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General entanglement-assisted quantum error-correcting codes

Min-Hsiu Hsieh, Igor Devetak, Todd Brun

arXiv:0708.2142v1quant-ph

TL;DR

Standard quantum codes require dual-containing classical codes, limiting the use of some efficient classical codes. This paper combines entanglement-assisted and operator quantum error correction in a unified formalism, presents construction approaches, and concludes that the scheme includes advantages of both extensions.

  • Problem

    Standard quantum-code construction requires dual-containing classical codes, a substantial barrier to using efficient Turbo and LDPC codes in quantum information theory.

    Source: §I · p. 1

  • Method

    The paper unifies entanglement-assisted and operator quantum error correction, with constructions from existing EAQECCs and directly from classical linear codes.

    Source: §I · p. 1, §V · p. 7

  • Results

    The authors present a unified scheme they describe as the most general quantum error-correction theory then known, combining entanglement-assisted and operator correction.

    Source: §I · p. 2, §V · p. 7

  • Takeaways & Limitations

    The paper gives several EAOQECC construction examples and states that classical linear codes can be used to construct them directly.

    Source: §V · p. 7

Abstract

from arXiv · show

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement-assisted quantum error correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.

I. INTRODUCTION

The stabilizer connection links quantum codes to classical codes, but standard quantum-code construction requires a dual-containing constraint that limits the use of efficient classical codes. This paper unifies entanglement assistance and operator quantum error correction in one formalism.

I. INTRODUCTION

  • The CSS construction derived quantum codes from classical binary codes, and the stabilizer formalism later generalized the connection to classical symplectic codes.The symplectic codes are closely related to linear codes over GF(4).Source: §I · p. 1
  • Standard quantum codes require dual-containing classical codes, a substantial barrier to using efficient Turbo and LDPC codes in quantum information theory.The passage notes that constructing dual-containing versions of these codes has slowed progress toward their quantum versions.Source: §I · p. 1
  • EAQECCs can use any classical quaternary code, while OQECCs add passive correction but do not let classical-code performance predict quantum-code performance.EAQECC performance, including minimum distance and net rate, is determined by the classical code's performance.Source: §I · p. 1
  • The paper combines entanglement-assisted and operator quantum error correction into one formalism, with conventional quantum error correction as a zero-entanglement special case.Standard QECCs are also a special case of OQECCs when the protected subsystem is the entire system.Source: §I · p. 1
  • The paper presents the framework, discusses relationships among the code families, and gives examples trading entanglement assistance against passive error correction.The stated structure covers the theory, examples, and concluding discussion.Source: §I · p. 2

II. REVIEW OF EAQECCS AND OQECCS

The review presents stabilizer codes through commuting Pauli operators, then describes how entanglement assistance and gauge subsystems extend that framework. Their code parameters and correction conditions are specified through subgroup structure.

II. REVIEW OF EAQECCS AND OQECCS

  • A conventional code space is the common +1 eigenspace of an abelian Pauli subgroup that excludes −I, compactly specified by generators.The generators play a role analogous to a classical code's parity-check matrix.Source: §II · p. 2
  • A conventional stabilizer code corrects an error set when each pairwise error product either lies outside the stabilizer's centralizer or belongs to the stabilizer.The centralizer is denoted Z(S).Source: §II · p. 2
  • An EAQECC encodes k = n − s − c logical qubits in n transmitted qubits, using c shared ebits and s ancillas; its net rate is (k − c)/n.Bob measures the transmitted qubits together with his halves of the shared ebits to correct errors and decode.Source: §II · p. 2
  • An OQECC's gauge subgroup identifies states that differ only by gauge operations as carrying the same information, adding passive error-correction capacity.The code parameters satisfy s + k + r = n, with r gauge qubits.Source: §II · p. 2

A. The canonical code

The canonical EAOQECC combines ancillas, shared entanglement, and gauge qubits in a single construction. It corrects errors using a reduced syndrome while treating gauge-only state changes as equivalent, with a tradeoff between encoded rate and error avoidance.

