Grover's algorithm is a well-known contribution to quantum computing. It searches one value within an unordered sequence faster than any classical algorithm. A fundamental part of this algorithm is the so-called oracle, a quantum circuit that marks the quantum state corresponding to the desired value. A generalization of it is the oracle for Amplitude Amplification, that marks multiple desired states. In this work we present a classical algorithm that builds a phase-marking oracle for Amplitude Amplification. This oracle performs a less-than operation, marking states representing natural numbers smaller than a given one. Results of both simulations and experiments are shown to prove its functionality. This less-than oracle implementation works on any number of qubits and does not require any ancilla qubits. Regarding depth, the proposed implementation is compared with the one generated by Qiskit automatic method, UnitaryGate. We show that the depth of our less-than oracle implementation is always lower. This difference is significant enough for our method to outperform UnitaryGate on real quantum hardware.
Grover's algorithm is a well-known contribution to quantum computing. It searches one value within an unordered sequence faster than any classical algorithm. A fundamental part of this algorithm is the so-called oracle, a quantum circuit that marks the quantum state corresponding to the desired value. A generalization of it is the oracle for Amplitude Amplification, that marks multiple desired states. In this work we present a classical algorithm that builds a phase-marking oracle for Amplitude Amplification. This oracle performs a less-than operation, marking states representing natural numbers smaller than a given one. Results of both simulations and experiments are shown to prove its functionality. This less-than oracle implementation works on any number of qubits and does not require any ancilla qubits. Regarding depth, the proposed implementation is compared with the one generated by Qiskit automatic method, UnitaryGate. We show that the depth of our less-than oracle implementation is always lower. This difference is significant enough for our method to outperform UnitaryGate on real quantum hardware.
Grover’s algorithm is a well-known contribution to quantum computing. It searches one value within an unordered sequence. A fundamental part of this algorithm is the so-called oracle, a quantum circuit that marks the quantum state corresponding to the desired value. In this work we present a classical algorithm that builds an oracle that performs a less-than operation, marking states representing natural numbers smaller than a given one. Results are shown to prove its functionality. This lessthan oracle implementation works on any number of qubits and does not require any ancilla qubits. Regarding depth, the proposed implementation is compared with Qiskit automatic method, Diagonal. We show that the depth of our less-than oracle implementation is always lower. In addition, a comparison with another method for oracle generation in terms of gate count is also conducted to prove the efficiency of our method. The result presented here is part of a research work that aims to achieve reusable quantum operations that can be composed to perform more complex ones.
Abstract The steadily growing research interest in quantum computing—together with the accompanying technological advances in the realization of quantum hardware—fuels the development of meaningful real-world applications, as well as implementations for well-known quantum algorithms. One of the most prominent examples till today is Grover’s algorithm, which can be used for efficient search in unstructured databases. Quantum oracles that are frequently masked as black boxes play an important role in Grover’s algorithm. Hence, the automatic generation of oracles is of paramount importance. Moreover, the automatic generation of the corresponding circuits for a Grover quantum oracle is deeply linked to the synthesis of reversible quantum logic, which—despite numerous advances in the field—still remains a challenge till today in terms of synthesizing efficient and scalable circuits for complex Boolean functions. In this paper, we present a flexible method for automatically encoding unstructured databases into oracles, which can then be efficiently searched with Grover’s algorithm. Furthermore, we develop a tailor-made method for quantum logic synthesis, which vastly improves circuit complexity over other current approaches. Finally, we present another logic synthesis method that considers the requirements of scaling onto real world backends. We compare our method with other approaches through evaluating the oracle generation for random databases and analyzing the resulting circuit complexities using various metrics.
The steadily growing research interest in quantum computing - together with the accompanying technological advances in the realization of quantum hardware - fuels the development of meaningful real-world applications, as well as implementations for well-known quantum algorithms. One of the most prominent examples till today is Grover's algorithm, which can be used for efficient search in unstructured databases. Quantum oracles that are frequently masked as black boxes play an important role in Grover's algorithm. Hence, the automatic generation of oracles is of paramount importance. Moreover, the automatic generation of the corresponding circuits for a Grover quantum oracle is deeply linked to the synthesis of reversible quantum logic, which - despite numerous advances in the field - still remains a challenge till today in terms of synthesizing efficient and scalable circuits for complex boolean functions. In this paper, we present a flexible method for automatically encoding unstructured databases into oracles, which can then be efficiently searched with Grover's algorithm. Furthermore, we develop a tailor-made method for quantum logic synthesis, which vastly improves circuit complexity over other current approaches. Finally, we present another logic synthesis method that considers the requirements of scaling onto real world backends. We compare our method with other approaches through evaluating the oracle generation for random databases and analyzing the resulting circuit complexities using various metrics.
Several prominent quantum computing algorithms—including Grover’s search algorithm and Shor’s algorithm for finding the prime factorization of an integer—employ subcircuits termed ‘oracles’ that embed a specific instance of a mathematical function into a corresponding bijective function that is then realized as a quantum circuit representation. Designing oracles, and particularly, designing them to be optimized for a particular use case, can be a non-trivial task. For example, the challenge of implementing quantum circuits in the current era of NISQ-based quantum computers generally dictates that they should be designed with a minimal number of qubits, as larger qubit counts increase the likelihood that computations will fail due to one or more of the qubits decohering. However, some quantum circuits require that function domain values be preserved, which can preclude using the minimal number of qubits in the oracle circuit. Thus, quantum oracles must be designed with a particular application in mind. In this work, we present two methods for automatic quantum oracle synthesis. One of these methods uses a minimal number of qubits, while the other preserves the function domain values while also minimizing the overall required number of qubits. For each method, we describe known quantum circuit use cases, and illustrate implementation using an automated quantum compilation and optimization tool to synthesize oracles for a set of benchmark functions; we can then compare the methods with metrics including required qubit count and quantum circuit complexity.
Grover's algorithm, which was discovered in 1996, is faster than any classical algorithm can ever be. Quantum systems are in a superposition of states and can examine multiple data simultaneously. This work conducted a comprehensive analysis to understand how Grover's algorithm works. Grover's search provides quadratic speed in comparison to the classical one; it explores target data in the unstructured database. Grover search starts with the superposition of all Qubits, which results in the same amplitude for all data points. To achieve superposition, the Hadamard gate was applied to all Qubits. Further, a black box function named Oracle is applied to mark the desired state. In the next phase, amplitude amplification is done, which enhances the probability of the desired data point. Thus, Grover's search yields results with an O (√ N) complexity, while its counterpart classical algorithm searches the desired data point with an O (N) complexity. Experiments have been performed on the Qiskit simulator using 2 and 3 qubits for the detailed analysis of Grover's search.
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.