{ "id": "epamqz726v", "slug": "amplitude-estimation", "title": "Amplitude estimation", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.algorithms" } ], "geo": [ { "id": "geo.nongeo.t1946.technology.computing.ai.algorithms", "label": "Non-geographic · 1946 to 2000: Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.algorithms", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing" }, { "id": "geo.nongeo.t1946.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai" }, { "id": "geo.nongeo.t1946.technology.computing.ai.algorithms", "label": "Algorithms and computational methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.algorithms" } ] } ], "excerpt": "Amplitude estimation, also known as quantum amplitude estimation, is a quantum algorithm that estimates the probability of a marked subspace, using O(1/ε) queries versus classical O(1/ε²) sampling.", "snippet": "Amplitude estimation, also known as quantum amplitude estimation, is a quantum algorithm that estimates the probability of a marked subspace, using O(1/ε) queries versus classical O(1/ε²) sampling.", "node": "technology.computing.ai.algorithms", "markdown": "# Amplitude estimation\n\nAmplitude estimation is a quantum algorithm that estimates the probability a of a marked (\"good\") subspace in a [quantum superposition](https://www.edgepedia.org/quantum-superposition), whose state-vector amplitude has magnitude √a, achieving a quadratic speedup over classical [Monte Carlo estimation](https://www.edgepedia.org/monte-carlo-estimation) of the same quantity. Given a state-preparation unitary and a predicate separating good from bad outcomes, the algorithm returns an estimate of the good amplitude a using \\( O(1/\\varepsilon) \\) oracle queries to reach additive error ε, where classical sampling needs \\( O(1/\\varepsilon^{2}) \\) repetitions.[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) • [2](https://www.nature.com/articles/s41534-021-00379-1)\n\n| Key fact | Value |\n|---|---|\n| Task | Estimate amplitude a of a good subspace prepared by a unitary[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) |\n| Canonical query complexity | \\( O(1/\\varepsilon) \\) versus classical \\( O(1/\\varepsilon^{2}) \\)[2](https://www.nature.com/articles/s41534-021-00379-1) |\n| Canonical error bound | \\|a − ã\\| ≤ 2π√(a(1−a))/M + π²/M² with probability ≥ 8/π² ≈ 81%[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) • [2](https://www.nature.com/articles/s41534-021-00379-1) |\n| ChebAE query count | versus \\( \\approx 9.93/\\varepsilon \\) for IQAE at ≥ 95% success[3](https://doi.org/10.22331/q-2023-03-02-937) |\n| Practical barrier | Depolarizing noise creates error floors; quantum time advantage appears only near precision \\( 10^{-13} \\)–\\( 10^{-11} \\) on simulated hardware[4](https://www.mdpi.com/2624-960X/6/1/1) |\n| Main applications | Monte Carlo integration, financial pricing and risk, machine learning, chemistry[5](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0) |\n\n## How it works\n\nThe setup is a unitary that prepares a superposition with amplitude a on the good part of a state space, where a Boolean predicate partitions outcomes into good and bad elements.[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) The Grover operator Q acts on the two-dimensional subspace spanned by the good and bad components with eigenvalues \\( \\lambda_{\\pm} = e^{\\pm i 2\\theta_{a}} \\), where \\( a = \\sin^{2}(\\theta_{a}) \\).[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) Because the amplitude is encoded in this eigenphase, any eigenvalue-estimation primitive recovers a. [Quantum phase estimation](https://www.edgepedia.org/quantum-phase-estimation) on controlled powers of Q converts the phase into a measurable integer, and the amplitude follows from the sine-squared relation.[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) • [2](https://www.nature.com/articles/s41534-021-00379-1) With M Grover iterations, the estimate lands within \\( 2\\pi\\sqrt{a(1-a)}/M + \\pi^{2}/M^{2} \\) of the true value with probability at least \\( 8/\\pi^{2} \\).[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)\n\nClassical Monte Carlo estimation of a probability to precision ε needs \\( M = O(1/\\varepsilon^{2}) \\) samples; amplitude estimation achieves \\( M = O(1/\\varepsilon) \\).