Implementation of standard quantum error correction codes for solid-state qubits

In quantum error-correcting code (QECC), many quantum operations and measurements are necessary to correct errors in logical qubits. In the stabilizer formalism, which is widely used in QECC, generators G_i \(i=1,2,..\) consist of multiples of Pauli matrices and perform encoding, decoding and measurement. In order to maintain encoding states, the stabilizer Hamiltonian H_{rm stab}=-sum_i G_i is suitable because its ground state corresponds to the code space. On the other hand, Hamiltonians of most solid-state qubits have two-body interactions and show their own dynamics. In addition solid-state qubits are fixed on substrate and qubit-qubit operation is restricted in their neighborhood. The main purpose of this paper is to show how to directly generate the stabilizer Hamiltonian H_{rm stab} from conventional two-body Hamiltonians with Ising interaction and XY interaction by applying a pulse control method such as an NMR technique. We show that generation times of H_{rm stab} for nine-qubit code, five-qubit code and Steane code are estimated to be less than 300 ns when typical experimental data of superconducting qubits are used, and sufficient pulse control is assumed. We also show how to prepare encoded states from an initial state |0....0>. In addition, we discuss an appropriate arrangement of two- or three-dimensional arrayed qubits.