Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.23787

Amplitude amplification and estimation require inverses

Ewin Tang, Wright, John

Published Jul 31, 2025• 0 citations
arXivHTMLPDF

Abstract

We prove that the generic quantum speedups for brute-force search and counting only hold when the process we apply them to can be efficiently inverted. The algorithms speeding up these problems, amplitude amplification and amplitude estimation, assume the ability to apply a state preparation unitary $U$ and its inverse $U^\dagger$; we give problem instances based on trace estimation where no algorithm which uses only $U$ beats the naive, quadratically slower approach. Our proof of this is simple and goes through the compressed oracle method introduced by Zhandry. Since these two subroutines are responsible for the ubiquity of the quadratic "Grover" speedup in quantum algorithms, our result explains why such speedups are far harder to come by in the settings of quantum learning, metrology, and sensing. In these settings, $U$ models the evolution of an experimental system, so implementing $U^\dagger$ can be much harder -- tantamount to reversing time within the system. Our result suggests a dichotomy: without inverse access, quantum speedups are scarce; with it, quantum speedups abound.

DOI
10.48550/arxiv.2507.23787

Related papers

Quantum Counting

We study some extensions of Grover's quantum searching algorithm. First, we generalize the Grover iteration in the light of a concept called amplitude amplification. Then, we show that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform approximate counting, which can be seen as an amplitude estimation process.

WORLD SCIENTIFIC (EUROPE) eBooks•Jul 7, 2017•245 citations

Practical Guide to Quantum Computing: Grover's Algorithm # 4

Mikhail Pavlov

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.

•Mar 4, 2026•0 citations

Chapter 4: Amplitude amplification and estimation

Giacomo Nannicini

In Chs. 2 and 3 we described several quantum algorithms that are exponentially faster than classical algorithms for the same problem, under some measure of complexity. Unfortunately, the algorithms of Ch. 2 apply to rather contrived problems that do not have known direct applications. We now describe a quantum algorithm that gives only a polynomial—more specifically, quadratic—speedup with respect to classical algorithms, but it applies to a large class of problems and finds many useful applications, both directly and as an algorithmic design tool. The algorithm is known as Grover’s search [89], and its generalization is known as amplitude amplification [43]. Amplitude amplification is widely used as part of many quantum algorithms. It also serves as the basis for amplitude estimation, a way of estimating probabilities quadratically faster than with classical Monte Carlo, and for quantum minimum finding. All these topics are discussed in this chapter.

Society for Industrial and Applied Mathematics eBooks•Dec 17, 2025•0 citations

Optimal fixed-point quantum amplitude amplification using Chebyshev polynomials

Theodore J. Yoder, Guang Hao Low, Isaac L. Chuang

Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction $\lambda$ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction, but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability, and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of $\lambda$.

•Sep 11, 2014•1 citation

Amplitude amplification and estimation

Alexander M. Dalzell, Sam McArdle, Mario Berta +10 more

This chapter covers the quantum algorithmic primitives of amplitude amplification and amplitude estimation. Amplitude amplification is a generalization of Grover’s quantum algorithm for the unstructured search problem. Amplitude estimation can be understood in a similar framework, where it utilizes quantum phase estimation to estimate the value of the amplitude or probability associated with a quantum state. Both amplitude amplification and amplitude estimation provide a quadratic speedup over their classical counterparts, and feature prominently as an ingredient in many end-to-end algorithms.

Cambridge University Press eBooks•Apr 24, 2025•0 citations

The power of time reversal in Hamiltonian property testing

Richard R. Allen, Matthias C. Caro, Yanlin Chen +2 more

Quantum algorithms enable Heisenberg-limited scaling across many tasks in sensing and learning, achieving precision $\varepsilon$ using $O(1/\varepsilon)$ total evolution time. However, in Hamiltonian property testing, the best known algorithms have $O(1/\varepsilon^2)$ total evolution time, unless it is possible to $\textit{reverse}$ the flow of time under the Hamiltonian. In this work, we prove that time reversal is necessary to close this gap. We establish evolution time lower bounds for Hamiltonian certification and locality testing, for a continuous family of error metrics ranging from average- to worst-case. We also prove matching upper bounds for Hamiltonian certification in all cases, both with and without time reversal, showing that the advantage from time reversal persists. Our results follow from a new Fourier-analytic approach to quantum lower bounds in continuous time, which is sensitive to time reversal and applicable beyond Hamiltonian property testing. This gives an alternative proof that inverses are required for amplitude amplification and estimation, improving the lower bounds of Tang and Wright and extending to fractional queries. We also show that, for unstructured search, uncertainty in the phase of the oracle destroys the quantum speedup without inverse queries, even when the oracle is fixed and coherent. Our work rigorously establishes time reversal as a resource governing both the precision of quantum measurement and the robustness of quantum algorithms.

