Distance Multiplication and Geometry of Quantum Codes in Product Manifolds

We present constructions of quantum error-correcting codes that can be embedded in submanifolds of the product of manifolds, characterizing the logical operations resulting from the homological structure of the base manifolds. We prove a distance multiplication theorem for the product codes, as well as estimates on locality and the weight of stabilizer checks. We then identify different submanifolds/subcomplexes in the product manifold to create quantum error-correcting codes and find conditions to perform code switching between them using lattice surgery. We use intersection pairing to establish bounds on logical operators in terms of systoles and cosystoles of the base codes. Finally, we also use relative homology to establish balanced distance properties for products of codes with boundaries. Throughout the paper, we provide multiple examples of quantum codes which guide our intuition and highlight their potential for near-term implementation in quantum hardware.