# Amplitude estimation

Amplitude 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.<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup><sup> • </sup><sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup>

| Key fact | Value |
|---|---|
| Task | Estimate amplitude a of a good subspace prepared by a unitary<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup> |
| Canonical query complexity | \( O(1/\varepsilon) \) versus classical \( O(1/\varepsilon^{2}) \)<sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup> |
| Canonical error bound | \|a − ã\| ≤ 2π√(a(1−a))/M + π²/M² with probability ≥ 8/π² ≈ 81%<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup><sup> • </sup><sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup> |
| ChebAE query count | versus \( \approx 9.93/\varepsilon \) for IQAE at ≥ 95% success<sup>[3](https://doi.org/10.22331/q-2023-03-02-937)</sup> |
| Practical barrier | Depolarizing noise creates error floors; quantum time advantage appears only near precision \( 10^{-13} \)–\( 10^{-11} \) on simulated hardware<sup>[4](https://www.mdpi.com/2624-960X/6/1/1)</sup> |
| Main applications | Monte Carlo integration, financial pricing and risk, machine learning, chemistry<sup>[5](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0)</sup> |

## How it works

The 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.<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup> 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 \( a = \sin^{2}(\theta_{a}) \).<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup> 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.<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup><sup> • </sup><sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup> 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 \( 8/\pi^{2} \).<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup>

Classical Monte Carlo estimation of a probability to precision ε needs \( M = O(1/\varepsilon^{2}) \) samples; amplitude estimation achieves \( M = O(1/\varepsilon) \).<sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup><sup> • </sup><sup>[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/)</sup> 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.<sup>[3](https://doi.org/10.22331/q-2023-03-02-937)</sup>

## How it is done

A practitioner runs four stages<sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup><sup> • </sup><sup>[4](https://www.mdpi.com/2624-960X/6/1/1)</sup>:

1. **State preparation.** Implement the unitary that encodes the quantity of interest, so that measuring the good outcome has probability a.
2. **Grover operator.** Construct the Grover operator Q used for amplification.
3. **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.
4. **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.<sup>[7](https://qiskit-community.github.io/qiskit-finance/tutorials/00_amplitude_estimation.html)</sup>

## Origin

The 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).<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup><sup> • </sup><sup>[3](https://doi.org/10.22331/q-2023-03-02-937)</sup> Their Est_Amp(A, χ, M) procedure outputs an estimate ã with 0 ≤ ã ≤ 1.<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup> The general concept of amplifying a subspace's amplitude is a generalization of the boosting technique in Grover's original search paper.<sup>[1](https://doi.org/10.48550/arxiv.quant-ph/0005055)</sup> 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.<sup>[8](https://doi.org/10.48550/arxiv.quant-ph/9805082)</sup>

## Variants

Starting around 2019, several alternatives to phase-estimation-based QAE appeared.<sup>[3](https://doi.org/10.22331/q-2023-03-02-937)</sup>

**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.<sup>[9](https://doi.org/10.1007/s11128-019-2565-2)</sup> 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.<sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup><sup> • </sup><sup>[10](https://link.springer.com/article/10.1007/s11128-021-03215-9)</sup>

**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.<sup>[11](https://doi.org/10.48550/arxiv.1912.05559)</sup><sup> • </sup><sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup>

**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.<sup>[12](https://doi.org/10.48550/arxiv.1908.10846)</sup><sup> • </sup><sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup>

**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.<sup>[3](https://doi.org/10.22331/q-2023-03-02-937)</sup>

**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 \( 1/\varepsilon \).<sup>[5](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0)</sup> 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.<sup>[13](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/)</sup> Variational quantum amplitude estimation, by Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch (Quantum, 2022), offers a variational alternative.<sup>[14](https://doi.org/10.22331/q-2022-03-17-670)</sup> Qiskit implements canonical, iterative, maximum-likelihood, and Faster Amplitude Estimation behind a common interface.<sup>[7](https://qiskit-community.github.io/qiskit-finance/tutorials/00_amplitude_estimation.html)</sup>