quant-pharXiv:2610.04996•Oct 6, 2026•0 citations
We prove that the generic quantum speedups for brute-force search and counting only hold when the process we apply them to can be efficiently inverted. The algorithms speeding up these problems, amplitude amplification and amplitude estimation, assume the ability to apply a state preparation unitary UUU and its inverse U†U^\daggerU†; we give problem instances based on trace estimation where no algorithm which uses only UUU beats the naive, quadratically slower approach. Our proof of this is simple and goes through the compressed oracle method introduced by Zhandry. Since these two subroutines are responsible for the ubiquity of the quadratic "Grover" speedup in quantum algorithms, our result explains why such speedups are far harder to come by in the settings of quantum learning, metrology, and sensing. In these settings, UUU models the evolution of an experimental system, so implementing U†U^\daggerU† can be much harder -- tantamount to reversing time within the system. Our result suggests a dichotomy: without inverse access, quantum speedups are scarce; with it, quantum speedups abound.
We study some extensions of Grover's quantum searching algorithm. First, we generalize the Grover iteration in the light of a concept called amplitude amplification. Then, we show that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform approximate counting, which can be seen as an amplitude estimation process.
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.
In Chs. 2 and 3 we described several quantum algorithms that are exponentially faster than classical algorithms for the same problem, under some measure of complexity. Unfortunately, the algorithms of Ch. 2 apply to rather contrived problems that do not have known direct applications. We now describe a quantum algorithm that gives only a polynomial—more specifically, quadratic—speedup with respect to classical algorithms, but it applies to a large class of problems and finds many useful applications, both directly and as an algorithmic design tool. The algorithm is known as Grover’s search [89], and its generalization is known as amplitude amplification [43]. Amplitude amplification is widely used as part of many quantum algorithms. It also serves as the basis for amplitude estimation, a way of estimating probabilities quadratically faster than with classical Monte Carlo, and for quantum minimum finding. All these topics are discussed in this chapter.
Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction λ\lambdaλ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction, but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability, and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ\lambdaλ.
This chapter covers the quantum algorithmic primitives of amplitude amplification and amplitude estimation. Amplitude amplification is a generalization of Grover’s quantum algorithm for the unstructured search problem. Amplitude estimation can be understood in a similar framework, where it utilizes quantum phase estimation to estimate the value of the amplitude or probability associated with a quantum state. Both amplitude amplification and amplitude estimation provide a quadratic speedup over their classical counterparts, and feature prominently as an ingredient in many end-to-end algorithms.
Quantum algorithms enable Heisenberg-limited scaling across many tasks in sensing and learning, achieving precision ε\varepsilonε using O(1/ε)O(1/\varepsilon)O(1/ε) total evolution time. However, in Hamiltonian property testing, the best known algorithms have O(1/ε2)O(1/\varepsilon^2)O(1/ε2) total evolution time, unless it is possible to reverse\textit{reverse}reverse the flow of time under the Hamiltonian. In this work, we prove that time reversal is necessary to close this gap. We establish evolution time lower bounds for Hamiltonian certification and locality testing, for a continuous family of error metrics ranging from average- to worst-case. We also prove matching upper bounds for Hamiltonian certification in all cases, both with and without time reversal, showing that the advantage from time reversal persists. Our results follow from a new Fourier-analytic approach to quantum lower bounds in continuous time, which is sensitive to time reversal and applicable beyond Hamiltonian property testing. This gives an alternative proof that inverses are required for amplitude amplification and estimation, improving the lower bounds of Tang and Wright and extending to fractional queries. We also show that, for unstructured search, uncertainty in the phase of the oracle destroys the quantum speedup without inverse queries, even when the oracle is fixed and coherent. Our work rigorously establishes time reversal as a resource governing both the precision of quantum measurement and the robustness of quantum algorithms.