Yifan Hong

Code switching is a powerful technique in quantum error correction that allows one to leverage the complementary strengths of different codes to achieve fault-tolerant universal quantum computation. However, existing code-switching protocols which encapsulate recent generalized lattice surgery approaches often either require many rounds of measurements to ensure fault-tolerance or suffer from low code rates. We present a single-shot, universal protocol that uses code-switching between high-rate quantum codes to perform fault-tolerant quantum computation. To our best knowledge, our work contains the first universal fault-tolerant quantum computation protocol that achieves what we term single-shot universality that is characterized by (i) single-shot error correction, (ii) single-shot state preparation, as well as (iii) logical gates and logical measurements with constant depth circuits. We achieve this by showing how to perform single-shot code switching between high-rate homological product codes by developing a generalization of Bombin's dimensional jump for color codes and Hillmann et al.'s single-shot lattice surgery for higher-dimensional topological codes. We introduce a vastly simpler recipe to construct 3D homological product codes with transversal CCZ gates that grants immense flexibility in the choice of expander graphs and local codes, allowing us to expand the search space for codes with good parameters and interesting logical gates. Our work opens an alternative path towards universal fault-tolerant quantum computation with low space-time overhead by circumventing the need for magic state distillation.

The homological product is a general-purpose recipe that forges new quantum codes from arbitrary classical or quantum input codes, often providing enhanced error-correcting properties. When the input codes are classical linear codes, it is also known as the hypergraph product. We investigate structured homological product codes that admit logical operations arising from permutation symmetries in their input codes. We present a broad theoretical framework that characterizes the logical operations resulting from these underlying automorphisms. In general, these logical operations can be performed by a combination of physical qubit permutations and a subsystem circuit. In special cases related to symmetries of the input Tanner graphs, logical operations can be performed solely through qubit permutations. We further demonstrate that these "automorphism gadgets" can possess inherent fault-tolerant properties such as effective distance preservation, assuming physical permutations are free. Finally, we survey the literature of classical linear codes with rich automorphism structures and show how various classical code families fit into our framework. Complementary to other fault-tolerant gadgets for homological product codes, our results further advance the search for practical fault tolerance beyond topological codes in platforms capable of long-range connectivity.