## Applications

Amplitude estimation is a central subroutine for applications in chemistry, finance, and machine learning.<sup>[5](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0)</sup> 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.<sup>[15](https://doi.org/10.1098/rspa.2015.0301)</sup> 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 \( O(\kappa) \).<sup>[13](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/)</sup>

## Limitations and alternatives

**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 \( 1/N \).<sup>[16](https://ar5iv.labs.arxiv.org/html/2006.14145)</sup> [Dephasing](https://www.edgepedia.org/dephasing) is less detrimental than amplitude damping, so runtimes should stay well below the \( T_{1} \) time.<sup>[16](https://ar5iv.labs.arxiv.org/html/2006.14145)</sup> 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.<sup>[10](https://link.springer.com/article/10.1007/s11128-021-03215-9)</sup> IQAE's iterative feedback is very sensitive to noise because errors propagate between iterations.<sup>[4](https://www.mdpi.com/2624-960X/6/1/1)</sup>

**Depth and overhead.** Canonical QAE runs the oracle circuit \( O(1/\varepsilon) \) times in series plus a QFT, a prohibitive depth for NISQ devices.<sup>[13](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/)</sup> 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.<sup>[17](https://arxiv.org/abs/2609.25992)</sup> The textbook variant is destructive, collapsing the state, and produces biased estimates.

**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.<sup>[2](https://www.nature.com/articles/s41534-021-00379-1)</sup> 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 \( 10^{-13} \)–\( 10^{-11} \).<sup>[4](https://www.mdpi.com/2624-960X/6/1/1)</sup>

**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.<sup>[17](https://arxiv.org/abs/2609.25992)</sup> 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.<sup>[17](https://arxiv.org/abs/2609.25992)</sup> 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.<sup>[4](https://www.mdpi.com/2624-960X/6/1/1)</sup>

## References

1. [Brassard, Gilles and colleagues (2000). Quantum Amplitude Amplification and Estimation. .](https://doi.org/10.48550/arxiv.quant-ph/0005055)
2. [Iterative quantum amplitude estimation (npj Quantum Information, 2021)](https://www.nature.com/articles/s41534-021-00379-1)
3. [Patrick Rall, Bryce Fuller (2023). Amplitude Estimation from Quantum Signal Processing. Quantum.](https://doi.org/10.22331/q-2023-03-02-937)
4. [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)
5. [Real quantum amplitude estimation (EPJ Quantum Technology, 2023)](https://link.springer.com/article/10.1140/epjqt/s40507-023-00159-0)
6. [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/)
7. [Quantum Amplitude Estimation, Qiskit Finance tutorial](https://qiskit-community.github.io/qiskit-finance/tutorials/00_amplitude_estimation.html)
8. [Brassard, Gilles, Hoyer, Peter, Tapp, Alain (1998). Quantum Counting. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.quant-ph/9805082)
9. [Yohichi Suzuki and colleagues (2020). Amplitude estimation without phase estimation. Quantum Information Processing.](https://doi.org/10.1007/s11128-019-2565-2)
10. [Amplitude estimation via maximum likelihood on noisy quantum computer (Quantum Information Processing)](https://link.springer.com/article/10.1007/s11128-021-03215-9)
11. [Grinko, Dmitry and colleagues (2019). Iterative Quantum Amplitude Estimation. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1912.05559)
12. [Aaronson, Scott, Rall, Patrick (2019). Quantum Approximate Counting, Simplified. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1908.10846)
13. [Low depth algorithms for quantum amplitude estimation (Quantum 6, 745, 2022)](https://quantum-journal.org/papers/q-2022-06-27-745/pdf/)
14. [Kirill Plekhanov and colleagues (2022). Variational quantum amplitude estimation. Quantum.](https://doi.org/10.22331/q-2022-03-17-670)
15. [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)
16. [Quantum Amplitude Estimation in the Presence of Noise](https://ar5iv.labs.arxiv.org/html/2006.14145)
17. [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)

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*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods*

*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*

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