A. The canonical code

  • The construction appends s ancillas, c shared maximally entangled pairs, and r gauge qubits to the k-qubit input, with s + k + r + c = n.States differing only in the gauge state encode the same quantum information.Source: A. The canonical code · p. 2
  • The encoding and decoding correct the specified error set for any fixed functions α and β mapping the reduced syndrome to logical-qubit Pauli actions.The canonical proposition states the functions' domain and codomain as (Z2)^s × (Z2)^c × (Z2)^c to (Z2)^k.Source: A. The canonical code · pp. 2–3
  • Only the reduced syndrome (a, a1, a2) is needed for correction; errors indexed by b or by (c, d) are passive because they do not alter encoded information.The (c, d) errors act only through the subsystem's gauge operations.Source: A. The canonical code · p. 3
  • The canonical construction assigns s generators to ancillas, c symplectic pairs to entanglement, and r symplectic pairs to gauge qubits.These subgroups define the code space and the operations allowed on gauge qubits.Source: A. The canonical code · p. 3
  • The canonical code fixes a 2^k+r-dimensional code space, treating states that differ only in their gauge state as carrying the same information.Its parameters are [[n, k; r, c]], with s + k + r = n.Source: A. The canonical code · p. 3
  • The code trades encoded rate against error avoidance: reducing rate can improve passive error-avoiding ability.The code fixes a 2^k+r-dimensional space in which gauge-state variants represent the same information.Source: A. The canonical code · p. 3
  • The canonical EAOQECC corrects an error set when every pairwise error product belongs to the isotropic-and-gauge subgroup times operators outside the entanglement subgroup's centralizer.This is the stated correctability condition for the canonical code.Source: A. The canonical code · p. 3

B. The general case

The general construction maps subgroups with the required commutation structure to canonical form, then defines an EAOQECC that combines entanglement assistance with passive error avoidance. The theorem gives a condition under which the resulting code corrects an error set.

The general case

  • A subgroup V of size 2^m can be generated by operators with canonical commutation relations, with parameters p and q satisfying p + 2q = m.The lemma supplies the generator structure used to organize the subgroups in the construction.Source: B. The general case · p. 4
  • A commutation-preserving one-to-one map between V and S is implemented by a unitary conjugation, up to a phase for each generator.This result links the general subgroup construction to the canonical subgroups.Source: B. The general case · p. 4
  • The [[n, k; r, c]] EAOQECC combines entanglement assistance with passive error-avoiding ability through the subgroups (SI, SS, SG).The general case links these subgroups to canonical form using a unitary that preserves commutation relations.Source: B. The general case · p. 4
  • Theorem III.5 defines an [[n, k; r, c]] EAOQECC from (SI, SE, SG) and states that it corrects E when E1†E2 lies in ⟨SI, SG⟩S(Gn − Z(⟨SI, SE⟩)) for all E1, E2 ∈ E.The encoding and decoding pair is obtained using a unitary that preserves the commutation relations.Source: B. The general case · p. 4

C. Properties of EAOQECCs

The section describes how EAOQECCs can trade transmission rate and active correction against passive error avoidance, while retaining the code-distance ordering stated in Theorem III.6. It also gives a distance-based correction criterion and compares criteria across code families.

Properties of EAOQECCs

  • An [[n, k, d; r, c]] EAOQECC has distance d when it corrects error sets whose operators each have weight t satisfying 2t + 1 ≤ d.The weight of a Pauli operator is the number of nonidentity single-qubit operators.Source: C. Properties of EAOQECCs · p. 4
  • The gauge subgroup may contain both isotropic and symplectic subgroups; adding symplectic partners to its isotropic generators can support construction from EAQECCs.The passage relates this approach to moving stabilizer generators and their symplectic partners into the gauge subgroup.Source: C. Properties of EAOQECCs · p. 4
  • d1 ≤ d2 ≤ d3 when an [[n, k+r, d1; 0, c]] code is transformed into an [[n, k, d2; r, c]] code and then an [[n, k, d3; 0, c]] code.Adding gauge qubits expands the correctable error set while sacrificing part of the transmission rate; removing half of each gauge symplectic pair and adding the remaining generators to SI gives the final code.Source: C. Properties of EAOQECCs · p. 4
  • The section lists distinct error-correcting criteria for conventional stabilizer QECCs, EAQECCs, OQECCs, and EAOQECCs.It presents the EAOQECC criterion alongside the criteria for the other three code types.Source: C. Properties of EAOQECCs · p. 5

