# Estimating Interception Density in the BB84 Protocol: A Study with a Noisy Quantum Simulator
## 3 Background
### Quantum Computer and Qubits  
A quantum computer employs quantum mechanical phenomena to enable the calculation of some problems exponentially faster than classical computers. The fundamental unit of such a computer is the qubit, which differs from a classical bit in that it exists in a superposition of two basis states The state of superposition implies that, upon measurement, the qubit will invariably assume one of two possible values, either 0 or 1, according to a probability law that is dependent on the relative weights of these two states [28].  
For the sake of simplicity, from here on in the paper, it is assumed that polarization encoding is used to represent classical bits (0 s and 1 s), hence employing the polarization states of photons. In this case, each qubit can exist in one of two bases: a + (\"rectilinear\") basis, with orthogonal states \(\ket{\uparrow}\) and \(\ket{\rightarrow}\), and an \(X\) (\"diagonal\") basis, with orthogonal states \(\ket{\
earrow}\) and \(\ket{\
warrow}\). An example of mapping between classical bits and qubits is \(\ket{0}=\ket{\uparrow}\) and \(\ket{1}=\ket{\rightarrow}\) for the \(+\) basis or \(\ket{0}=\ket{\
earrow}\) and \(\ket{1}=\ket{\
warrow}\) for the \(X\) basis, as summarized in Table 1.  
A quantum computer employs quantum logic gates to manipulate the states of qubits. Of particular interest are the Pauli-X gate, denoted as X, which functions in a manner analogous to a NOT gate, and the Hadamard gate, denoted as H, which induces a rotation that changes the basis, e.g., from \(+\) to \(X\), and vice versa, since its functional matrix is hermetian and unitary. Both gates operate on individual qubits, allowing for transitions between different states and bases as desired [29]. The quantum indeterminacy principle precludes the possibility of distinguishing between the four states with absolute reliability, as they are not all orthogonal to each other. Rather, they can only be distinguished in pairs. Indeed, once the basis has been set, the measurement is only possible between the orthogonal states.  
If you tried to measure a qubit with the wrong basis (i.e., different from the one used for encoding), the resulting reading would be random, either 0 or 1, with equal probability \(\frac{1}{2}\). Conversely, if the measurement was conducted with the same basis, the output would always be the original state (of course, in the absence of external noise sources). In brief, one fundamental property of quantum mechanics is that the act of measurement itself alters the state of the observed object. Related to this property is the no-cloning theorem, which states that an arbitrary quantum state cannot be duplicated perfectly [30]. In other words, this theorem implies that creating an independent and identical copy of an unknown qubit is impossible.  
These two properties form the basis of QKD protocols, as they prevent an adversary, referred to as Eve, from intercepting a stream of qubits, modifying them, and forwarding them without being visible.  
One reason why quantum computation is so important today is the discovery, by [PERSON], of a uniform family of quantum networks that solve the factoring problem. [PERSON]'s factoring algorithm has demonstrated that the security of many public cryptographic systems is compromised once a quantum computer is available. On a quantum computer, [PERSON]'s algorithm is capable of factoring an integer, \(N\), in polynomial time. This means that the time taken is polynomial in \(logN\), which is significantly faster than the most efficient known classical algorithm [31]. [PERSON]'s algorithm has indicated the necessity of the development of new protocols that can withstand the mainstream implementation of quantum computers. One potential solution is the use of QKD protocols as a secure alternative to classical public-key distribution schemes.