Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians

Implementing robust quantum error correction (QEC) is imperative for harnessing the promise of quantum technologies. We introduce a framework that takes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>y</mml:mi></mml:math> classical code and explicitly constructs the corresponding QEC code. Our framework can be seen to generalize the CSS codes, and goes beyond the stabilizer formalism (Fig. 1). A concrete advantage is that the desirable properties of a classical code are automatically incorporated in the design of the resulting quantum code. We reify the theory by various illustrations some of which outperform the best previous constructions. We then introduce a local quantum spin-chain Hamiltonian whose ground space we analytically completely characterize. We utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem.