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Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes
Kenza Guenda, Somphong Jitman, T. Aaron Gulliver
TL;DR
EAQECC constructions need a way to determine how much pre-shared entanglement a classical code requires. This paper links that amount to the code’s hull, uses the link to design flexible constructions, and obtains MDS, maximal-entanglement, and asymptotically good families under stated conditions.
Problem
The number of shared entangled states required to construct an EAQECC from a classical code is generally difficult to determine.
Method
The paper relates Euclidean and Hermitian entanglement requirements to classical-code hulls and uses this relationship to construct EAQECCs from suitable classical codes.
Results
The constructions include MDS maximal-entanglement EAQECCs, an asymptotically good family in odd characteristic, and parameter-flexible codes with desirable entanglement amounts.
Takeaways & Limitations
Hull-based entanglement accounting supports EAQECC designs with controllable shared entanglement and good error performance.
Takeaways & Limitations
One construction assumes q = 2 and c is odd.
Abstract
from arXiv · showhide
Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.
1 Introduction
EAQECCs extend classical-to-quantum code constructions by allowing arbitrary classical codes through pre-shared entanglement. The paper relates entanglement requirements to classical-code structure and develops constructions with flexible parameters, including LCD-based and asymptotically good families.
- EAQECCs use pre-shared entanglement to construct quantum codes from arbitrary classical codes, without requiring self-orthogonality.
- The paper relates the number of required maximally shared qubits to the hull of the underlying classical code.
- The proposed methods provide flexibility in choosing EAQECC parameters, including MDS or near-MDS codes with few shared entangled states.
- The paper constructs EAQECCs from Reed-Solomon and generalized Reed-Solomon codes and develops maximal-entanglement codes from LCD codes.
- An asymptotically good family of EAQECCs is obtained in the odd-characteristic case.
2 Preliminaries
The preliminaries define classical linear codes, duality, cyclic and GRS codes, and the EAQECC construction framework. They specify how parity-check matrices determine entanglement consumption and quantum-code parameters.
- An [n, k, d]q linear code is a k-dimensional subspace of Fq^n with minimum Hamming distance d.
- An MDS code satisfies d = n − k + 1, attaining the classical Singleton bound.
- A cyclic code is invariant under coordinate shifts and is generated by a monic polynomial dividing x^n − 1.
- A GRS code evaluates scaled polynomials at distinct field elements and is an MDS code with parameters [n, k, n − k + 1]ℓ.
- An [[n, k, d; c]]q EAQECC encodes k logical qudits into n physical qudits using c maximally entangled states.
- Given parity-check matrices H1 and H2, the construction requires c = rank(H1H2^t) shared states and yields dimension k1 + k2 − n + c.
- In the Hermitian construction, c = rank(HH†), and the resulting code has parameters [[n, 2k − n + c, d; c]]q.
- Maximal-entanglement EAQECCs satisfy c = n − k, while codes attaining the EAQECC Singleton bound are called MDS EAQECCs.
3 The Number of Maximally Entangled States
The paper shows that the entanglement required by Euclidean and Hermitian EAQECC constructions is determined by the corresponding hull dimension. This yields explicit EAQECC parameters and preserves MDS status when the classical code is MDS.
- 3.1 The Euclidean Case: For a classical code C with parity-check matrix H, rank(HH^t) is independent of H and is determined by the Euclidean hull Hull(C) = C ∩ C⊥.
- 3.1 The Euclidean Case: The Euclidean construction yields [[n, k − dim(Hull(C)), d; n − k − dim(Hull(C))]]q EAQECCs.
- 3.1 The Euclidean Case: When C is MDS, the two Euclidean EAQECCs obtained from C and C⊥ are also MDS.
- 3.2 The Hermitian Case: For codes over Fq2, rank(HH†) is independent of H and is determined by the Hermitian hull Hullh(C) = C ∩ C⊥h.
- 3.2 The Hermitian Case: The Hermitian construction yields [[n, k − dim(Hullh(C)), d; n − k − dim(Hullh(C))]]q2 EAQECCs.
