A generic architecture of n-bit quantum operators is proposed for cost-effective transpilation, based on the layouts and the number of n neighbor physical qubits for IBM quantum computers, where n >= 3. This proposed architecture is termed "GALA-n quantum operator". The GALA-n quantum operator is designed using the visual approach of the Bloch sphere, from the visual representations of the rotational quantum operations for IBM native gates (square root of X, X, RZ, and CNOT). In this paper, we also proposed a new formula for the quantum cost, which calculates the total numbers of native gates, SWAP gates, and the depth of the final transpiled quantum circuits. This formula is termed the "transpilation quantum cost". After transpilation, our proposed GALA-n quantum operator always has a lower transpilation quantum cost than that of conventional n-bit quantum operators, which are mainly constructed from costly n-bit Toffoli gates.
A generic architecture of n-bit quantum operators is proposed for cost-effective transpilation, based on the layouts and the number of n neighbor physical qubits for IBM quantum computers, where n >= 3. This proposed architecture is termed "GALA-n quantum operator". The GALA-n quantum operator is designed using the visual approach of the Bloch sphere, from the visual representations of the rotational quantum operations for IBM native gates (square root of X, X, RZ, and CNOT). In this paper, we also proposed a new formula for the quantum cost, which calculates the total numbers of native gates, SWAP gates, and the depth of the final transpiled quantum circuits. This formula is termed the "transpilation quantum cost". After transpilation, our proposed GALA-n quantum operator always has a lower transpilation quantum cost than that of conventional n-bit quantum operators, which are mainly constructed from costly n-bit Toffoli gates.
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.
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.
Lov K. Grover introduced, in 1996, Grover's algorithm as a quantum search algorithm to find all solutions for quantum oracles representing classical problems. My research observed that the Grover diffusion operator of Grover's algorithm gives wrong solutions when Boolean oracles are designed in some logical structures. Therefore, I invented the new "controlled-diffusion operator" for Boolean oracles as a new approach for Grover's algorithm that always correctly solves the problem of Grover's algorithm. Another important problem in quantum computing is designing reliable and cost-effective quantum gates using reversible binary logic. In classical logic design, the stage of logic design can be separated from the stage of mapping topological circuits to geometrical architecture. In contrast to classical logic design, the stages of quantum circuit design are not separable. For practical quantum applications, a quantum circuit is realized (transpiled) into the layout of a real quantum device. For a real quantum device, two factors affect the cost-effective transpilation: the utilization of supported (native) gates and the geometrical connectivity pattern of n neighboring physical qubits, where native gates ensure cost-effective transpilation and n ≥ 2. In this dissertation, I introduced two new approaches to leverage these two factors for real IBM quantum devices. The first approach initializes the uniform distribution for all qubits using IBM native √X gates, instead of using the conventional Hadamard gates, which are IBM non-native gates producing cost-expensive transpilation. The second approach geometrically constructs generic and cost-effective quantum gates using IBM native gates. In literature the Bloch sphere is used as a visualization tool for a quantum gate, which is initially designed using unitary matrices computations. In contrast, I introduced the Bloch sphere approach (BSA) as a "geometrical design tool" to design a quantum gate using the Bloch sphere, which leads to fewer unitary matrices computations. I employed the BSA to design various generic and cost-effective quantum gates and libraries based on the layouts of IBM quantum devices. Using the BSA, I invented two quantum libraries to investigate the trade-off between scalability and reliability, for the purpose of how the input qubits of a quantum gate are routed into the layout of an IBM quantum device and how many ancilla qubits are utilized. Experimentally, for the same layout of an IBM quantum device, I prove that the transpiled quantum circuits of oracles utilizing my two approaches always have lower quantum costs and shorter depths than those of the transpiled quantum
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.
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.