[2](https://www.nature.com/articles/s41534-021-00379-1) • [6](https://lucidalu.github.io/Quantum-algorithms-A-survey-of-applications-and-end-to-end-complexities/quantum-algorithmic-primitives/amplitude-amplification-and-estimation/amplitude-estimation/) The speedup is never more than quadratic: when the amplitude is near 0 or 1 \\( p \\approx O(\\varepsilon) \\) or \\( p \\approx 1 - O(\\varepsilon) \\)), the classical approach also achieves \\( M \\sim 1/\\varepsilon \\). Asymptotic query complexity is Θ(ε⁻¹), so constant-factor improvements are what distinguish variants in practice.[3](https://doi.org/10.22331/q-2023-03-02-937)\n\n## How it is done\n\nA practitioner runs four stages[2](https://www.nature.com/articles/s41534-021-00379-1) • [4](https://www.mdpi.com/2624-960X/6/1/1):\n\n1. **State preparation.** Implement the unitary that encodes the quantity of interest, so that measuring the good outcome has probability a.\n2. **Grover operator.** Construct the Grover operator Q used for amplification.\n3. **Controlled powers and inverse QFT.** In canonical QAE, m evaluation qubits encode a phase estimate with \\( M = 2^{m} \\) possible outcomes, and controlled powers of Q implement the phase estimation; an inverse quantum [Fourier transform](https://www.edgepedia.org/fourier-transform) (QFT) is then applied to the ancillas.\n4. **Readout.** Measure the ancillas to obtain an integer y ∈ {0, …, M − 1}, map it to \\( \\tilde{\\theta}_{a} = y \\cdot \\pi / M \\), and output \\( \\tilde{a} = \\sin^{2}(\\tilde{\\theta}_{a}) \\). The canonical algorithm is restricted to a discrete grid of amplitude values set by the number of evaluation qubits; interpolated maximum-likelihood post-processing recovers intermediate values.[7](https://qiskit-community.github.io/qiskit-finance/tutorials/00_amplitude_estimation.html)\n\n## Origin\n\nThe first amplitude estimation algorithm was given by [Gilles Brassard](https://www.edgepedia.org/gilles-brassard) and colleagues in the paper *Quantum Amplitude Amplification and Estimation* (arXiv, 2000; also published in an AMS Contemporary Mathematics volume).[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) • [3](https://doi.org/10.22331/q-2023-03-02-937) Their Est_Amp(A, χ, M) procedure outputs an estimate ã with 0 ≤ ã ≤ 1.[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) The general concept of amplifying a subspace's amplitude is a generalization of the boosting technique in Grover's original search paper.[1](https://doi.org/10.48550/arxiv.quant-ph/0005055) An immediate precursor was *Quantum Counting*, which combined ideas from Grover's and Shor's algorithms to perform approximate counting, describable as an amplitude estimation process.[8](https://doi.org/10.48550/arxiv.quant-ph/9805082)\n\n## Variants\n\nStarting around 2019, several alternatives to phase-estimation-based QAE appeared.[3](https://doi.org/10.22331/q-2023-03-02-937)\n\n**MLAE.** Maximum Likelihood Amplitude Estimation, introduced by Yohichi Suzuki and colleagues in *Amplitude estimation without phase estimation* (Quantum Information Processing, 2020), replaces phase estimation with Grover iterations at several depths plus classical maximum-likelihood estimation.[9](https://doi.org/10.1007/s11128-019-2565-2) It needs neither ancilla qubits nor controlled operations, drastically reducing gate counts, and is the only variant allowing parallel execution of queries; its [Fisher information](https://www.edgepedia.org/fisher-information) scales as \\( O(N_{\\mathrm{shots}} \\cdot M^{2}) \\), giving an error lower bound \\( \\Omega(1/(\\sqrt{N_{\\mathrm{shots}}} \\cdot M)) \\) with no upper bound provided.[2](https://www.nature.com/articles/s41534-021-00379-1) • [10](https://link.springer.com/article/10.1007/s11128-021-03215-9)\n\n**IQAE.** Iterative Quantum Amplitude Estimation, by Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner (2019), uses only Grover iterations with a classical feedback loop, provably achieving the quadratic speedup up to a double-logarithmic factor while reducing required qubits and gates.[11](https://doi.org/10.48550/arxiv.1912.05559) • [2](https://www.nature.com/articles/s41534-021-00379-1)\n\n**QAES.** Quantum Approximate Counting, Simplified, by [Scott Aaronson](https://www.edgepedia.org/scott-aaronson) and Patrick Rall (2019), was the first variant rigorously proven to achieve quadratic speedup without QPE, with optimal asymptotic query complexity, but with very large constants: empirically about \\( 10^{8} \\) times more oracle queries than IQAE, worse than classical [Monte Carlo](https://www.edgepedia.org/monte-carlo) at the tested accuracies.