- 3.2 The Hermitian Case: When C is MDS, the two Hermitian EAQECCs obtained from C and its Hermitian dual are also MDS.
4 The New Constructions
The paper develops Euclidean and Hermitian constructions that control entanglement consumption while preserving or improving classical-code distance, including MDS EAQECC families derived from dual-containing codes and Hermitian hull dimensions.
- 4.1 The Euclidean Case: For q > 3, Euclidean dual-containing [n, k, d]_q codes yield EAQECCs with c adjustable from 0 to ℓ and distance d′ satisfying d ≤ d′ ≤ d + c.Here ℓ = dim(C) − dim(C⊥).
- 4.1 The Euclidean Case: Reed–Solomon codes provide an example family for these Euclidean constructions, producing codes with parameters [[q + c − 1, 2k + 1 − q, d′ ≥ n − k + 1; c]]_q for c ≤ ℓ.The construction applies when the relevant Reed–Solomon code is dual containing.
- 4.2 The Hermitian Case: For q > 2, Hermitian dual-containing [n, k, d]_{q^2} codes yield [[n + c, 2k − n, d′; c]]_q EAQECCs for 0 ≤ c ≤ ℓ, with d ≤ d′ ≤ d + c.A second Hermitian construction yields [[n + 1, 2k − n − 1 + c, d′; c]]_q with d′ ∈ {d, d + 1}.
- 4.2 The Hermitian Case: Hermitian dual-containing GRS codes produce MDS EAQECCs, including [[n + 1, 2k − n, n − k + 2; 1]]_q and [[n + 1, 1, k + 1; 2k − n − 1]]_q families.These constructions use an [n + 1, k, n − k + 2]_{q^2} MDS code whose Hermitian hull dimension is n − k.
- 4.3 MDS EAQECCs from the Hermitian Hulls of GRS Codes: Hermitian hull dimensions of GRS codes enable further MDS families with one, two, or three logical qudits in the stated ranges of k.The resulting entanglement parameters are n − 2k + 1, n − 2k + 2, and n − 2k + 3, respectively.
5 EAQECCs from LCD codes
LCD codes yield maximal-entanglement EAQECCs, including MDS families, and broader constructions transform arbitrary classical codes into EAQECCs with controlled parameters. The section also establishes asymptotically good EAQECC families over fields of odd characteristic.
- LCD constructions: An LCD code is characterized by a nonsingular HH^t for a parity-check matrix H, enabling constructions with maximal entanglement.LCD codes satisfy C ∩ C⊥ = {0}; their duals are also LCD, and the largest entanglement occurs with LCD codes.
- LCD constructions: An [n, k, d]q LCD code produces maximal-entanglement EAQECCs [[n, k, d, n − k]]q and [[n, n − k, d⊥, k]]q.The second code uses the minimum distance d⊥ of the classical dual.
- MDS families: MDS maximal-entanglement EAQECCs with parameters [[q + 1, k, q − k + 2, q + 1 − k]]q exist for all permitted k, including all k for even q and odd k for odd q.These families are constructed from cyclic LCD codes generated by self-reciprocal polynomials.
- General constructions: An [n, k, d]q classical code yields EAQECCs with (N, c) equal to (2n − k, 2n − 2k), (3n − 2k, 3n − 3k), (4n − 3k, 4n − 4k), or (5n − 4k, 5n − 5k), depending on q and s.The corresponding cases are q even with s = 2, q ≡ 1 mod 4 with s = 3, q ≡ 3 mod 4 with s = 4, and arbitrary q with s = 5; d ≤ d′ ≤ sd − 1.
- Rate properties: These LCD-derived EAQECCs have positive net rate and rate larger than 1/2 under stated classical-code rate conditions, including k/n > 2/3 for even q and k/n > 4/5 when q ≡ 3 mod 4.The conditions are sufficient for the rate properties established in the corollary.
- Asymptotically good families: For q = l^2 with l an odd prime, a family of EAQECCs with parameters [[n_j, k_j, d_j; c_j]]q exists that is asymptotically good.The construction uses q-ary expansions of transitive code families and the q ≡ 1 mod 4 case of the general construction.