  Recently, Bravyi et al. [1] proposed a set of small quantum Bivariate Bicycle
(BB) codes that achieve a similar circuit-level error rate to the surface code
but with an improved encoding rate. In this work, we generalise a novel
parity-check circuit design principle that we call morphing circuits (first
introduced in [2]) and apply this methodology to BB codes. Our construction
generates a new family of BB codes -- including a new $[[144,12,12]]$ "gross"
code -- whose parity check circuits require a qubit connectivity of degree five
instead of six. Intriguingly, each parity check circuit requires only 6 rounds
of CNOT gates -- one fewer than in Ref. [1] -- even though our new codes have
weight-9 stabilisers. We also show how to perform logical input/output circuits
to an ancillary rotated surface code using morphing circuits, all within a
biplanar layout. The new codes perform at least as well as those of Ref. [1]
under uniform circuit-level noise when decoded using BP-OSD. Finally, we
develop a general framework for designing morphing circuits and present a
sufficient condition for its applicability to two-block group algebra codes.
  [1] S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J.
Yoder, Nature 627, 778 (2024).
  [2] C. Gidney and C. Jones, New circuits and an open source decoder for the
color code (2023), arXiv:2312.08813.
