How is a state of a qubit measured physically on a quantum computer?
It depends on technology used for a qubit implementation.
For example, here is an explanation provided by IBM Q team:
We must perform the qubit measurements in a way that does not destroy the qubit quantum state. One method is to weakly couple each qubit to a microwave resonator whose resonance characteristics depend on the state of the qubit. Once the qubit operations are completed in your score, you can measure the qubits by sending a microwave tone to their resonators and analyzing the signal it reflects back. The phase and amplitude of this reflected signal will be different depending on the qubit state. These signals in the resonator are boosted via a chain of amplifiers inside of our dilution refrigerator, including a quantum-limited amplifier at 15 mK, and a high-electron mobility transistor amplifier at 4 K.
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