Transversal dimension jump for product qLDPC codes

We introduce transversal dimension jump, a code-switching protocol for lifted product (LP) quantum low-density parity-check (qLDPC) codes across different chain-complex dimensions, enabling universal fault-tolerant quantum computation with low overhead. The construction leverages the product structure of LP codes to implement one-way transversal CNOTs between a 3D code and its 2D component codes, enabling teleportation-based switching with geometrically nonlocal gates. Combined with constant-depth CCZ gates in 3D LP codes and low-overhead transversal Clifford gates in 2D LP codes, this yields universal, high-rate quantum logical computation with high thresholds and low space-time costs. Beyond asymptotic schemes, we identify explicit 3D-2D LP code pairs supporting cup-product CCZ gates, including bivariate tricycle-bicycle families such as the $[[81,3,5]]$-$[[54,2,6]]$ pair, where the 3D tricycle codes admit depth-2 CCZ, weight-6 stabilizers, and pseudo-thresholds $\gtrsim 0.4\%$. As a byproduct, we show that the 3D codes enable highly efficient magic-state preparation: a single round of stabilizer measurements followed by depth-2 CCZ and postselection produces states with error $<10^{-9}$ and success probability $\sim 35\%$. Our results establish a native integration of qLDPC codes with complementary transversal gates-covering nearly all practically relevant families known so far-and open a broad design space for scalable, low-overhead universal quantum computation.