 2n−k 2^{n-k} 2n−k where k=n−rank(S) k = n - \text{rank}(\mathcal{S}) k=n−rank(S). Distance d d d corrects t=⌊(d−1)/2⌋ t = \lfloor (d-1)/2 \rfloor t=⌊(d−1)/2⌋ errors. Key bound: k≤n−log2∣S∣ k \le n - \text{log}_2 |\mathcal{S}| k≤n−log2​∣S∣. The Knill–Laflamme condition: ⟨ψi∣Ea†Eb∣ψj⟩=δijCab \langle \psi_i | E_a^\dagger E_b | \psi_j \rangle = \delta_{ij} C_{ab} ⟨ψi​∣Ea†​Eb​∣ψj​⟩=δij​Cab​, where Ea E_a Ea​ are error operators. | CONNECTION: The stabilizer group S \mathcal{S} S is a **binary linear code** over F22n \mathbb{F}_2^{2n} F22n​ — a **lattice structure** in symplectic space. The symplectic inner product ⟨(a∣b),(a′∣b′)⟩=a⋅b′+a′⋅b(mod2) \langle (a|b), (a'|b') \rangle = a \cdot b' + a' \cdot b \pmod{2} ⟨(a∣b),(a′∣b′)⟩=a⋅b′+a′⋅b(mod2) defines a **self-d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
In quantum communication systems, quantum error-correcting codes (QECCs) are known to exhibit improved performance with the use of error-free entanglement bits (ebits). In practical situations, ebits inevitably suffer from errors, and as a result, the error-correcting capability of the code is diminished. Previous studies have proposed two different schemes as a solution. One study uses only one QECC to correct errors on the receiver side (i.e., Bob) and sender side (i.e., Alice). The other uses different QECCs on each side. In this paper, we present a method to correct errors on both sides by using single nonadditive entanglement-assisted codeword stabilized quantum error-correcting code (EACWS QECC). We use the property that the number of effective error patterns decreases as much as the number of ebits. This property results in a greater number of logical codewords using the same number of physical qubits.