# The Encoding and Decoding Complexities of Entanglement-Assisted Quantum Stabilizer Codes

## TL;DR

This paper analyzes the encoding and decoding complexities of entanglement-assisted quantum stabilizer codes, providing practical implementation bounds and discussing the computational hardness of decoding in this context.

## Contribution

It introduces a practical encoding circuit complexity for entanglement-assisted quantum stabilizer codes and extends decoding hardness results to these codes.

## Key findings

- Encoding complexity for $[[n,k;c]]$ codes is $O(n(n-k+c)/ ext{log} n$.
- Special case $c=0$ reduces to known $O(n^2/ ext{log} n)$ complexity.
- Shared entanglement increases encoding complexity but offers benefits.

## Abstract

Quantum error-correcting codes are used to protect quantum information from decoherence. A raw state is mapped, by an encoding circuit, to a codeword so that the most likely quantum errors from a noisy quantum channel can be removed after a decoding process. A good encoding circuit should have some desired features, such as low depth, few gates, and so on. In this paper, we show how to practically implement an encoding circuit of gate complexity $O(n(n-k+c)/\log n)$ for an $[[n,k;c]]$ quantum stabilizer code with the help of $c$ pairs of maximally-entangled states. For the special case of an $[[n,k]]$ stabilizer code with $c=0$, the encoding complexity is $O(n(n-k)/\log n)$, which is previously known to be $O(n^2/\log n)$. For $c>0,$ this suggests that the benefits from shared entanglement come at an additional cost of encoding complexity. Finally we discuss decoding of entanglement-assisted quantum stabilizer codes and extend previously known computational hardness results on decoding quantum stabilizer codes.