Research

We develop a coding-theoretic framework for stabilizer codes on mixed-register quantum devices, whose quantum locations have non-uniform local dimensions. We characterize the algebraic structure governing mixed-register Pauli commutation and prove no-go theorems for coprime trans-dimensional Clifford entanglement, oscillator–qudit stabilizer codes, and stabilizer codes over pairwise coprime local dimensions.

We also give two constructions for mixed-register stabilizer codes. The first uses a structure theorem for non-Abelian Pauli subgroups to determine the minimum number and dimensions of additional registers needed to resolve noncommutativity. The second combines stabilizer codes over coprime local dimensions through shared registers, preserves the minimum constituent distance and stabilizer sparsity, and yields coding-theoretically optimal mixed-register qLDPC code families from known good qLDPC codes.

We develop a modular probabilistic framework for analyzing the minimum distance of encode-accumulate code families over arbitrary finite fields. The analysis separates three properties of the input encoding: the probability of producing a fixed output weight, the distribution of its nonzero values, and the distribution of its zero and nonzero positions.

We instantiate the framework for repeat-accumulate (RA), repeat-accumulate-accumulate (RAA), and encode-accumulate (EA) codes, whose simple encoders make them relevant to hash-based zk-SNARKs. We derive explicit finite-length distance guarantees for all three families and show that q-ary RA codes attain the same asymptotic minimum-distance lower bounds as their binary counterparts. We also evaluate RA codes in small parameter regimes and implement the three encoders to compare encoding time across fields and parameter choices. A manuscript is in preparation.