Towards addressable fault-tolerant logical computation on lifted-product codes

Quantum error correction (QEC) codes are central to fault-tolerantly process quantum information on quantum computers. Among many candidates of families of QEC codes, the quantum low-density parity-check (qLDPC) codes have captured many recent attentions and a key question towards efficient computation using the qLDPC codes is to design protocols for fault-tolerant addressable logical operations. Furthermore, in near-time applications, the Abelian group algebra codes or the lifted-product codes have been drawing significant attention, due to its low space overhead in memory level and excellent error suppression capability. However, little is known regarding efficient, fault-tolerant logical computation in the lifted product codes. In this work, we provide a systematic framework for performing logical computations on lifted-product codes. Firstly, we characterize explicitly the logical basis set for lifted product codes from the theory of homological algebra, which is essential to realize any logical computation. Secondly, using the basis characterization we demonstrate a family of lifted-product codes that support arbitrarily addressable Pauli-based logical measurement operations with constant-compilation overhead. Thirdly, we demonstrate the spacetime volume cost in performing logical operations for the candidates of lifted-product codes and compare with that from IBM's gross code.