[12](https://doi.org/10.48550/arxiv.1908.10846) • [2](https://www.nature.com/articles/s41534-021-00379-1)\n\n**ChebAE.** Amplitude Estimation from Quantum Signal Processing, by Patrick Rall and Bryce Fuller (Quantum, 2023), samples [Chebyshev polynomials](https://www.edgepedia.org/chebyshev-polynomials) and reaches error ε with ≥ 95% success in about \\( 4.66/\\varepsilon \\) queries, 45–65% of IQAE's count, and supports non-destructive, unbiased, and depth–repetition tradeoffs.[3](https://doi.org/10.22331/q-2023-03-02-937)\n\n**Others.** Real Quantum Amplitude Estimation (RQAE) extends QAE to estimate the sign of a real amplitude, with an adjustable amplification policy and oracle calls growing approximately as \\( 1/\\varepsilon \\).[5](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0) Power-law amplitude estimation uses schedules with circuit depth \\( O(1/\\varepsilon^{1-\\beta}) \\) and total oracle calls \\( O(1/\\varepsilon^{1+\\beta}) \\), trading speedup for shallower circuits.[13](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/) Variational quantum amplitude estimation, by Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch (Quantum, 2022), offers a variational alternative.[14](https://doi.org/10.22331/q-2022-03-17-670) Qiskit implements canonical, iterative, maximum-likelihood, and Faster Amplitude Estimation behind a common interface.[7](https://qiskit-community.github.io/qiskit-finance/tutorials/00_amplitude_estimation.html)\n\n## Applications\n\nAmplitude estimation is a central subroutine for applications in chemistry, finance, and machine learning.[5](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0) Ashley Montanaro's *Quantum speedup of Monte Carlo methods* (Proceedings of the Royal Society A, 2015) extended the quadratic speedup to a broad class of Monte Carlo estimators, underpinning financial asset pricing and risk applications.[15](https://doi.org/10.1098/rspa.2015.0301) Specific uses include overlap estimation for quantum chemistry observables and inner-product estimation for quantum machine learning. Amplitude estimation variants also reduce the condition-number dependence in quantum linear system solvers from \\( O(\\kappa^{2}) \\) to \\( O(\\kappa) \\).[13](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/)\n\n## Limitations and alternatives\n\n**Noise.** Under depolarizing noise, the exponential Grover schedule underlying canonical QAE requires exponentially many samples to beat the noise level, making Heisenberg scaling unlikely on near-term hardware; a linear schedule yields error \\( \\sim 1/N^{3/4} \\), between the classical \\( 1/N^{1/2} \\) and Heisenberg \\( 1/N \\).[16](https://ar5iv.labs.arxiv.org/html/2006.14145) [Dephasing](https://www.edgepedia.org/dephasing) is less detrimental than amplitude damping, so runtimes should stay well below the \\( T_{1} \\) time.[16](https://ar5iv.labs.arxiv.org/html/2006.14145) For MLAE, reaching ε = 10⁻⁴ with O(1/ε) queries requires depolarizing noise κ ≲ 10⁻³, and there exist anomalous target values where the Fisher information matrix degenerates so that more amplification does not help.[10](https://link.springer.com/article/10.1007/s11128-021-03215-9) IQAE's iterative feedback is very sensitive to noise because errors propagate between iterations.[4](https://www.mdpi.com/2624-960X/6/1/1)\n\n**Depth and overhead.** Canonical QAE runs the oracle circuit \\( O(1/\\varepsilon) \\) times in series plus a QFT, a prohibitive depth for NISQ devices.[13](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/) State preparation matters: if the initial state is expensive to prepare, non-destructive variants are preferred, and preparation via the Grover–Rudolph method can erase the asymptotic speedup.[17](https://arxiv.org/abs/2609.25992) The textbook variant is destructive, collapsing the state, and produces biased estimates.\n\n**Comparisons.** In an empirical comparison estimating \\( a = 1/2 \\) at 95% confidence with 100 shots, IQAE outperformed MLAE, canonical QAE, and classical Monte Carlo; applying QPE in this setting gave no advantage and only increased complexity.[2](https://www.nature.com/articles/s41534-021-00379-1) The wall-clock advantage is harder to reach: in one benchmark, generating a Monte Carlo sample on a laptop took four orders of magnitude less time than MLQAE on a simulator, and a quantum time advantage appeared only around precision \\( 10^{-13} \\)–\\( 10^{-11} \\).