A. EAOQECC from EAQECC

The example starts with an [[8,1,3;1]] EAQECC that encodes one qubit into eight physical qubits with one ebit and corrects arbitrary single-qubit errors. Recombining its stabilizers yields an EAOQECC with two gauge qubits while retaining distance 3.

EAOQECC from EAQECC

  • The original [[8,1,3;1]] EAQECC encodes one qubit into eight physical qubits with one ebit and corrects arbitrary single-qubit errors.The code is degenerate.Source: A. EAOQECC from EAQECC · p. 5
  • Recombining the first four stabilizers into two isotropic generators and two gauge generators with their symplectic partners yields an [[8,1,3;2,1]] EAOQECC.The resulting generators are listed in Table II.Source: A. EAOQECC from EAQECC · p. 5, Table II · p. 5
  • The construction yields an [[8,1,3;2,1]] EAOQECC with two qubits of gauge symmetry, and Table II labels the parameters c=2 and r=1.Source: A. EAOQECC from EAQECC · p. 5, Table II · p. 5

B. EAOQECCs from classical BCH codes

The section derives a construction of EAOQECCs from classical binary codes and illustrates it with a [63,39,9] BCH code. Moving symplectic pairs into the gauge subgroup yields codes with positive net rate, while the distance remains lower bounded by 7 in this example.

B. EAOQECCs from classical BCH codes

  • Theorem IV.1 constructs an [[n, 2k −n + c; c]] EAQECC from a binary parity-check matrix, with c = rank(HHT) ebits required.The proof uses the CSS construction and determines the number of symplectic pairs from the rank of HHT.Source: B. EAOQECCs from classical BCH codes · p. 5
  • Replacing GF(2^6) elements with binary representations yields a binary [63,39,9] BCH code with a 24×63 parity-check matrix; its first 18 rows give a [63,45,7] dual-containing BCH code.Source: B. EAOQECCs from classical BCH codes · p. 6
  • The [63,39,9] code gives a [[63,21,9;6]] EAQECC, with six symplectic pairs from the last six rows of H2 transferred one at a time to the gauge subgroup.The resulting EAOQECC parameters are listed in Table III.Source: B. EAOQECCs from classical BCH codes · p. 6, Table III · p. 6
  • For this BCH example, any first symplectic-pair removal leaves the distance lower bounded by 7, and the resulting codes have positive net rate.The paper notes that these positive-net-rate EAOQECCs could be used as catalytic codes.Source: B. EAOQECCs from classical BCH codes · p. 6

C. EAOQECCs from classical quaternary codes

A classical quaternary code provides an EAQECC starting point for an EAOQECC construction. A random search over symplectic pairs reduces entanglement by one ebit while retaining correction of all one-qubit errors and adding passive correction.

C. EAOQECCs from classical quaternary codes

V. CONCLUSION

The conclusion presents EAOQEC as combining entanglement assistance with operator quantum error correction. It reports construction routes from existing quantum codes and directly from classical linear codes.

V. CONCLUSION

  • EAOQEC combines the advantages of entanglement-assisted and operator quantum error correction in one general scheme.Source: §V · p. 7
  • Methods for constructing OQECCs from standard QECCs can be applied to construct EAOQECCs from EAQECCs.Source: §V · p. 7
  • The paper also presents direct construction of EAOQECCs from classical linear codes.Source: §V · p. 7
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