A
quantum computational method is provided. The
quantum computational method includes providing a physical
quantum system (100) comprising constituents (50). The quantum computational method includes performing an encoded quantum computation on the physical
quantum system. Performing the encoded quantum computation includes preparing at least a portion of the physical
quantum system in an initial
quantum state. Performing the encoded quantum computation includes evolving at least a portion of the physical
quantum system to a final
quantum state. Performing the encoded quantum computation includes measuring at least a portion of the physical quantum
system to provide a read-out. During at least a portion of the encoded quantum computation, the
quantum state of at least a portion of the quantum
system is an encoded quantum state corresponding to a quantum error-
correction code. The quantum error-
correction code is a parity code that encodes logical qubits (10) into code qubits (20). The code qubits include parity qubits, wherein each parity
qubit represents the parity of an associated subset of logical qubits. Each code
qubit is physically implemented in a corresponding subsystem (50, 350) of the physical quantum system, wherein the subsystem comprises one or more constituents. During a first portion of the encoded quantum computation, the code qubits include a first code
qubit duplication set (1350, 1450, 1550, 1651-1653, 1750) associated with a first logical qubit. The first code qubit duplication set includes at least three code qubits, wherein either (a) each code qubit in the first code qubit duplication set is a parity qubit representing the parity of a same first subset of logical qubits that includes the first logical qubit or (b) each code qubit in the first code qubit duplication set is a data qubit representing a quantum state of the first logical qubit. The quantum computational method further includes performing a first sequence of physical quantum operations on the physical quantum system during the first portion of the encoded quantum computation. The first sequence of physical quantum operations acts at least on the subsystems of the physical quantum system corresponding to the code qubits in the first code qubit duplication set. The first sequence of physical quantum operations is an encoded realization, via the parity code, of a first logical quantum operation (501-505, 1310, 1410, 1310, 1510, 1610, 1710, 1810) acting at least on the first logical qubit.