## 2 Quantum Uncertainty: The Problem and the Solution
### The Problem
This randomness in the output values can be quantified through the probability distribution \(P(I)=|\langle\psi_{I}(\vec{r})|\psi_{out}(\vec{r},t)\rangle|^{2}\) given by [PERSON]'s law. From a quantum engineering point of view, this quantum uncertainty (seen as noise in the current) is inconvenient for efficiently processing logical (either classical or quantum) information. In classical computations, the uncertainty on the electrical current can be eliminated by using the ensemble value of the current \(\langle I\rangle\) computed from a large number of identical experiments, each one giving \(I^{i}\), with the subindex \(i\) identifying the experiment. The ensemble value is defined as \(\langle I\rangle=(\sum_{i=1}^{N_{exp}}I^{i})/N_{exp}\), where \(N_{exp}\rightarrow\infty\) is the number of experiments. In principle, this ensemble value would require repeating the same experiment for a large set of \(N_{exp}\) identical quantum electron devices. In practice, by invoking ergodic arguments, the repetition of the experiment is substituted by measuring at different times in the same quantum electron device. Thus, instead of defining the signal of the output logical value as the instantaneous current \(I^{1}\) (which has noise) one defines it as the DC value of the electrical current \(\langle I\rangle\) computed during a large time interval (which has no noise). This solution is efficient for reducing the noise, but it requires a large measuring time. (In our simulation example with an RTD with a device active region length of 10 nm, the injection time of 0.05 ps and the velocity of electrons as \(10^{4}\) m/s, the time after which we get the non-fluctuating value of the current is around 50 ps. In any case, the measuring time is again a parameter that depends on many factors, like injection time, velocity of electrons, electron density, level of tolerable uncertainty: etc., and that can be enlarged or reduced as desired by manipulating these parameters.) The quantum uncertainty described above represents also a problem for quantum computing. In fact, although the logical output information in quantum computing is encoded in the final wave function \(\psi_{out}(\vec{r},t)\) (not in an observable \(I^{i}\)), the quantum state \(\psi_{out}(\vec{r},t)\) is not itself an observable  
Figure 1: **(a)** Schematic of classical computing exemplified with RTD where only the active device is governed by unitary quantum evolutions (enclosed in the cyan color dashed line), while the contacts and the cable leads to quantum decoherence which provides a fixed value of the current obtained in the measuring apparatus (shown at the right end). **(b)** Schematic of quantum computing exemplified with an MZI where the quantum wholeness require that a coherent unitary evolution appears in all the gates (enclosed in the cyan color dashed line). Only at the end, when the wave function is measured, decoherence can be accepted. (i.e., it cannot be measured in a single shot measurement). Thus, the quantum state of the system needs to be deduced from the expectation value \(\langle I\rangle\). Again, the measurement process of such observable \(\langle I\rangle\) has the same inconveniences mentioned above for classical computing, due to the quantum uncertainty. We notice that in a quantum computing algorithm, with many interconnected quantum gates, the measurement of the observable is done only at the last gate. In fact, trying to measure at an intermediate gate would be understood as a type of decoherent phenomena that would dramatically perturb the unitary evolution required in typical quantum algorithms. In Figure 1 we encircle the regions of the connected gates where the dynamics of electrons are supposed to be governed by unitary quantum evolutions. From Figure 1, one can understand why decoherence is a serious problem for quantum computing, but not for classical computing.