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Entanglement-assisted quantum error-correcting codes over arbitrary finite fields
Carlos Galindo, Fernando Hernando, Ryutaroh Matsumoto, Diego Ruano
TL;DR
The paper addresses whether binary-field formulae for the entanglement required by EAQECCs remain valid over arbitrary finite fields. It proves the extensions using symplectic, Hermitian, and Euclidean methods, and adds a finite-field Gilbert-Varshamov bound and code constructions. The resulting formulas and constructions apply across the stated finite-field settings, with practical computation of c available for suitable bases.
Problem
The paper investigates the lack of proofs for formulae used to determine the minimum entanglement required by EAQECCs over arbitrary finite fields.
Method
The paper uses symplectic forms, Hermitian and Euclidean inner products, finite-field basis maps, and geometric decompositions to derive EAQECC formulas and constructions.
Results
The paper proves that the entanglement formulas extend to arbitrary finite fields and supplies a valid finite-field Gilbert-Varshamov bound and EAQECC constructions.
Takeaways & Limitations
The results provide finite-field methods for computing required entanglement and constructing EAQECCs through symplectic, Hermitian, Euclidean, and self-orthogonal-code frameworks.
Takeaways & Limitations
The notation treats q as a prime power, with the Hermitian setting restricted to even prime powers in the geometric-decomposition subsection.
Abstract
from arXiv · showhide
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.
1. Introduction
The paper extends entanglement-assisted quantum error-correction beyond binary codes, proving finite-field formulae and adding existence bounds and constructions valid over arbitrary finite fields.
- 1. Introduction: Entanglement assistance allows EAQECCs to be constructed from any classical linear code, avoiding self-orthogonality restrictions imposed by standard constructions.This broadens the classical-code choices available for quantum code construction.
- 1. Introduction: The paper fills a literature gap by proving formulae for the minimum number c of maximally entangled pairs required by EAQECCs over arbitrary finite fields.The results use symplectic forms and Hermitian or Euclidean inner products.
- 1. Introduction: The paper gives a Gilbert-Varshamov-type formula for EAQECC existence that is valid over any finite field.This extends the previously presented binary-field result to the general finite-field setting.
- 1. Introduction: It also provides existence conditions and parameters for EAQECCs derived from classical self-orthogonal codes over arbitrary finite fields.The constructions include the broader finite-field setting previously considered in the binary case.
- 1. Introduction: The paper organizes these contributions around entanglement formulae, a Gilbert-Varshamov bound, and constructions based on symplectic, Hermitian, or Euclidean duality.These topics are treated in Sections 2–4.
2. EAQECCs over Fq
The paper develops finite-field constructions of EAQECCs and proves that entanglement formulas extend across symplectic, Hermitian, and Euclidean settings. It also gives practical computation procedures and a sufficient Gilbert-Varshamov existence condition.
- 2.1. The symplectic case.: For arbitrary finite fields, the required entanglement in the symplectic case is determined by the rank of HXH_X^T and depends only on the code and its symplectic dual.The construction yields an [[n, k+c, d; c]]_q EAQECC.
- 2.2. The Hermitian case.: In the Hermitian case, an [n, k, d]_{q^2} code produces an [[n, 2k−n+c, d; c]]_q EAQECC with c=rank(HH^*).Here H^* is the qth-power transpose of H.
- 2.3. The Euclidean case.: In the Euclidean CSS construction, two length-n codes with dimensions k1 and k2 yield an [[n, n−k1−k2+c, d; c]]_q EAQECC using the minimum required c pairs.The minimum distance is bounded below by the stated Euclidean-code distance expression.
3. Gilbert-Varshamov-type sufficient condition of existence of entanglement-assisted codes
The paper establishes a Gilbert–Varshamov-type sufficient condition for EAQECC existence over arbitrary finite fields and derives an asymptotic form of that condition.
- The resulting Gilbert–Varshamov-type bound extends a binary-field result to EAQECCs over arbitrary finite fields.
- Theorem 5 gives a sufficient condition for constructing an EAQECC from a code with prescribed dimension, distance, and entanglement-related dimension difference.The condition requires dim_Fq C = n − k, distance at least δ on the specified symplectic complement difference, and dimension difference 2c.
- The proof uses the symplectic group’s transitive action to count spaces and error vectors satisfying the required conditions.The argument defines a family of F_q-linear spaces and compares the relevant bad spaces through symplectic transformations.
- Combining the counting equalities shows that Inequality (3) is sufficient to ensure existence of a code satisfying the theorem’s conditions.
- The section derives an asymptotic version of Theorem 5 using the q-ary entropy function for sufficiently large block length.Theorem 6 assumes R ≤ 1, ε < 1/2, and λ ≤ (1 − R)/2.
4. EAQECCs coming from punctured QECCs
The paper constructs EAQECCs over finite fields by puncturing self-orthogonal codes under symplectic, Hermitian, and Euclidean dualities. These constructions preserve distance relationships while reducing transmitted qudits.
- Overview: Punctured self-orthogonal codes provide EAQECC constructions under symplectic, Hermitian, and Euclidean dualities.The paper presents these as constructions from punctured codes and notes that fewer transmitted qudits yield better performance.
- Symplectic form: For symplectic codes, the constructed entanglement-assisted code has minimum distance at least ds(C⊥s \ C).The constructed distance is ds(S(C⊥s) \ S(C)), which is bounded below by the corresponding unshortened-code distance.
- Symplectic form: If 2c ≤ dH(C⊥s \ {⃗0}) −1, puncturing a self-orthogonal code C with dimension n−k provides an entanglement-assisted code.The code is Fq-linear and satisfies C ⊆ C⊥s.
- Hermitian inner product: Hermitian puncturing and shortening are defined by deleting the final c coordinates and retaining words that extend with zeros, respectively.The h-shortening consists of length n−c words whose zero-padded extensions lie in C.
- Hermitian inner product: Under a Hermitian self-orthogonality assumption, puncturing yields an entanglement-assisted code when c is bounded by the dual code's minimum Hamming distance.The proof uses unchanged dimension after puncturing, shortening dimension reduced by c, and a distance comparison.