Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to use the quantum phase estimation algorithm to solve the search problem in this generalized case, completely bypassing the Grover algorithm altogether. We obtain a time complexity of this case of sqrt[D/M^{α}], where D is the search space dimension, M is the number of target states, and α≈1, which is close to the optimal scaling.
Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of
Is, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator
Ds with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of
O(B), where
B is a characteristic of the eigenspectrum of
Ds and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still
O(B) times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.
Grover's quantum search algorithm is an important landmark discovery in the area of quantum computing that drives a quantum computer from a known initial state(source state) to an unknown final state(target state) by using selective phase inversions of these states. The algorithm is simply a successive iteration of the Grover's search operator, which is a product of selective inversion operators of source and target states. In its original form, Grover's operator performs the selective inversions only on a unique source state and a unique target state. In the generalized version, we make our search operator to perform selective inversions on more than one source and target states. First, we find that the case of selective inversions of two source and two target states can be easily analysed somewhat similar to the case of Grover's algorithm. But unlike Grover's operator, the selective inversions of three source and three target states is not so easy to study analytically and hence we perform numerical calculations on some specific cases. We find that these generalized search operators also give us a successful quantum search as expected.
A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\ensuremath{\rangle}$ to an unknown target state $|t\ensuremath{\rangle}$. It uses a diffusion operator
Ds that has the source state as one of its eigenstates and
It, where ${I}_{\ensuremath{\psi}}$ denotes the selective phase inversion of the $|\ensuremath{\psi}\ensuremath{\rangle}$ state. It evolves $|s\ensuremath{\rangle}$ to a particular state $|w\ensuremath{\rangle}$, which we will call the
w state, in $O(B/\ensuremath{\alpha})$ time steps, where $\ensuremath{\alpha}$ is $|\ensuremath{\langle}t|s\ensuremath{\rangle}|$ and
B is a characteristic of the diffusion operator. Measuring the
w state gives the target state with a success probability of
O(1/B2), and
O(B2) applications of the algorithm can boost it from
O(1/B2) to
O(1), making the total time complexity $O({B}^{3}/\ensuremath{\alpha})$. In the special case of Grover's algorithm,
Ds is
Is and
B is very close to 1. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement
Iw. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using a phase estimation algorithm. This algorithm is used to efficiently approximate
Iw, which reduces the time complexity of general algorithm to $O(B/\ensuremath{\alpha})$. Although $O(B/\ensuremath{\alpha})$ algorithms are known to exist, our algorithm offers physical implementation advantages.
Grover's algorithm achieves a quadratic speedup for unstructured search when a global oracle for the target set is available. However, its success probability is generally below
100%. To overcome this issue, Long presented a modified version of Grover's search algorithm by introducing a phase-matching condition, which can search for the target set with zero theoretical failure rate. Nevertheless, in many applications, the target set is determined by the intersection of multiple constrained sets. Most existing Grover-based methods are limited to the intersection of only two sets and are not always successful in searching for the target set. In this work, based on Long's method, we propose an algorithm for exactly searching the intersection of multiple sets. We derive and prove a phase-matching condition for searching the intersection exactly. Furthermore, we present a quantum circuit that implements exact intersection search. Finally, we support our theoretical results with numerical simulations.
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.
Since Grover’s algorithm was first introduced, it has become a category of quantum algorithms that can be applied to many problems through the exploitation of quantum parallelism. The original application was the unstructured search problems with the time complexity of O(). In Grover’s algorithm, the key is Oracle and Amplitude Amplification. In this paper, our purpose is to show through examples that, in general, the time complexity of the Oracle Phase is O(N), not O(1). As a result, the time complexity of Grover’s algorithm is O(N), not O(). As a secondary purpose, we also attempt to restore the time complexity of Grover’s algorithm to its original form, O(), by introducing an O(1) parallel algorithm for unstructured search without repeated items, which will work for most cases. In the worst-case scenarios where the number of repeated items is O(N), the time complexity of the Oracle Phase is still O(N) even after additional preprocessing.