Pastawski-Yoshida-Harlow-Preskill (HaPPY) code[1]

Also known as a hyperbolic pentagon code (HyPeC). Holographic code constructed out of a network of perfect tensors that tesselates hyperbolic space. Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality and potentially a dF/CFT duality [2]. The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [3].

Encoding is accomplished using a tensor network of five-qubit encoding isometries, which are six-legged perfect tensors (with five legs corresponding to the physical qubits and one for the encoded logical qubit). A \(2n\)-legged perfect tensor is proportional to an isometry for any bipartition of its indices into a set \(A\) and a complementary set \(A^{\perp}\) such that \(|A|\leq|A^{\perp}|\).

To construct the encoding, one first uniformly tiles the hyperbolic AdS/CFT disc using pentagons and hexagons. Then, one places a 6-legged five-qubit encoding tensor at each hexagon and pentagon, contracting legs between neighboring shapes and leaving one leg uncontracted at each pentagon. This construction forms an encoding isometry from the uncontracted legs in the bulk to the uncontracted legs at the boundary.