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Four electron spin qubits in quantum dots are studied by means of an exchangeinteraction Hamiltonian. The time-independent Schr\"odinger equation is exactlyanalytically solved for the symmetric case, that is equal qubit frequencies andequal couplings. The spin system can be used as two data-qubits and twoancillas, thereby realizing a $X$ and $Z$ stabilizing circuit exploited inerror correction and surface codes. The unitary evolution of correspondingstable states is calculated and consequences for error syndromes and fidelitiesare assessed. It is also shown that applying the evolution operator for aprescribed duration and subsequent measurement of the ancillas lead to a`verifiable' two-qubit phase gate.

Four electron spin qubits with exchange interaction