[4](https://www.mdpi.com/2624-960X/6/1/1)\n\n**Hardware results.** A 2025/2026 study ran 60 hardware trials of an MLAE Monte Carlo pipeline on IonQ Forte-1 plus 100 trials on noise models.[17](https://arxiv.org/abs/2609.25992) In the noiseless limit the pipeline attained error scaling \\( \\varepsilon \\propto N^{-0.88} \\), close to Heisenberg scaling, but every noisy backend saturated at an error floor of \\( (2\\text{–}5) \\times 10^{-2} \\), so deeper amplification beyond a shallow optimal depth gives no sustained accuracy gain.[17](https://arxiv.org/abs/2609.25992) Earlier, an 11-qubit trapped-ion benchmark found MLQAE gave the best trade-off between circuit length and precision, achieving average error below \\( 10^{-3} \\) from just two qubits.[4](https://www.mdpi.com/2624-960X/6/1/1)\n\n## References\n\n1. [Brassard, Gilles and colleagues (2000). Quantum Amplitude Amplification and Estimation. .](https://doi.org/10.48550/arxiv.quant-ph/0005055)\n2. [Iterative quantum amplitude estimation (npj Quantum Information, 2021)](https://www.nature.com/articles/s41534-021-00379-1)\n3. [Patrick Rall, Bryce Fuller (2023). Amplitude Estimation from Quantum Signal Processing. Quantum.](https://doi.org/10.22331/q-2023-03-02-937)\n4. [The Quantum Amplitude Estimation Algorithms on Near-Term Devices: A Practical Guide (MDPI Quantum Reports, 2024)](https://www.mdpi.com/2624-960X/6/1/1)\n5. [Real quantum amplitude estimation (EPJ Quantum Technology, 2023)](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0)\n6. [Amplitude estimation, Quantum algorithms: A survey of applications and end-to-end complexities](https://lucidalu.github.io/Quantum-algorithms-A-survey-of-applications-and-end-to-end-complexities/quantum-algorithmic-primitives/amplitude-amplification-and-estimation/amplitude-estimation/)\n7. [Quantum Amplitude Estimation, Qiskit Finance tutorial](https://qiskit-community.github.io/qiskit-finance/tutorials/00_amplitude_estimation.html)\n8. [Brassard, Gilles, Hoyer, Peter, Tapp, Alain (1998). Quantum Counting. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.quant-ph/9805082)\n9. [Yohichi Suzuki and colleagues (2020). Amplitude estimation without phase estimation. Quantum Information Processing.](https://doi.org/10.1007/s11128-019-2565-2)\n10. [Amplitude estimation via maximum likelihood on noisy quantum computer (Quantum Information Processing)](https://link.springer.com/article/10.1007/s11128-021-03215-9)\n11. [Grinko, Dmitry and colleagues (2019). Iterative Quantum Amplitude Estimation. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1912.05559)\n12. [Aaronson, Scott, Rall, Patrick (2019). Quantum Approximate Counting, Simplified. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1908.10846)\n13. [Low depth algorithms for quantum amplitude estimation (Quantum 6, 745, 2022)](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/)\n14. [Kirill Plekhanov and colleagues (2022). Variational quantum amplitude estimation. Quantum.](https://doi.org/10.22331/q-2022-03-17-670)\n15. [Ashley Montanaro (2015). Quantum speedup of Monte Carlo methods. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences.](https://doi.org/10.1098/rspa.2015.0301)\n16. [Quantum Amplitude Estimation in the Presence of Noise](https://ar5iv.labs.arxiv.org/html/2006.14145)\n17. [Maximum-Likelihood Amplitude Estimation for Quantum Monte Carlo Integration on Trapped-Ion Hardware: Convergence, Noise-Floor Saturation, Depth-Dependent Bias](https://arxiv.org/abs/2609.25992)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/amplitude-estimation", "markdown_url": "https://www.edgepedia.org/amplitude-estimation.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Amplitude estimation\", Edgepedia (EdgeChat), https://www.edgepedia.org/amplitude-estimation. Edgepedia Community License 1.0.", "credit_md": "\"[Amplitude estimation](https://www.edgepedia.org/amplitude-estimation)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/amplitude-estimation](https://www.edgepedia.org/amplitude-estimation). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Amplitude estimation\", Edgepedia (EdgeChat), https://www.edgepedia.org/amplitude-estimation. Edgepedia Community License 1.0.", "speakable": "Amplitude estimation, also known as quantum amplitude estimation, is a quantum algorithm that estimates the probability of a marked subspace, using O queries versus classical O sampling." }