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arXiv:2512.15352

Amplitude-amplified coherence detection and estimation

Rhea Alexander, Michalis Skotiniotis, Daniel Manzano

quant-phSubmitted Dec 17, 2025• Announced Dec 18, 2025• 0 citations
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Abstract

The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state $|ψ\rangle$. Having access to $m$ copies of an unknown pure state $|ψ\rangle$ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success $\ge 2/3$ is $Θ(c(|ψ\rangle)^{-1})$, where $c(|ψ\rangle)$ is the geometric measure of coherence of $|ψ\rangle$. However, assuming access to the unitary $U_ψ$ which prepares the unknown state $|ψ\rangle$, and its inverse $U_ψ^\dagger$, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number $O(c(|ψ\rangle)^{-1/2})$ of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error $\varepsilon$ with a sample complexity that scales as $O(1/\varepsilon)$ as compared to the $O(1/\varepsilon^2)$ sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.

Comments
11 pages, 2 figures. Comments welcome!

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Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.
Optimal single-copy estimation of quantum state moments: why Tr⁡(ρ3)\operatorname{Tr}(ρ^3)Tr(ρ3) and Tr⁡(ρ4)\operatorname{Tr}(ρ^4)Tr(ρ4) are equally hard
Estimating nonlinear properties of an unknown quantum state with restrictive experimental accessibility is a fundamental problem in quantum learning. We study the sample complexity of estimating the state moments Tr⁡(ρt)\operatorname{Tr}(ρ^t)Tr(ρt), allowing arbitrary adaptive single-copy measurements. While purity estimation has optimal sample complexity Θ(d)Θ(\sqrt d)Θ(d​) at constant additive error, the optimal dimension dependence for higher moments has remained unclear. We resolve this question for every fixed integer t≥2t\ge 2t≥2: for any sufficiently small constant additive error, the optimal sample complexity is Θ ⁣(d⌈log⁡2t⌉1+⌈log⁡2t⌉) Θ\!\left(d^{\frac{\lceil\log_2 t\rceil}{1+\lceil\log_2 t\rceil}}\right) Θ(d1+⌈log2​t⌉⌈log2​t⌉​) with constant success probability. The complexity thus follows a dyadic hierarchy: for each integer h≥1h\geq1h≥1, all moment orders 2h−1<t≤2h2^{h-1}<t\leq2^h2h−1<t≤2h share the same exponent. In particular, the third and fourth moments both require Θ(d2/3)Θ(d^{2/3})Θ(d2/3) copies. Our results also apply to multistate overlaps, weighted and partial-transpose moments, fixed-rank spectrum estimation, and distributed estimation of nonlinear quantities. We also establish an Ω(d1−1/t)Ω(d^{1-1/t})Ω(d1−1/t) lower bound for arbitrary nonadaptive single-copy protocols, matching the known upper bound and demonstrating an advantage from adaptivity for every t≥4t\geq4t≥4. Our upper bound uses adaptive state filtering to reduce higher-order moment estimation to lower-order moment estimation in a smaller state space. For the lower bound, we directly compare moment-matched ensembles through a smooth interpolation, bypassing the maximally mixed state as an intermediate hypothesis and yielding matching bounds beyond d\sqrt dd​.
Quantum Amplitude Estimation (QAE) -- a technique by which the amplitude of a given quantum state can be estimated with quadratically fewer queries than by standard sampling -- is a key sub-routine in several important quantum algorithms, including Grover search and Quantum Monte-Carlo methods. An obstacle to implementing QAE in near-term noisy intermediate-scale quantum (NISQ) devices has been the need to perform Quantum Phase Estimation (QPE) -- a costly procedure -- as a sub-routine. This impediment was lifted with various QPE-free methods of QAE, wherein Grover queries of varying depths / powers (often according to a "schedule") are followed immediately by measurements and classical post-processing techniques like maximum likelihood estimation (MLE). Existing analyses as to the optimality of various query schedules in these QPE-free QAE schemes have hitherto assumed noise-free systems. In this work, we analyse QPE-free QAE under common noise models that may afflict NISQ devices and report on the optimality of various query schedules in the noisy regime. We demonstrate that, given an accurate noise characterization of one's system, one must choose a schedule that balances the trade-off between the greater ideal performance achieved by higher-depth circuits, and the correspondingly greater accumulation of noise-induced error.
Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given mmm Bernoulli distributions with unknown biases p1,…,pmp_1,\dots,p_mp1​,…,pm​ and a gap parameter ΔΔΔ, the goal is to estimate the fraction ρρρ of \emph{positive} distributions (pi≥1/2+Δp_i\ge1/2+Δpi​≥1/2+Δ) to within additive error εεε. We prove an upper bound of O~ ⁣(ρΔε+1Δε)\tilde{O}\!\big(\frac{\sqrtρ}{Δε}+\frac{1}{Δ\sqrtε}\big)O~(Δερ​​+Δε​1​) queries, achieving a quadratic speedup over the classical sample complexity. % of Θ(ρ/Δ2ε2)Θ(ρ/Δ^2ε^2)Θ(ρ/Δ2ε2). Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of Ω(ρ/(Δε))Ω(\sqrtρ/(Δε))Ω(ρ​/(Δε)) via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap Δ=Θ(1)Δ=Θ(1)Δ=Θ(1), which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to O~(ρε+1ε)\tilde{O}\big(\frac{\sqrtρ}ε+\frac{1}{\sqrtε}\big)O~(ερ​​+ε​1​) and Ω(ρε)Ω(\frac{\sqrtρ}ε)Ω(ερ​​), thereby characterizing the query complexity of quantum counting with bounded-error oracles.