The quantum efficiency, which characterizes the quality of information gain against information loss, is an important figure of merit for any realistic quantum detectors in the gradual process of collapsing the state being measured. In this work we consider the problem of solid-state charge qubit measurements with a single-electron-transistor (SET). We analyze two models: one corresponds to a strong response SET, and the other is a tunable one in response strength. We find that the response strength would essentially bound the quantum efficiency, making the detector non-quantum-limited. Quantum limited measurements, however, can be achieved in the limits of strong response and asymmetric tunneling. The present study is also associated with appropriate justifications for the measurement and backaction-dephasing rates, which were usually evaluated in controversial methods.Keywords: quantum qubit, quantum measurement, single electron transistor.PACS numbers: 73.63. Kv, 03.65.Ta, 03.65.Yz, In quantum mechanics the Copenghagen's postulate assumes that a quantum measurement would instantaneously collapse the state being measured onto one of the eigenstates of the observable. This is the concept of perfect projective measurement. In practice, however, any realistic quantum detectors cannot realize such type of measurements. Actually, a realistic quantum measurement is a process of gradually collapsing the state being measured. In this context, the quantum efficiency is an important figure of merit for a quantum detector. To be more specific, let us consider the measurement of a twostate (qubit) system. Assuming the qubit is in an idle state which is simply a superposition of the logic basis states, but experiences no rotational operation between them. Further, we focus on a quantum measurement using the so-called quantum non-demolition (QND) detector, which only dephases the quantum coherence defined by the superposition, but does not flip the basis states. This situation coincides with the fact that the measurement operator is commutable with the qubit Hamiltonian, which is one of the major criterions of the QND measurement in general [1,2]. While for more general case the QND measurements of a qubit were discussed in Refs. [3,4], we restrict us in the present work to a simpler case as mentioned above with, however, a particular interest in the quantum efficiency. Now, consider the qubit measurements with a QND detector. During the (gradual) collapse process, one can get the measurement result only after some time until the signal-to-noise ratio reaches unity, owing to the stochastic nature of the elementary events leading to the collapse (such as tunneling or excitations in the detector). This * E-mail: lixinqi@bnu.edu.cn consideration actually defines a measurement time (τ m ).On the other hand, the qubit state being measured would be inevitably dephased because of the detector's backaction, from which we can define a dephasing rate (Γ d ). Then, the quantum efficiency of a realistic detector is defined by η = 1/(2Γ...