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
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
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.
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 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.
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.
Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on Grover’s algorithm, a quantum algorithm for unstructured search problems that provides a quadratic speedup over classical approaches. In an unstructured search problem, we are given a function (often represented as an oracle) that marks one or more “correct” solutions among NNN possible candidates. A classical algorithm requires on the order of O(N)O(N)O(N) evaluations in the worst case. In contrast, Grover’s algorithm finds a solution using only O(N)O(\sqrt{N})O(N) oracle queries. This quadratic improvement implies: The quantum algorithm requires roughly the square root of the number of operations needed classically. Any classical algorithm for unstructured search must have cost at least quadratic relative to Grover’s quantum procedure. Core Idea of the Algorithm Grover’s algorithm relies on: Preparation of a uniform superposition (typically using Hadamard gates). An oracle that marks valid solutions via phase inversion. The Grover diffusion operator, which amplifies the probability amplitude of the marked states through interference. By repeatedly applying the Grover operator, the amplitude of the desired solution grows while others diminish, allowing measurement to reveal the correct answer with high probability. Practical Considerations Despite its theoretical elegance and wide applicability, the quadratic speedup of Grover’s algorithm is unlikely to yield near-term practical advantage. Modern classical hardware operates at extremely high clock rates and is vastly more mature than current quantum devices. For problem sizes accessible to today’s quantum computers, classical systems remain dominant. However, history shows that even sub-quadratic classical improvements — such as: Fast Fourier Transform (FFT) Efficient sorting algorithms like quicksort and mergesort have had transformative practical impact. The crucial distinction is that Grover’s algorithm depends on fundamentally different hardware — quantum processors — which are still in early development. Long-Term Perspective As quantum hardware advances in qubit count, coherence time, and error correction, Grover’s quadratic advantage may become practically significant for: Cryptographic key search Optimization heuristics Combinatorial search problems Amplitude amplification subroutines in broader quantum algorithms Through hands-on experimentation on IBM Q, this lesson demonstrates how theoretical amplitude amplification translates into executable quantum circuits, reinforcing the relationship between interference, probability amplitudes, and computational complexity.
A controlled-diffusion operator for Boolean oracles is designed as a new approach for Grover's algorithm to search for solutions for arbitrary logical structures of such oracles, since the Grover diffusion operator is not able to find correct solutions for some logical structures of Boolean oracles. We also show that the Phase oracles do not work sometimes correctly using the Grover diffusion operator. Our proposed controlled-diffusion operator relies on the states of output qubit, as the reflection of Boolean decisions from a Boolean oracle without relying on the phase kickback. We prove that on many examples of Boolean and Phase oracles the Grover diffusion operator is not working correctly. The oracles in these examples are constructed using different structures of POS, SOP, ESOP, CSP-SAT, and XOR-SAT. Our mathematical models and experiments prove that the proposed controlled-diffusion operator successfully searches for all solutions for all Boolean oracles regardless of their different logical structures.