New Circuits for Simultaneously Initiating Two Different Quantum Superpositions

This article presents, for the first time, a new approach to building quantum circuits for the initialization of two multi-qubit superpositions, namely, two different superpositions in one circuit, not in two separate circuits. For this, we introduce the concept of the discrete two signal-induced heap transformation (D2siHT). This transformation is generated by two signals, or vectors, which we call generators. The quantum analogue of the D2siHT is described. It allows us to build a quantum circuit to transform two superpositions $|\mathit{x}⟩$ and $|\mathit{y}⟩$ into the first conventual basis states $|000\dots 0⟩$ and $|010\dots 0⟩$, respectively. Therefore, we can build a single quantum circuit to initiate two multi-qubit superpositions $|\mathit{x}⟩$ and $|\mathit{y}⟩$ from the basis states $|000\dots 0⟩$ and $|010\dots 0⟩$, respectively. Examples with quantum circuits for the preparation and transformation of two 2- and 3-qubit superpositions are described in detail. The results of circuit simulation using Qiskit are also presented. The main characteristic of the D2siHT is its path of processing the data of two generators and input qubits. We consider different paths to effectively compute the D2siHT. Such paths can reduce, for instance, the depth of the resulting quantum circuits, which can lead to a reduction in execution times and susceptibility to decoherence and noise. Multi-qubit superpositions are considered with real amplitudes, but the presented approach can be extended to initiate two such superpositions with complex amplitudes, as well.