  It is known that a quantum computer operating on electron-spin qubits with
single-electron Hamiltonians and assisted by single-spin measurements can be
simulated efficiently on a classical computer. We show that the exponential
speed-up of quantum algorithms is restored if single-charge measurements are
added. These enable the construction of a CNOT (controlled NOT) gate for free
fermions, using only beam splitters and spin rotations. The gate is nearly
deterministic if the charge detector counts the number of electrons in a mode,
and fully deterministic if it only measures the parity of that number.
