A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.
A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.
A classical limit of Grover's algorithm is discussed by assuming a very rapid decoherence (or dephasing) between consecutive Grover's unitary operations, which leads pure quantum states to completely decohered mixed states. One can identify a specific element among N unsorted elements by a probability of the order of unity after k∼N/4 steps of classical amplification defined by the decohered mixed states, in contrast to Grover's k∼πN/4 steps in quantum mechanical amplification. This difference is caused by the loss of quantum coherence with or without the loss of entanglement depending on each case.
In this paper, we study decoherence in Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers ${\ensuremath{\sigma}}_{z}$ error with probability p(0<p<1) independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of 2Mnp and derive its matrix element order by order under the $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ limit. (In this large-n limit, we assume p is small enough, so that 2Mnp(>0) can take any real positive value or 0.) This approach gives us an interpretation about creation of new modes caused by ${\ensuremath{\sigma}}_{z}$ error and an asymptotic form of an arbitrary-order correction. Calculating the matrix element up to the fifth-order term numerically, we investigate a region of 2Mnp (perturbative parameter), where the algorithm finds the correct item with a threshold of probability Pth or more. It satisfies $2Mnp<(8/5)(1\ensuremath{-}{P}_{\mathrm{th}})$ around $2Mnp\ensuremath{\simeq}0$ and ${P}_{\mathrm{th}}\ensuremath{\simeq}1,$ and this linear relation is applied to a wide range of Pth approximately. This observation is similar to a result obtained by Bernstein and Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum-to-classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.
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.
Grover's search algorithm is the optimal quantum algorithm that can search an unstructured database quadratically faster than any known classical algorithm. The role of entanglement and correlations in the search algorithm have been studied in great detail and it is known that entanglement between the qubits is necessary to gain a quadratic speedup, for pure state implementation of the Grover search algorithm. Here, we systematically investigate the behavior of quantum coherence and monogamy of entanglement in the discrete analogue of the analog analogue of Grover search algorithm. The analog analogue of Grover search is a continuous time quantum algorithm based on the adiabatic Hamiltonian evolution that gives a quadratic speedup, similar to the original Grover search algorithm. We show that the decrease of quantum coherence, quantified using various coherence monotones, is a clear signature of attaining the maximum success probability in the analog Grover search. We also show that for any two qubit reduced density matrix of the system, the concurrence evolves in close vicinity to the increasing rate of success probability. Furthermore, we show that the system satisfies a n-party monogamy inequality for arbitrary times, hence bounding the amount of n-qubit entanglement during the quantum search.
In this paper, we study decoherence in Grover's quantum search algorithm using a perturbative method. We assume that each two-state system (qubit) that belongs to a register suffers a phase-flip error (${\ensuremath{\sigma}}_{z}$ error) with probability p independently at every step in the algorithm, where $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. Considering an n-qubit density operator to which Grover's iterative operation is applied M times, we expand it in powers of 2Mnp and derive its matrix element order by order under the large-n limit. [In this large-n limit, we assume p is small enough, so that 2Mnp can take any real positive value or zero. We regard $x\ensuremath{\equiv}2Mnp$ $(\ensuremath{\geqslant}0)$ as a perturbative parameter.] We obtain recurrence relations between terms in the perturbative expansion. By these relations, we compute higher orders of the perturbation efficiently, so that we extend the range of the perturbative parameter that provides a reliable analysis. Calculating the matrix element numerically by this method, we derive the maximum value of the perturbative parameter x at which the algorithm finds a correct item with a given threshold of probability Pth or more. (We refer to this maximum value of x as xc, a critical point of x.) We obtain a curve of xc as a function of Pth by repeating this numerical calculation for many points of Pth and find the following facts: a tangent of the obtained curve at Pth=1 is given by $x=(8∕5)(1\ensuremath{-}{P}_{\mathrm{th}})$, and we have ${x}_{\mathrm{c}}>\ensuremath{-}(8∕5){\mathrm{log}}_{e}\phantom{\rule{0.2em}{0ex}}{P}_{\mathrm{th}}$ near Pth=0.
Abstract. Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. Classical search procedures for an N element database require at most O(N) time complexity. Grover’s algorithm is able to find a solution with high prob-ability in O( N) time through an amplitude amplification scheme. In this work we draw elements from both classical and quantum computation to develop an alternative search proposal based on quantum entanglement detection schemes. In 2002, Horodecki and Ekert proposed an efficient method for direct detection of quantum entanglement. Our proposition to quantum search combines quantum entanglement detection along-side entanglement inducing operators. Grover’s quantum search relies on measuring a quantum superposition after having applied a unitary evolution. We deviate from the standard method by focusing on fine-tuning a unitary operator in order to infer the solution with certainty. Our proposal sacrifices space for speed and depends on the mathematical properties of linear positive maps Λ which have not been operationally characterized. Whether such a Λ can be easily determined remains an open question.