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