Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Dualities and Identities for Maximal-Entanglement Quantum Codes

Ching–Yi Lai, Todd A. Brun, Mark M. Wilde

Published Oct 26, 2010• 1 citation
OpenAlex

Abstract

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglementassisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for non-degenerate EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the GilbertVarshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximalentanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel.

Related papers

1Dualities and Identities for Entanglement-Assisted Quantum Codes

Ching–Yi Lai, Todd A. Brun, Mark M. Wilde

Abstract—The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from ex-changing the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglement-assisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel. Index Terms—quantum dual code, entanglement-assisted quantum error correction, MacWilliams identity, linear programming bound, entanglement-assisted repetition codes, entanglement-assisted accumulator codes I.

Civil War Book Review•Jan 1, 2014•29 citations

Duality in Entanglement-Assisted Quantum Error Correction

Ching–Yi Lai, Todd A. Brun, Mark M. Wilde

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.

arXiv:1302.4150•IEEE Transactions on Information Theory•Feb 8, 2013•79 citations

Entanglement-Assisted Quantum Codes From Algebraic Geometry Codes

Francisco Revson F. Pereira, Ruud Pellikaan, Giuliano G. La Guardia +1 more

Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.

arXiv:1907.06357•IEEE Transactions on Information Theory•Sep 16, 2021•3 citations

Entanglement-assisted quantum error-correcting codes with imperfect ebits

Ching–Yi Lai, Todd A. Brun

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.

arXiv:1204.0302•Physical Review A•Sep 17, 2012•61 citations

Entanglement-assisted quantum error-correcting codes and asymmetric entanglement-assisted quantum error-correcting codes of length 5pt over 𝔽pm

Hai Quang Dinh, Bac Trong Nguyen, Van‐Nui Nguyen +1 more

Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.

International Journal of Modern Physics C•Feb 27, 2026•0 citations

Bounds on concatenated entanglement-assisted quantum error-correcting codes

Nihar Ranjan Dash, Sanjoy Dutta, R. Srikanth +1 more

Code concatenation combines two or more component codes to design larger codes with greater noise resilience. Introducing entanglement assistance to concatenated codes provides a further advantage in terms of improved error rates and beating certain bounds on codes that would otherwise be unbeatable. First, we derive the general expression for the shared entanglement of a concatenated code and show that the number of ebits can depend on the order of concatenating the component entanglement-assisted quantum error-correcting codes (EAQECCs). We further construct families of pairs of EAQECCs such that the number of ebits of the resultant of concatenating the two codes in a given pair is order independent. Second, we derive conditions on code distance under which nonmaximal-entanglement EAQECCs obtained from a classical quaternary Griesmer or Plotkin code saturate the entanglement-assisted (EA) Griesmer or linear EA Plotkin bound, respectively, extending the known result for maximal-entanglement EAQECCs. Furthermore, we present several families of such nonmaximal-entanglement EAQECCs. Third, we derive an EA version of the quantum Griesmer-Rains bound on the number of correctable errors for EAQECCs. Finally, we present families of pairs of EAQECCs such that the violation of the EA Hamming bound by the resultant of concatenating the two codes in a given pair is order dependent.

arXiv:2412.16082•Physical Review A•Nov 10, 2025•1 citation
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglementassisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for non-degenerate EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the GilbertVarshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximalentanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel.
Abstract—The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from ex-changing the original code’s information qubits for its ebits and vice versa. As an introduction to this notion, we show how entanglement-assisted repetition codes and entanglement-assisted accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes. We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We obtain linear programming bounds for EAQEC codes, by exploiting general forms of these dualities and corresponding MacWilliams identities. We establish the Gilbert-Varshamov bound and the Plotkin bound for EAQEC codes, and all of these bounds allow us to formulate a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits. Finally, we provide an upper bound on the block error probability when transmitting maximal-entanglement EAQEC codes through the depolarizing channel. Index Terms—quantum dual code, entanglement-assisted quantum error correction, MacWilliams identity, linear programming bound, entanglement-assisted repetition codes, entanglement-assisted accumulator codes I.
Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.
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.
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.
Code concatenation combines two or more component codes to design larger codes with greater noise resilience. Introducing entanglement assistance to concatenated codes provides a further advantage in terms of improved error rates and beating certain bounds on codes that would otherwise be unbeatable. First, we derive the general expression for the shared entanglement of a concatenated code and show that the number of ebits can depend on the order of concatenating the component entanglement-assisted quantum error-correcting codes (EAQECCs). We further construct families of pairs of EAQECCs such that the number of ebits of the resultant of concatenating the two codes in a given pair is order independent. Second, we derive conditions on code distance under which nonmaximal-entanglement EAQECCs obtained from a classical quaternary Griesmer or Plotkin code saturate the entanglement-assisted (EA) Griesmer or linear EA Plotkin bound, respectively, extending the known result for maximal-entanglement EAQECCs. Furthermore, we present several families of such nonmaximal-entanglement EAQECCs. Third, we derive an EA version of the quantum Griesmer-Rains bound on the number of correctable errors for EAQECCs. Finally, we present families of pairs of EAQECCs such that the violation of the EA Hamming bound by the resultant of concatenating the two codes in a given pair is order dependent.