Linear programming bounds for quantum channels acting on quantum error-correcting codes

TL;DR

This paper develops a linear programming framework based on generalized quantum weight enumerators to evaluate and bound the performance of quantum error-correcting codes under arbitrary noise channels, extending beyond traditional bounds.

Contribution

It introduces generalized quantum weight enumerators for any quantum channel and establishes a linear program to bound code parameters, especially under approximate error correction.

Findings

Bound the maximum dimension of permutation-invariant quantum codes.

Prove non-existence of certain codes correcting amplitude damping errors.

Generalize quantum weight enumerators to arbitrary channels.

Abstract

While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds are not optimized to quantify the performance of quantum codes under the effect of arbitrary quantum channels that describe bespoke noise models. Herein, for any Kraus decomposition of any given quantum channel, we introduce corresponding quantum weight enumerators that naturally generalize the Shor-Laflamme quantum weight enumerators. We establish an indirect linear relationship between these generalized quantum weight enumerators by introducing an auxiliary exact weight enumerator that completely quantifies the quantum code's projector, and is independent of the underlying noise process. By additionally working within the framework of approximate quantum error correction, we establish a general framework for constructing a linear program that is infeasible whenever approximate quantum error correcting codes with corresponding parameters do not exist. Our linear programming framework allows us to establish the non-existence of certain quantum codes that approximately correct amplitude damping errors, and obtain non-trivial upper bounds on the maximum dimension of a broad family of permutation-invariant quantum codes.