Implementing single-qubit POVMs on a circuit-based quantum computer

We present a deterministic protocol to implement general single-qubit POVMs 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 are positive, and suggest that implementing non-trivial POVMs could be within the reach of current noisy intermediate scale quantum computing devices.