Efficient Tensor Network Representations of Quantum Error Correcting Codes

Abstract

The design space of quantum error-correcting codes is vast and evaluating the performance of large codes is computationally difficult. Although weight enumerator polynomials (WEPs) are valuable for characterizing codes, their exact calculation scales exponentially with the size of the code using brute-force enumeration. The Quantum LEGO (QL) framework addresses this by representing quantum error-correcting codes as tensor networks (TNs). Calculating the WEP of the code is then done through TN contraction, offering super-polynomial speedups in some cases. However, realizing this advantage depends on resolving the \#P-hard problem of finding a good contraction schedule and choosing a good TN representation since they are not unique.This thesis addresses these issues by optimizing contraction schedules for stabilizer code TNs and evaluating the performance of different TNs for compass codes. First, we demonstrate that the intermediate tensors of the QL networks are incredibly sparse, which invalidates the dense tensor cost functions of standard optimizers. To resolve this, we introduce the Sparse Stabilizer Tensor (SST) cost function, a polynomial-time algorithm based on the rank of the parity check matrix that computes the exact sparse contraction cost. Integrating the SST cost function in the optimizers significantly improves the contraction schedules found and reduces the variance.Second, we evaluate the impact of choosing different TNs to represent the compass code family. We introduce a novel layout for compass codes, called the Qubit-wise layout. This layout outperforms all known TN constructions for compass codes, including the universal representations. This novel layout allows for the efficient calculation of WEPs, where a distance-15 WEP calculation takes approximately 15 minutes to compute. Ultimately, this research demonstrates that while universal representations are broadly applicable, designing highly tailored and code-specific layouts is critical for maximizing the computational advantages of the QL framework. This research, along with the open-source software tools we developed, provides an efficient pipeline for the design and analysis of QEC codes.