We construct a new family of permutationally invariant codes that correct t Pauli errors for any t≥1 . We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new ((4,2,2)) optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct t Pauli errors from the previously known results for t=1 to any number of errors. For small t , these conditions can be used to construct new examples of codes by computer.We construct a new family of permutationally invariant codes that correct t Pauli errors for any t≥1 . We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new ((4,2,2)) optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct t Pauli errors from the previously known results for t=1 to any number of errors. For small t , these conditions can be used to construct new examples of codes by computer.

arXiv:2310.05358QuantumApr 30, 202418 citations

The equivalence of quantum deletion and insertion errors on permutation-invariant codes

Lewis Bulled, Yingkai Ouyang

Quantum synchronisation errors are a class of errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are $t$ -insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are $(t,s)$ -insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.Quantum synchronisation errors are a class of errors that change the number of qubits in a quantum system. The classical error correction of synchronisation errors has been well-studied, including an insertion-deletion equivalence more than a half-century ago, but little progress has been made towards the quantum counterpart since the birth of quantum error correction. We address the longstanding problem of a quantum insertion-deletion equivalence on permutation-invariant codes, detailing the conditions under which such codes are $t$-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are $(t,s)$-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.

quant-pharXiv:2602.08780npj Quantum InformationFeb 10, 20260 citations

A Family of Permutationally Invariant Quantum Codes

Arda Aydin, Max A. Alekseyev, Alexander Barg

We construct a new family of permutationally in-variant codes that correct $t$ Pauli errors for any $t\geqslant 1$ . We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. This is an extended abstract of the preprint [1].We construct a new family of permutationally in-variant codes that correct$t$Pauli errors for any$t\geqslant 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. This is an extended abstract of the preprint [1].

IEEE International Symposium on Information Theory (ISIT)Jul 7, 20240 citations

A new take on permutation-invariant quantum codes

Yingkai Ouyang

May 13, 20242 citations

A theory of quantum error correction for permutation-invariant codes

Yingkai Ouyang, Gavin K. Brennen

We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.

quant-pharXiv:2602.13638Feb 17, 20260 citations

t

-insertion error-correctable. We extend these conditions to quantum insertion-deletion errors, formulating a more restrictive set of conditions under which permutation-invariant codes are

(t,s)

-insdel error-correctable. For completeness, we extend our approach to qudit errors on qudit permutation-invariant codes. Our work resolves many of the outstanding questions regarding the quantum error correction of synchronisation errors.

We construct a new family of permutationally in-variant codes that correct

t

Pauli errors for any

t\geqslant 1

. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. This is an extended abstract of the preprint [1].