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. For that, 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-plane, which is the 2D circular top-view of the Bloch sphere. Collectively, the Bloch sphere and its XY-plane 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 gates ( , , , and ), as follows. 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 "basis" gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, we propose the XY-plane of the Bloch sphere 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 different projectional planes of the Bloch sphere, e.g., the XZ-plane and the YZ-plane, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as an open geometrical framework for prospective quantum computing research.
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. For that, 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-plane, which is the 2D circular top-view of the Bloch sphere. Collectively, the Bloch sphere and its XY-plane 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 gates ( , , , and ), as follows. 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 "basis" gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, we propose the XY-plane of the Bloch sphere 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 different projectional planes of the Bloch sphere, e.g., the XZ-plane and the YZ-plane, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as an open geometrical framework for prospective quantum computing research.
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.
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.
The conventional design technique of any n-bit quantum gate is mainly achieved using unitary matrices multiplication, where n >= 2 and 1 <= m <= n-1 for m target qubits and n-m control qubits. These matrices represent quantum rotations by an n-bit quantum gate. For a quantum designer, such a conventional technique requires extensive computational time and effort, which may generate an n-bit quantum gate with a too high quantum cost. The Bloch sphere is only utilized as a visualization tool to verify the conventional design correctness for quantum rotations by a quantum gate. In contrast, this paper introduces a new concept of using the Bloch sphere as a "geometrical design tool" to build cost-effective n-bit quantum gates with lower quantum costs. This concept is termed the "Bloch sphere approach (BSA)". In BSA, a cost-effective n-bit quantum gate is built without using any unitary matrices multiplication. Instead, the quantum rotations for such a gate are visually selected using the geometrical planar intersections of the Bloch sphere. The BSA can efficiently map m targets among n-m controls for an n-bit quantum gate, to satisfy the limited layout connectivity for the physical neighboring qubits of a quantum computer. Experimentally, n-bit quantum gates built using the BSA always have lower quantum costs than those for such gates built using the conventional quantum design techniques.
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 conventional design technique of any n-bit quantum gate is mainly achieved using the multiplication of unitary matrices for control qubits and target qubits, where and .These matrices represent quantum rotations by an n-bit quantum gate.For a quantum designer, such a conventional technique requires extensive computational time and effort, which may generate an n-bit quantum gate with too high quantum cost.In general, the Bloch sphere is only utilized as a visualization tool to verify the conventional design correctness for quantum rotations by a quantum gate.In contrast, this paper introduces a new concept of using the Bloch sphere as a "geometrical design tool" to build cost-effective n-bit quantum gates with lower quantum costs.This concept is termed the "Bloch sphere approach (BSA)."In BSA, a cost-effective n-bit quantum gate is built without using any multiplication of unitary matrices.Instead, the quantum rotations for such a gate are visually selected using the geometrical planar intersections of the Bloch sphere.The BSA can efficiently map m targets among controls for an n-bit quantum gate to satisfy the limited layout connectivity for the physical neighboring qubits of a superconducting quantum computer.Experimentally, n-bit quantum gates built using the BSA always have lower quantum costs than those for such gates built using the conventional quantum design techniques, by utilizing the same transpilation settings and the restricted layout connectivity of a real IBM superconducting quantum computer, where qubits.
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.