Symplectic cone code[1]

Description

Qubit stabilizer code obtained by attaching a CSS ancilla code to an input stabilizer code represented as a symplectic complex, with the output logical operators determined by an embedded column complex [1]. Gluing maps extend the ancilla \(X\)-type checks onto the input qubits and the input checks onto the ancilla qubits. A defect map then adds \(Z\)-type support on the ancilla’s own qubits to the ancilla \(X\)-type checks so that all extended checks commute. The construction generalizes the mapping cone framework of quantum code embedding from CSS codes to arbitrary stabilizer codes [1].

The generalization extends fault-tolerant logical measurement, geometrically local embedding, and weight reduction to codes and logical operators that admit no CSS structure [1]. Formally, a symplectic cone is a height-two cone whose middle level is the input-code symplectic complex \(D\). Its top and bottom levels are the cochain complex and the chain complex of the ancilla code \(A\), respectively. Its stabilizer map is block lower triangular, with the level maps on the diagonal, the gluing maps below the diagonal, and the defect map in the corner [1; Eq. (10)]. The gluing maps form a chain map compatible with the defect map, and this compatibility is what makes the cone a symplectic complex, i.e., keeps the extended checks commuting [1]. If \(H_1(A)=0\), the logical space \(H_1(C)\) is naturally isomorphic to that of the embedded column complex \(H^0(A)\to H_1(D)\to H_0(A)\) [1; Thm. 2]. This column-complex homology need not equal the logical space of the input code \(D\). The Layer code and weight-reduction applications choose the column complex to preserve the input logical space. For measurement of one logical operator \(\ell^{\star}\), it instead gives \(H_1(C)\cong[\ell^{\star}]^{\perp}/[\ell^{\star}]\), so the deformed code has one fewer logical qubit [1; Thm. 32]. The ancilla is CSS in isolation, but once glued its \(X\)-type checks are generically dressed with \(Z\)-type support by the defect map. The output code is therefore generally non-CSS even when the input code is CSS [1].