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Bulletin of the American Physical Society

Parity-Heap Transformations for Simulation of Fermionic Wavefunctions

M.M. Curtis, Nicholas C. Rubin, Eyob Alebachew Sete, William J. Zeng

Published Mar 14, 2017• 0 citations
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The multi-qubit transformation is the important step in quantum computation in many applications. In recent works, this problem is addressed by solving by using the QR decomposition with elementary rotations with one- and two-qubit CNOT gates. Reducing the number of these gates is critical since operations on three or more qubits are difficult to implement on the current quantum computers. The quantum signal induced heap transform (QsiHT) provides an efficient method for state preparation in the realm of quantum image representation by amplitudes. This paper presents a detailed analysis of how the QsiHT, particularly through the strong path, is utilized to construct quantum states to represent images. The algorithm behind this method and the realization of the corresponding quantum circuits is outlined. Practical examples of the QsiHT on 2 and 3-qubit systems are given using the Qiskit Framework for quantum circuit simulation.

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Unitary Operations with Five and More Qubits: Roadmaps and Effective Quantum Circuits

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This article presents the general method of QR decomposition of r-qubit operations, r≥ 3, by means of quantum signal-induced heap transformations (QsiHT). These are quantum analogues of discrete signal-induced heap transformations, which are generated by given signals and paths, or orders, of processing the data. The case of the 5-qubit operations is described in detail, and a recurrent form of calculation of all 5-qubit QsiHTs from the 4-qubit QsiHTs is given. For that, roadmaps and quantum circuits are presented for all 31 5-point QsiHTs that are used in the QR decomposition. New roadmaps, namely the schemes with paths for performing basic operations on qubits, and corresponding quantum circuits, are also described for the 4- and 3-qubit transformations. All QsiHTs use fast paths, which allow us to calculate the QR decomposition only on disjoint bit planes. As a result, we build the quantum circuits for 5- and more-qubit operations without permutations, only elementary rotations. Unitary operations with real numbers are considered. In the general case, the method of compositing all the roadmaps and quantum schemes for the calculation of any r-qubit operation by only (2r−1)2r−1 elementary rotations is described.

Information•Feb 6, 2026•0 citations
The slant transformation extends the Fourier transformation by incorporating a linear phase modulation that introduces directional sensitivity and localized frequency response. This makes the slant transformation a valuable tool for image processing, pattern recognition, and signal encoding. In this work, we present a new implementation for the slant transformation using the quantum signal-induced heap transformation (QsiHT) and its corresponding QsiHT-based QR decomposition to realize the quantum slant transformation (QST). The QsiHT-QR approach decomposes the slant transformation into a sequence of unitary heap-based rotational transformations that encode the required linear phase modulation. This decomposition minimizes the use of multi-controlled rotations and eliminates redundant CNOT gates, which significantly reduces the circuit depth. The quantum algorithms for the 2- and 3-qubit cases are described in detail, providing the necessary foundations for generalizing the implementation to higher-order qubit systems. Using IBM’s Qiskit framework for quantum simulation, the proposed quantum algorithms are validated through a 4-qubit quantum neural network (QNN) architecture.
The multi-qubit transformation is the important step in quantum computation in many applications. In recent works, this problem is addressed by solving by using the QR decomposition with elementary rotations with one- and two-qubit CNOT gates. Reducing the number of these gates is critical since operations on three or more qubits are difficult to implement on the current quantum computers. The quantum signal induced heap transform (QsiHT) provides an efficient method for state preparation in the realm of quantum image representation by amplitudes. This paper presents a detailed analysis of how the QsiHT, particularly through the strong path, is utilized to construct quantum states to represent images. The algorithm behind this method and the realization of the corresponding quantum circuits is outlined. Practical examples of the QsiHT on 2 and 3-qubit systems are given using the Qiskit Framework for quantum circuit simulation.
This article presents the general method of QR decomposition of r-qubit operations, r≥ 3, by means of quantum signal-induced heap transformations (QsiHT). These are quantum analogues of discrete signal-induced heap transformations, which are generated by given signals and paths, or orders, of processing the data. The case of the 5-qubit operations is described in detail, and a recurrent form of calculation of all 5-qubit QsiHTs from the 4-qubit QsiHTs is given. For that, roadmaps and quantum circuits are presented for all 31 5-point QsiHTs that are used in the QR decomposition. New roadmaps, namely the schemes with paths for performing basic operations on qubits, and corresponding quantum circuits, are also described for the 4- and 3-qubit transformations. All QsiHTs use fast paths, which allow us to calculate the QR decomposition only on disjoint bit planes. As a result, we build the quantum circuits for 5- and more-qubit operations without permutations, only elementary rotations. Unitary operations with real numbers are considered. In the general case, the method of compositing all the roadmaps and quantum schemes for the calculation of any r-qubit operation by only (2r−1)2r−1 elementary rotations is described.