Abstract In the previous chapter, we have seen that a photon in an interferometer can be a prototype for a qubit. Might there be any other prototypes for a qubit arising from other particles that we might know? In fact, an electron is another prototype for a qubit. An electron has many measurable properties such as energy, mass, momentum. But, for the purposes of creating a qubit, we want to focus on a property with only two measurable values. An electron has a two-state property which is called spin.
The use of the Bngroup operators is developed for implementing entangling processes for large n-qubits systems (n > 2). By operating with the Bngroup operators on the computational basis of states, entangled states are obtained with properties analogous to those of the 2-qubits Bell states. Tracing any qubit from the 3-qubits Bnentangled states, we get mixed states which are separable with zero concurrence, and correspondingly with only classical correlations. These properties are generalized to larger Bnentangled states (n > 3). The properties of the Bnentangled states are analyzed by the use of Hilbert–Schmidt (HS) decompositions.
We show how to prepare any graph state of up to 12 qubits with (a) the minimum number of controlled-
Z gates and (b) the minimum preparation depth. We assume only one-qubit and controlled-
Z gates. The method exploits the fact that any graph state belongs to an equivalence class under local Clifford operations. We extend up to 12 qubits the classification of graph states according to their entanglement properties, and identify each class using only a reduced set of invariants. For any state, we provide a circuit with both properties (a) and (b), if it does exist, or, if it does not, one circuit with property (a) and one with property (b), including the explicit one-qubit gates needed.