  We derive a deterministic protocol to implement a general single-qubit POVM
on near-term circuit-based quantum computers. The protocol has a modular
structure, such that an $n$-element POVM is implemented as a sequence of
$(n-1)$ circuit modules. Each module performs a $2$-element POVM. Two
variations of the protocol are suggested, one optimal in terms of number of
ancilla qubits, the other optimal in terms of number of qubit gate operations
and quantum circuit depth. We use the protocol to implement $2$- and
$3$-element POVMs on two publicly available quantum computing devices. The
results we obtain suggest that implementing non-trivial POVMs could be within
the reach of the current noisy quantum computing devices.
