NP-hardness of decoding quantum error-correction codes

TL;DR

Decoding quantum error-correction codes is computationally NP-hard, even with degeneracy, indicating no efficient decoding algorithms exist and supporting potential quantum cryptography applications.

Contribution

The paper proves that the general quantum decoding problem is NP-hard regardless of degeneracy, highlighting fundamental computational complexity in quantum error correction.

Findings

Quantum decoding is NP-hard for all QECCs.

Degeneracy does not simplify the decoding complexity.

Supports potential for quantum cryptosystems based on decoding hardness.

Abstract

Though the theory of quantum error correction is intimately related to the classical coding theory, in particular, one can construct quantum error correction codes (QECCs) from classical codes with the dual containing property, this does not necessarily imply that the computational complexity of decoding QECCs is the same as their classical counterparts. Instead, decoding QECCs can be very much different from decoding classical codes due to the degeneracy property. Intuitively, one expect degeneracy would simplify the decoding since two different errors might not and need not be distinguished in order to correct them. However, we show that general quantum decoding problem is NP-hard regardless of the quantum codes being degenerate or non-degenerate. This finding implies that no considerably fast decoding algorithm exists for the general quantum decoding problems, and suggests the existence of a quantum cryptosystem based on the hardness of decoding QECCs.