Grover's algorithm searches an unstructured database of $N$ entries, finding a marked element in time $\mathcal{O}(\sqrt{N})$. The algorithm was later generalised to amplitude amplification. The original algorithm succeeds with probability close to one and needs to run for a number of iterations chosen ahead of time. We discuss an implementation of amplitude amplification that, instead, runs for a number of iterations unspecified a priori and only halts on success. We prove that this approach maintains the quadratic speed-up of the original quantum algorithm. The key component is a feedback loop controlled by a weak measurement that allows limited monitoring of the evolution of the quantum state.
Grover's algorithm searches an unstructured database of N entries, finding a marked element in time O(N). The algorithm was later generalised to amplitude amplification. The original algorithm succeeds with probability close to one and needs to run for a number of iterations chosen ahead of time. We discuss an implementation of amplitude amplification that, instead, runs for a number of iterations unspecified a priori and only halts on success. We prove that this approach maintains the quadratic speed-up of the original quantum algorithm. The key component is a feedback loop controlled by a weak measurement that allows limited monitoring of the evolution of the quantum state.
This paper concerns the Grover algorithm that permits to make amplification of quantum states previously tagged by an Oracle. Grover's algorithm allows searches in an unstructure database of n entries finding a marked element with a quadratic speedup. The algorithm requires a predefined number of runs to succeed with probability close to one.This article provides a description of the amplitude amplification quantum algorithm mechanism in a very short computational way, based on tensor products and provides a geometric presentation of the successive system states. All the basis changes are fully described to provide an alternative to the wide spread Grover description based only on matrices and complex tensor computation. Our experiments encompass numerical evaluations of circuit using the Qiskit library of IBM that meet the theoretical considerations
We present a family of quantum amplitude amplification algorithms that interleave ordinary Grover rotations with tunable weak measurements in loops. A successful measurement yields the target state and stops the algorithm, while after a failure, the loop resumes from the post-measurement state without restarting. We express the leading costs in expected Grover iterations as p→0. For a known target probability p, an exact weak measurement-conditioned loop for states near the angle π/4 uses (π/8+1/4+o(1))/p iterations, improving on the standard Grover constant π/4 and on optimised restart Grover, and matching the corresponding infimum in our continuous-angle analysis. When only a lower bound 0<p0≤p<1/2 is available, a discrete Lyapunov equation gives the exact expected oracle cost for every fixed measurement strength and a closed-form optimal strength. Using the strength selected from p0, the expected number of Grover iterations is at most 221(p1+p01), where the expected cost decreases as the actual p increases, and gives the leading constant 1/2 at the promise boundary p=p0. For completely unknown p, we identify measurement strength κ(t)=Θ(1/t) as the critical scale within the regular schedules considered here and analyse κb(t)=min{1/2,b/t}. For each fixed b>2, we rigorously derive a closed-form Gamma-function expression C(b) giving (C(b)/2+o(1))/p expected iterations. Numerical minimisation of this explicit expression gives b≈5.2 and C(b)/2≈1.01. The results show that weak measurements preserve the Θ(1/p) search scale while adding an explicit fixed-point control mechanism and provable oracle savings.
Abstract A while loop tests a termination condition on every iteration. On a quantum computer, such measurements perturb the evolution of the algorithm. We define a while loop primitive using weak measurements, offering a trade-off between the perturbation caused and the amount of information gained per iteration. This trade-off is adjusted with a parameter set by the programmer. We provide sufficient conditions that let us determine, with arbitrarily high probability, a worst-case estimate of the number of iterations the loop will run for. As an example, we solve Grover’s search problem using a while loop and prove the quadratic quantum speed-up is maintained.
Grover's algorithm is one of the pioneering demonstrations of the advantages of quantum computing over its classical counterpart, providing - at most - a quadratic speed-up over the classical solution for unstructured database search. The original formulation of Grover's algorithm is non-deterministic, finding the answer with a probability that varies with the size of the search space and the number of marked elements. A recent reformulation introduced a deterministic form of Grover's algorithm that - in principle - finds the answer with certainty. Here we realise the deterministic Grover's algorithm on a programmable photonic integrated circuit, finding that it not only outperforms the original Grover's algorithm as predicted, but is also markedly more robust against technological imperfections. We explore databases of 4 to 10 elements, with every choice of a single marked element, achieving an average success probability of 99.77±0.05%.
Grover’s algorithm harnesses the power of quantum computing to swiftly locate specific elements in an unstructured database, outperforming classical computers in tasks like database searching. This algorithm capitalizes on the unique ability of qubits to be in both 0 and 1 states simultaneously (superposition), allowing it to scan the entire search space at once. It then boosts the probability of the target element, making it more prominent. While the foundational concepts of Grover’s algorithm are well-documented, practical implementation using quantum operators, especially for large search spaces, remains less explored beyond basic examples with a small number of qubits. Existing general synthesis techniques often involve numerous operators or are time-consuming. Our proposed methods specifically address the amplitude-amplification component of Grover’s algorithm for any size of search space. These methods detail the required types and quantities of qubits and operators, emphasizing minimal usage and efficient assembly. Developed and evaluated in Python, our methods consistently identified target elements with over 95% accuracy and achieved configurations comparably compact as those in the existing literature but at a faster pace. We anticipate that these methods facilitate practical implementations of Grover’s algorithm across various domains.
Searching a database is a central task in computer science and is paradigmatic of transport and optimization problems in physics. For an unstructured search, Grover's algorithm predicts a quadratic speedup, with the search time τ(N)=Θ(N) and N the database size. Numerical studies suggest that the time complexity can change in the presence of feedback, injecting information during the search. Here, we determine the time complexity of the quantum analog of a randomized algorithm, which implements feedback in a simple form. The search is a continuous-time quantum walk on a complete graph, where the target is continuously monitored by a detector. Additionally, the quantum state is reset if the detector does not click within a specified time interval. This yields a non-unitary, non-Markovian dynamics. We optimize the search time as a function of the hopping amplitude, detection rate, and resetting rate, and identify the conditions under which time complexity could outperform Grover's scaling. The overall search time does not violate Grover's optimality bound when including the time budget of the physical implementation of the measurement. For databases of finite sizes monitoring can warrant rapid convergence and provides a promising avenue for fault-tolerant quantum searches.