A proposed method for preparing the superposition states of qubits using different axes of the Bloch sphere. This method utilizes the Y-axis of the Bloch sphere using IBM native (square root of X) gates, instead of utilizing the X-axis of the Bloch sphere using IBM non-native Hadamard gates, for transpiling cost-effective quantum circuits on IBM quantum computers. In this paper, our presented method ensures that the final transpiled quantum circuits always have a lower quantum cost than that of the transpiled quantum circuits using the Hadamard gates.
A proposed method for preparing the superposition states of qubits using different axes of the Bloch sphere. This method utilizes the Y-axis of the Bloch sphere using IBM native (square root of X) gates, instead of utilizing the X-axis of the Bloch sphere using IBM non-native Hadamard gates, for transpiling cost-effective quantum circuits on IBM quantum computers. In this paper, our presented method ensures that the final transpiled quantum circuits always have a lower quantum cost than that of the transpiled quantum circuits using the Hadamard gates.
Conventionally, the quantum oracular algorithms, such as Deutsch-Jozsa, Bernstein-Vazirani, and Grover, utilize Hadamard gates to create uniform superposition states for the input qubits of an oracle. However, Hadamard gates are non-native (non-supported) gates in all real quantum computers. For this reason, Hadamard gates are considered cost-expensive gates when realizing (transpiling) such algorithms into a real quantum computer. This paper introduces a new methodology for cost-effective transpilation of Grover's algorithm into real quantum computers, by replacing all Hadamard gates with X gates. In quantum computing, the Hadamard and X gates create uniform superposition states of a qubit on the X-axis and Y-axis of the Bloch sphere, respectively. Hence, we utilize a different axis of the Bloch sphere to construct a cost-effective realization for the final transpiled circuit of Grover's algorithm. Theoretically, the final transpiled circuit of Grover's algorithm using X gates has been minimized to approximately 64% than the circuit of Grover's algorithm using Hadamard gates. Experimentally, because of the X is an IBM native gate, it has been proven that the final transpiled circuit of Grover's algorithm using X gates always has a lower quantum cost of 6% than the circuit of Grover's algorithm using Hadamard gates, due to the limited layout (architecture) connectivity of the physical neighboring qubits for IBM quantum computers.
A cost-effective n-bit Toffoli gate is proposed to be realized (or transpiled) based on the layouts (linear, T-like, and I-like) and the number of n physical qubits for IBM quantum computers. This proposed gate is termed the "layout-aware n-bit Toffoli gate". The layout-aware n-bit Toffoli gate is designed using the visual approach of the Bloch sphere, from the visual representations of the rotational quantum operations for IBM native gates. In this paper, we also proposed a new formula for the quantum cost, which calculates the total number of native gates, the crossing connections, and the depth of the final transpiled quantum circuit. This formula is termed the "transpilation quantum cost". After transpilation, our proposed layout-aware n-bit Toffoli gate always has a much lower transpilation quantum cost than that of the conventional n-bit Toffoli gate, where 3 <= n <= 7 qubits, for different IBM quantum computers.
The Bloch sphere is a geometrical 3D sphere that visualizes the states of a qubit after a series of quantum gates are applied to it. In quantum computing, the Bloch sphere is mainly used as a geometrical visualization (and verification) tool. On the other hand, in this protocol, we introduce the Bloch sphere as a geometrical design tool for building cost-effective quantum gates based on their rotational quantum operations in the XY, XZ, and/or YZ planes, which are the 2D circular projections along the axes of the Bloch sphere. Collectively, the Bloch sphere and its projectional planes are termed the Bloch sphere approach (BSA). With the BSA, various generic and cost-effective quantum gates and libraries are designed for IBM quantum computers, using the symmetrical and semi-symmetrical structures [1-5], Clifford+T gates, and IBM native "basis" gates ( , , , and ). Our designed generic and cost-effective quantum gates and libraries are listed as follows, where 2 n 5 qubits. Quantum libraries: GALA-n [3, 6] and CALA-n [4, 7], which have become part of the IBM Qiskit ecosystem [8] n-bit Toffoli gate [2-4] n-bit Boolean gates (AND, NAND, OR, NOR, Implication, and Inhibition) [3, 4] n-bit controlled- ( ) and controlled- ( ) gates [3, 4] n-bit Fredkin gate [3, 4] n-bit Miller gate [3, 4] Boolean-Phase SWAP gate: p-SWAP [4, 5] Because the quantum operations of all IBM native gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, the XY-plane of the Bloch sphere is utilized here for the BSA. However, the BSA can also be utilized to build generic and cost-effective quantum gates for other quantum computers, e.g., Intel, Google, and Rigetti, using other projectional planes of the Bloch sphere, e.g., the XZ and YZ planes, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as a generic and open geometrical framework for prospective quantum computing research and building interesting and innovative quantum gates and circuits.
