The Grover controlled-diffuser (CUs) for quantum Boolean oracles (Uω) is introduced as a new approach for Grover’s algorithm [1-3], to search for all solutions for arbitrary logical structures of such oracles, since the standard Grover diffuser (Us) [1-3] is not able to find all correct solutions for some logical structures of Uω. This protocol constructs the quantum circuit of the CUs operator [4] of Grover's algorithm, which relies on the states of the output qubit (as the reflection of Boolean decisions from a Uω) without relying on the conventional phase kickback mechanism. The CUs operator successfully searches for all correct solutions for all Uω regardless of their different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few.
The Grover controlled-diffuser (CUs) for quantum Boolean oracles (Uω) is introduced as a new approach for Grover’s algorithm [1-3], to search for all solutions for arbitrary logical structures of such oracles, since the standard Grover diffuser (Us) [1-3] is not able to find all correct solutions for some logical structures of Uω. This protocol constructs the quantum circuit of the CUs operator [4] of Grover's algorithm, which relies on the states of the output qubit (as the reflection of Boolean decisions from a Uω) without relying on the conventional phase kickback mechanism. The CUs operator successfully searches for all correct solutions for all Uω regardless of their different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few.
The Grover controlled-diffuser (CUs) for quantum Boolean oracles (Uω) is introduced as a new approach for Grover’s algorithm [1-3], to search for all solutions for arbitrary logical structures of such oracles, since the standard Grover diffuser (Us) is not able to find all correct solutions for some logical structures of Uω. This protocol constructs the quantum circuit of the CUs operator [4] of Grover's algorithm, which relies on the states of the output qubit (as the reflection of Boolean decisions from a Uω) without relying on the conventional phase kickback mechanism. The CUs operator successfully searches for all correct solutions for all Uω regardless of their different logical structures, such as POS, SOP, ESOP, CSP-SAT, XOR-SAT, just to name a few.
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.
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
Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2 n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024
In this paper, we propose a quantum circuit based on Grover’s algorithm to solve the exact cover problem (ECP), which is NP-hard. A novel controlled counter is designed and used to implement the ECP oracle to check if every element is covered by a set exactly once. If so, a feasible solution is found. The phase of the qubit state representing a feasible solution is flipped by calling the oracle, and the state’s amplitude is amplified through the diffusion operator o(2n) times, where n is the number of sets given in the ECP. Thanks to Grover’s algorithm, the proposed quantum circuit achieves quadratic speedup compared with the classical algorithm that needs to call the oracle O(2n) times. To validate our approach, we execute the proposed quantum circuit using the IBM Quantum Lab service.
Abstract This laboratory work presents a practical implementation of Grover’s search algorithm using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and is intended to provide hands-on experience with amplitude amplification and oracle-based quantum search. In Laboratory Work No. 7, participants construct Grover’s algorithm step by step. The implementation begins with defining a function that initializes the quantum registers and prepares the initial stage of the circuit. The output qubit associated with the oracle is set to the state ∣1⟩|1\rangle∣1⟩ to enable phase inversion through the phase kickback mechanism. Hadamard gates are then applied to create a uniform superposition over the search space. The oracle is designed to mark the target state by applying a phase shift, effectively flipping the sign of the amplitude corresponding to the solution. This is followed by the diffusion operator (inversion about the mean), which amplifies the probability amplitude of the marked state. The Grover iteration—composed of the oracle and diffusion operator—is repeated an optimal number of times to maximize the probability of measuring the correct result. The laboratory exercise emphasizes the principles of quantum amplitude amplification, circuit modularization, oracle construction, and measurement-based verification. By executing the algorithm on simulators and cloud-accessible quantum hardware, participants observe the quadratic speedup of Grover’s algorithm compared to classical linear search methods. This practical guide bridges theoretical understanding of quantum search with executable quantum circuits, reinforcing the operational aspects of quantum algorithm design within modern quantum computing