## 2 Quantum Uncertainty: The Problem and the Solution
### The Problem  
The first step to implement a classical or quantum computing algorithm using quantum devices is the initial preparation of the quantum state associated with the quantum electron device \(\psi_{in}(\vec{r},0)\). In quantum computing, the initial state is directly linked to a combination of two states, \(\psi_{1}(\vec{r},0)\) and \(\psi_{0}(\vec{r},0)\), respectively associated with the logical values '1' and '0', whereas in classical computing, the link between logical information and initial quantum state is not direct. Typically, a quantum device for classical computation is connected to the external world through the contacts (also know as reservoirs) that determine the electron wave function depending on temperature and doping conditions. The input logical information is then linked to a value of an observable \(I\), not directly to the quantum state. We have used the symbol \(I\) to remind readers that hereafter, we will consider the electrical current as the physical magnitude where information is encoded. The second step is the manipulation of the initial quantum state through the quantum electron device (also known as gate in the literature). Typically, such manipulation, whether in classical or quantum computing is done through a (usually unitary) operator \(\hat{U}(t,0)\). In quantum computing, the final state \(\psi_{out}(\vec{r},t)=\hat{U}(t,0)\psi_{in}(\vec{r},0)\) is directly linked to the output logical information, while in classical computing the output logical information is linked to an observable \(I\) associated with this final state through a measurement process that we describe below. See Figure 1a where different gates of an RTD which are connected to exemplify a classical computation gate connected through output values of the electrical current (or voltage), while the connection among different gates of the quantum computing device is done through the wave function itself as depicted in Figure 1b. The third and last step, both in classical or quantum computing algorithms, is the measurement step. To get the final observable value \(I\) in classical computing, the quantum electron device has to be measured through a (non-unitary) process. Such non-unitary process is depicted as an ammeter in Figure 1. The evolution from the final state to the measured state, \(\psi_{out}(\vec{r},t)\rightarrow\psi_{I}(\vec{r},t)\), is called collapse or reduction of the wave function. The subindex \(I\) here refers to the measured state of the current which corresponds to an eigenvalue \(I\) of the eigenfunction \(\psi_{I}(\vec{r},t)\) associated with the operator \(\hat{I}\). In quantum computing, the final wave function \(\psi_{out}(\vec{r},t)\) is not directly measurable in a single shot measurement. Instead, the logical information assigned to this final quantum state has to be indirectly deduced from the measurement of an observable assigned to such final state. See Figure 1b. Notice that the measurement in quantum computing has to be done only once, at the end of the gate, because each measurement collapses the wave function, destroying the required superpositions of different states in the quantum computing algorithms. The process of measurement involves a quantum uncertainty which is a consequence of the fact that each time a quantum measurement is done, the wave function collapses into an eigenstate of the operator \(\hat{I}\) associated with the measuring apparatus. The observable output \(I_{out}\) is a random value equal to the eigenvalue associated with the mentioned eigenstate. In general, and this is true for the measurement of the electrical current, the final state before measurement \(\psi_{out}(\vec{r},t)\) is not an eigenstate of the current \(\psi_{I}(\vec{r},t)\
eq\psi_{out}(\vec{r},t)\). In fact, the final state can be written as a superposition of many different current eigenstates. Thus, each time we repeat an experiment to obtain information about the output current, we get different values.