From Logical Gates to LDPC Codes: Augmenting Automorphisms for Efficient Compilation

Given an error-correcting code (ECC), determining the logical operations arising from its symmetries is well studied. These symmetries may lift to efficient fault-tolerant gadgets in quantum ECCs when conditions are met, realized through qubit permutation or transversal operations. However, since useful quantum computations often share subroutines that rely on restricted families of transformations, the inverse question--which remains less studied even classically--is often more pragmatic: given a logical action, how can we design a code to implement it efficiently?

In this talk, we investigate co-designs of compute-efficient hypergraph product codes. Instead of searching for logical gadgets from existing symmetries of the code, we explore ways in which symmetries can be adaptively augmented to match computation needs. In doing so, we obtain qLDPC codes whose logical automorphism groups implement key logical subroutines, such as magic state distillation, adders, and CNOT Fan-outs, efficiently and fault-tolerantly, without sacrificing asymptotic thresholds or LDPC properties.

These constructions outline the basis for a Complex-Instruction-Set fault-tolerant quantum computing architecture, in which functional-level primitives are natively supported on co-optimized QECCs. By compiling algorithmic subroutines into physical schedules on reconfigurable neutral-atom hardware, and using circuit-level simulations to obtain realistic logical error rates, we also demonstrate preliminary evidence of end-to-end rescrouces advantages.