A Novel Approach to State-to-State Transformation in Quantum Computing

This article presents a new approach to the problem of transforming one quantum state into another. It is shown that an r-qubit superposition |x can be obtained from another r-qubit superposition |y, by using only (2r−1) rotations, each presented by one controlled rotation gate. The quantum superpositions with real amplitudes are considered. The traditional two-stage approach Uy−1Ux:|x→|0⊕r→|y requires twice as many rotations. Here, both transformations to the conventual basis state, Ux: |x→ |0⊕r and Uy: |y→ |0⊕r, use (2r−1) rotations each on two binary planes, and many of these rotations require additional sets of CNOTs to be represented as 1- or 2-qubit-controlled gates. The proposed method is based on the concept of the discrete signal-induced heap transform (DsiHT) which is unitary and generated by a vector and a set of angular equations with given parameters. The quantum analog of this transform is described. The main characteristic of the DsiHT is the path of processing the data. It is shown that there exist such fast paths that allow for effective computing of the DsiHT, which leads to the simple quantum circuits for state preparation and transformation. Examples of such paths are given and quantum circuits for preparation and transformation of 2-, 3-, and 4-qubits are described in detail. CNOT gates are not used, but only controlled gates of elementary rotations around the y-axis. It is shown that the transformation and, in particular, only rotation gates with control qubits are required for initialization of 2-, 3-, and 4-qubits. The quantum circuits are simple and have a recursive form, which makes them easy to implement for arbitrary r-qubit superposition, with r≥2. This approach significantly reduces the complexity of quantum state transformations, paving the way for more efficient quantum algorithms and practical implementations on near-term quantum devices.