Theory of (Co)homological Invariants on Quantum LDPC Codes

Abstract

With recent breakthroughs in the construction of good qLDPC codes and nearly good qLTCs, the study of (co)homological invariants of quantum code complexes, which fundamentally underlie their logical operations, has become evidently important. In this work, we establish a systematic framework for mathematically analyzing these invariants across a broad spectrum of constructions, from HGP codes to sheaf codes, by synthesizing advanced math tools. We generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a long-standing challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, we prove that $\tildeΘ(N)$ independent cup products can be supported on almost good qLDPC codes and qLTCs of length N, opening the possibility of achieving linearly many parallel, nontrivial, constant-depth multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes via graph lifts, we propose a scheme that inductively generates families of both HGP and sheaf codes in an interlaced fashion from a constant-size HGP code. Notably, the induction preserves all (co)homological invariants of the initial code. This provides a general framework for lifting invariants or logical gates from small codes to infinite code families, and enables efficient verification of such features by checking on small instances. Our theory provides a substantive methodology for studying invariants in HGP codes and extends it to sheaf codes. In doing so, we reveal deep and unexpected connections between qLDPC codes and math, thereby laying the groundwork for future advances in quantum coding, fault tolerance, and physics.