Quantum Lego Expansion Pack: Enumerators from Tensor Networks

TL;DR

This paper introduces a tensor network method for efficiently computing quantum weight enumerator polynomials, enabling faster analysis of quantum codes, decoding, and error rate estimation, especially for complex or large codes.

Contribution

It presents the first tensor network approach for quantum weight enumerators, including novel enumerators and systematic code decomposition methods, improving efficiency over existing algorithms.

Findings

Efficient computation of quantum weight enumerators for various codes.

Exact analysis of complex quantum codes under noise models.

Faster decoding and error rate estimation for large or degenerate codes.

Abstract

We provide the first tensor network method for computing quantum weight enumerator polynomials in the most general form. If a quantum code has a known tensor network construction of its encoding map, our method is far more efficient, and in some cases exponentially faster than the existing approach. As a corollary, it produces decoders and an algorithm that computes the code distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can be substantially faster compared to the current methods. We also introduce novel weight enumerators and their applications. In particular, we show that these enumerators can be used to compute logical error rates exactly and thus construct (optimal) decoders for any i.i.d. single qubit or qudit error channels. The enumerators also provide a more efficient method for computing non-stabilizerness in quantum many-body states. As the power for these speedups rely on a Quantum Lego decomposition of quantum codes, we further provide systematic methods for decomposing quantum codes and graph states into a modular construction for which our technique applies. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the 2d Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force.