Long-range-enhanced Dynamic Automorphism Codes

Abstract

Topological quantum error-correcting codes such as the surface code and color code encode only O(1) logical qubits. Long-range-enhanced constructions improve this scaling by gluing topological code patches, producing qLDPC families such as [[n,Θ(n),Θ(log n)]] quantum rainbow codes [35] and long-range enhanced surface codes [22]. Separately, dynamic au- tomorphism codes [15, 20] implement logical gates through sequences of low-weight Pauli measurements rather than through a fixed static codespace, providing capabilities of new error correction protocols. This thesis combines these ideas by constructing long-range-enhanced dynamic auto- morphism codes on glued toric and glued color codes. The glued codes are described as pushouts of CSS chain complexes, with seam relations modifying the global logical data while preserving the local topological structure. In this setting, the relevant automorphism groups take the form Aut(X) = US ⋊ (GL(S) ×O(W,qW)), where S records the condensed seam subspace and W= S⊥/S carries the residual quadratic form. We prove that these automorphisms are dynamically implementable in the first nontriv- ial cases: the glued toric code with dim(S) = 1 and the glued color code with dim(S) = 2. For TC ∪S1 TC and CC ∪S1×S1 CC, we find that we can implement generators of their automorphism group via Pauli measurement sequences. In the glued color-code case, the generators yield logical Clifford operations including S and CZ gates, which builds upon results of [15].