  We propose a simplified version of the Kitaev's surface code in which error
correction requires only three-qubit parity measurements for Pauli operators
XXX and ZZZ. The new code belongs to the class of subsystem stabilizer codes.
It inherits many favorable properties of the standard surface code such as
encoding of multiple logical qubits on a planar lattice with punctured holes,
efficient decoding by either minimum-weight matching or renormalization group
methods, and high error threshold. The new subsystem surface code (SSC) gives
rise to an exactly solvable Hamiltonian with 3-qubit interactions,
topologically ordered ground state, and a constant energy gap. We construct a
local unitary transformation mapping the SSC Hamiltonian to the one of the
ordinary surface code thus showing that the two Hamiltonians belong to the same
topological class. We describe error correction protocols for the SSC and
determine its error thresholds under several natural error models. In
particular, we show that the SSC has error threshold approximately 0.6% for the
standard circuit-based error model studied in the literature. We also consider
a model in which three-qubit parity operators can be measured directly. We show
that the SSC has error threshold approximately 0.97% in this setting.
