A New Family of Permutation-Invariant Quantum Codes that Correct Qubit Errors and Deletions

Core Concepts

A new family of permutation-invariant quantum codes is constructed that can correct any t qubit errors and 2t deletion errors, with shorter code lengths compared to previous constructions.

Abstract

The authors construct a new family of permutation-invariant quantum codes, defined by the parameters g, m, δ, and ϵ. These codes encode one logical qubit into n = 2gm + δ + 1 physical qubits and can correct any t qubit errors and 2t deletion errors.
The key highlights are:

The new code family contains some previously known permutation-invariant codes as special cases, and in many cases, the new codes are shorter than the best known explicit permutation-invariant codes for Pauli errors and deletions.

The authors provide a detailed analysis of the error correction conditions for permutation-invariant codes, extending previous results from the single-error case to the general t-error case.

As a separate result, the authors generalize the conditions for permutation-invariant codes to correct t Pauli errors from the previously known results for t = 1 to any number of errors.

The authors show that the new code family includes a new (4, 2, 2) optimal single-deletion-correcting code.

The construction and analysis rely on several combinatorial identities proved in the appendix.

Stats

The code length n is given by n = 2gm + δ + 1, where g, m, δ are integer parameters.
The codes can correct any t qubit errors and 2t deletion errors, where t is a nonnegative integer.

Quotes

"A new family of permutation-invariant quantum codes is constructed that can correct any t qubit errors and 2t deletion errors, with shorter code lengths compared to previous constructions."
"The new code family contains some previously known permutation-invariant codes as special cases, and in many cases, the new codes are shorter than the best known explicit permutation-invariant codes for Pauli errors and deletions."

Key Insights Distilled From

A family of permutationally invariant quantum codes

by Arda Aydin,M... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2310.05358.pdf

Deeper Inquiries

How can the new permutation-invariant code family be generalized to encode more than one logical qubit?

The new permutation-invariant code family, denoted as Qg,m,δ,ϵ, can be generalized to encode more than one logical qubit by extending the construction to include additional basis states that correspond to the extra logical qubits. Each additional logical qubit would introduce a new set of coefficients α and β, similar to those defined for the single logical qubit case. The coefficients would need to satisfy the same conditions (C1)-(C4) as outlined in the construction of the code for a single logical qubit. By expanding the basis states and coefficients to accommodate multiple logical qubits, the code family can be scaled to encode and protect a larger quantum information space.

What are the limitations of permutation-invariant codes in terms of their ability to correct errors compared to more general quantum codes?

Permutation-invariant codes have certain limitations in terms of their error correction capabilities compared to more general quantum codes. One limitation is related to the types of errors that permutation-invariant codes can effectively correct. Permutation-invariant codes are specifically designed to correct errors that preserve the symmetry of the encoded quantum states under permutations. This means that they are particularly effective at correcting errors like deletions and certain types of Pauli errors that respect the symmetry of the code states. However, permutation-invariant codes may not be as efficient in correcting more general types of errors, such as coherent errors or correlated errors, which do not necessarily preserve the permutation symmetry of the code states.
Another limitation of permutation-invariant codes is their scalability and efficiency in encoding multiple logical qubits. While permutation-invariant codes are well-suited for encoding a single logical qubit due to their symmetry properties, extending them to encode multiple logical qubits can be challenging. The construction and optimization of permutation-invariant codes for multiple logical qubits may require more complex and computationally intensive techniques compared to more general quantum codes designed for multi-qubit systems.

Can the techniques used in this work be applied to construct efficient quantum codes for other types of errors, such as coherent errors or correlated errors?

The techniques used in the construction of the new family of permutation-invariant codes can potentially be adapted and applied to construct efficient quantum codes for other types of errors, such as coherent errors or correlated errors. However, the applicability and effectiveness of these techniques may vary depending on the specific characteristics of the errors and the desired error correction properties.
For coherent errors, which involve systematic deviations in the quantum state due to external factors, techniques from quantum error correction theory, such as encoding in larger Hilbert spaces and implementing error-correcting codes with transversal gates, can be explored. By incorporating symmetry considerations and optimization strategies similar to those used in permutation-invariant codes, efficient quantum codes for correcting coherent errors can be developed.
Similarly, for correlated errors that exhibit dependencies between different qubits or error channels, techniques for encoding and error correction that take into account these correlations can be investigated. By adapting the principles of symmetry and error correction conditions from permutation-invariant codes, tailored quantum codes can be designed to effectively mitigate the impact of correlated errors on quantum information processing tasks.