CALA-n: A Quantum Library for Realizing Cost-Effective 2-, 3-, 4-, and 5-bit Gates on IBM Quantum Computers using Bloch Sphere Approach, Clifford+T Gates, and Layouts
We introduce a new quantum layout-aware approach to realize cost-effective n-bit gates using the Bloch sphere, for 2≤n≤5 qubits. These n-bit gates are entirely constructed from the Clifford+T gates, in the approach of selecting sequences of rotations visualized on the Bloch sphere. This Bloch sphere approach ensures to match the quantum layout for synthesizing (transpiling) these n-bit gates into an IBM quantum computer. Various standard n-bit gates (Toffoli, Fredkin, etc.) and their operational equivalent of our proposed n-bit gates are examined and evaluated, in the context of the final quantum costs, as the final counts of generated IBM native gates. In this paper, we demonstrate that all our n-bit gates always have lower quantum costs than those of standard n-bit gates after transpilation. Hence, our Bloch sphere approach can be used to build a quantum library of various cost-effective n-bit gates for different layouts of IBM quantum computers.
The Bloch sphere is a geometrical 3D sphere that visualizes the states of a qubit after a series of quantum gates are applied to it. In quantum computing, the Bloch sphere is mainly used as a geometrical visualization (and verification) tool. On the other hand, in this protocol, we introduce the Bloch sphere as a geometrical design tool for building cost-effective quantum gates based on their rotational quantum operations in the XY, XZ, and/or YZ planes, which are the 2D circular projections of the Bloch sphere. Collectively, the Bloch sphere and its planes are termed the Bloch sphere approach (BSA). With the BSA, various generic and cost-effective quantum gates and libraries are designed for IBM quantum computers, using the symmetrical and semi-symmetrical structures [1-5], Clifford+T gates, and IBM native "basis" gates ( , , , and ). Our designed generic and cost-effective quantum gates and libraries are listed as follows, where 2 n 5 qubits. Quantum libraries: GALA-n [3, 6] and CALA-n [4, 7], which have become part of the IBM Qiskit ecosystem [8] n-bit Toffoli gate [2-4] n-bit Boolean gates (AND, NAND, OR, NOR, implication, and inhibition) [3, 4] n-bit controlled- ( ) and controlled- ( ) gates [3, 4] n-bit Fredkin gate [3, 4] n-bit Miller gate [3, 4] Boolean-Phase SWAP gate: p-SWAP [4, 5] Because the quantum operations of all IBM native gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, the XY-plane of the Bloch sphere is utilized here for the BSA. However, the BSA can also be utilized to build generic and cost-effective quantum gates for other quantum computers, e.g., Intel, Google, and Rigetti, using other projectional planes of the Bloch sphere, i.e., the XZ and YZ planes, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as a generic and open geometrical framework for prospective quantum computing research for building interesting and innovative quantum gates and circuits.
The Bloch sphere is a geometrical 3D sphere that visualizes the states of a qubit after a series of quantum gates are applied to it. In quantum computing, the Bloch sphere is mainly used as a geometrical visualization (and verification) tool. On the other hand, in this protocol, we introduce the Bloch sphere as a geometrical design tool for building cost-effective quantum gates based on their rotational quantum operations in the XY, XZ, and/or YZ planes, which are the 2D circular top and side views of the Bloch sphere. Collectively, the Bloch sphere and its planes are termed the Bloch sphere approach (BSA). With the BSA, various generic and cost-effective quantum gates and libraries are designed for IBM quantum computers, using the symmetrical and semi-symmetrical structures [1-5], Clifford+T gates, and IBM native "basis" gates ( , , , and ). Our designed generic and cost-effective quantum gates and libraries are listed as follows, where 2 n 5 qubits. Quantum libraries: GALA-n [3, 6] and CALA-n [4, 7], which have become part of the IBM Qiskit ecosystem [8] n-bit Toffoli gate [2-4] n-bit Boolean gates (AND, NAND, OR, NOR, implication, and inhibition) [3, 4] n-bit controlled- ( ) and controlled- ( ) gates [3, 4] n-bit Fredkin gate [3, 4] n-bit Miller gate [3, 4] Boolean-Phase SWAP gate: p-SWAP [4, 5] Because the quantum operations of all IBM native gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, the XY-plane of the Bloch sphere is utilized here for the BSA. However, the BSA can also be utilized to build generic and cost-effective quantum gates for other quantum computers, e.g., Intel, Google, and Rigetti, using other projectional planes of the Bloch sphere, i.e., the XZ and YZ planes, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as a generic and open geometrical framework for prospective quantum computing research for building interesting and innovative quantum gates and circuits.