Decoding tensor network codes

This thesis explores the theoretical convergence of condensed matter physics and high-energy physics through the framework of tensor networks and holographic quantum error-correcting codes (QECC). By establishing that the “efficient corner” of Hilbert space in tensor networks corresponds to the “robust codespace” of QECCs, the research illustrates how spacetime geometry emerges from entanglement structure. The study utilizes a “code tensor” formalism to construct stabilizer codes, demonstrating how larger codes, such as a $[[6, 4]]$ stabilizer code, can be built via the concatenation and contraction of simpler tensor components.

A significant portion of the work addresses the challenge of efficient decoding, proposing that Maximum Likelihood Decoding (MLD) can be approximated through tensor network contraction, thereby making optimal decoding computationally tractable for specific code families. Furthermore, the thesis analyzes the AdS/CFT correspondence, linking the Ryu-Takayanagi entropy formula to network connectivity and examining the reconstruction of bulk operators on the boundary. Finally, the research evaluates the resilience of these holographic codes against erasure errors, utilizing perfect tensors to protect bulk logical information from local boundary noise.