## 4 Application to Quantum Computing Devices: Mach-Zehnder Interferometer  
We provide now an example of our protocol for reducing the quantum uncertainty in a quantum computing device. Quantum computing algorithms require a suitable set of quantum gates to reproduce logical operations [18; 19; 20]. With respect to most recent implementations, mainly based on superconducting [21] or single-ion qubits [22], solid-state devices are promising candidates because of their scalability and potential to be integrated into classical circuitry. In this section we will study how the quantum uncertainty can be eliminated in a solid-state MZI acting as quantum gate. It is important to notice that any practical implementation of a quantum gate tends to have a non-negligible interaction between the quantum system and the environment (in terms of scattering with photons, background charges, impurities etc.) even when no measurement is designed. This interaction affects the expected unitary evolution of the state of the quantum system and produces loss of the logical information encoded in the state (decoherence). In solid-state MZI proposal, decoherence can be successfully minimized by injected electrons in _edge states_, chiral conductive channels generated in the Integer Quantum Hall regime [23]. Then, a strong enough magnetic field \(B\) is applied perpendicularly to a 2 DEG, so that the band structure is discretized into Landau levels. In proximity to the confining potential of the device, the system eigenstates form chiral channels where an electron propagates coherently for large distances [24]. The chirality of such _edge states_ prevents the electron to be back-reflected by eventual impurities on its path, unless it is scattered to the counterpropagating edge channel by a narrow quantum point contact [25; 26]. Coherent transport  
Figure 3: (**a**) Plot between the relative error, the transmission coefficient and the average current with the Fermi function fixed to unity. The relative error which is an indicator of quantum randomness, goes to zero, when the number of electrons in the device quantified by transmission coefficient and the mean current becomes very large (blue shaded region) while the relative probability of error is greater (in the red region) when the number of particles in the device is very small. In the green region the uncertainty already starts to disappear. (**b**) The same plot but with a constant transmission probability as unity and varying Fermi function also demonstrate the same outcome. of electrons in _edge states_ has been tested in a large variety of semiconductor devices, as Fabry-Perot interferometers [27], Hanbury-Brown-Twiss [28] and Hong-Ou-Mandel interferometers [29], thus validating the Integer Quantum Hall regime as an ideal framework to implement solid-state quantum logic gates [30]. Our MZI is schematically depicted in Figure 4. To simplify the discussion, as done along the whole paper, only one degree of freedom (one qubit) is considered for the unmodified quantum device. The generalization of the present protocol to a realistic quantum computing gate with more qubits, is explained in Appendix A. We consider that a contact or reservoir (as elaborated in Appendix B) is connected to the source contact (S1) in Figure 4a filling only one Landau level and all injected electrons have the same wave function \(|0\rangle\), but displaced in time by the distance \(\tau_{in}v_{x}\), as depicted in Figure 4b. At this point, we have not yet constructed the superposition of states to get our initial qubit. In our device setup, a potential dip or a quantum point contact acts as a half-reflecting beam splitter by randomly scattering the electron in one of the two available edge channels [31; 32; 33] so that the description of the quantum electron in the central region of our MZI is given by the quantum bit \(a|0\rangle+b|1\rangle\). This conclude the first step of the preparation of the quantum state.