Weakly Measured Loops for Quantum Amplitude Amplification: Oracle Savings and Adaptive Search with Unknown Target Probability
Abstract
We present a family of quantum amplitude amplification algorithms that interleave ordinary Grover rotations with tunable weak measurements in loops. A successful measurement yields the target state and stops the algorithm, while after a failure, the loop resumes from the post-measurement state without restarting. We express the leading costs in expected Grover iterations as p → 0. For a known target probability p, an exact weak measurement-conditioned loop for states near the angle π/4 uses (π/8+1/4+o(1))/√p iterations, improving on the standard Grover constant π/4 and on optimised restart Grover, and matching the corresponding infimum in our continuous-angle analysis. When only a lower bound 0 < p0 ≤ p < 1/2 is available, a discrete Lyapunov equation gives the exact expected oracle cost for every fixed measurement strength and a closed-form optimal strength. Using the strength selected from p0, the expected number of Grover iterations is at most 1/2√2(1/√p+1/√(p0)), where the expected cost decreases as the actual p increases, and gives the leading constant 1/√2 at the promise boundary p = p0. For completely unknown p, we identify measurement strength κ(t)=Θ(1/t) as the critical scale within the regular schedules considered here and analyse κb(t)=min{1/2,b/t}. For each fixed b > 2, we rigorously derive a closed-form Gamma-function expression C(b) giving (C(b)/2+o(1))/√p expected iterations. Numerical minimisation of this explicit expression gives b ≈ 5.2 and C(b)/2 ≈ 1.01. The results show that weak measurements preserve the Θ(1/√p) search scale while adding an explicit fixed-point control mechanism and provable oracle savings.
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BibTeX
@misc{fang2026weakly,
title = {{Weakly Measured Loops for Quantum Amplitude Amplification: Oracle Savings and Adaptive Search with Unknown Target Probability}},
author = {Wang Fang and Chris Heunen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39621},
eprint = {2609.39621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39621},
}
RIS
TY - UNPB
TI - Weakly Measured Loops for Quantum Amplitude Amplification: Oracle Savings and Adaptive Search with Unknown Target Probability
AU - Wang Fang
AU - Chris Heunen
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39621
UR - https://arxiv.org/abs/2609.39621
AB - We present a family of quantum amplitude amplification algorithms that interleave ordinary Grover rotations with tunable weak measurements in loops. A successful measurement yields the target state and stops the algorithm, while after a failure, the loop resumes from the post-measurement state without restarting. We express the leading costs in expected Grover iterations as p → 0. For a known target probability p, an exact weak measurement-conditioned loop for states near the angle π/4 uses (π/8+1/4+o(1))/√p iterations, improving on the standard Grover constant π/4 and on optimised restart Grover, and matching the corresponding infimum in our continuous-angle analysis. When only a lower bound 0 < p_0 ≤ p < 1/2 is available, a discrete Lyapunov equation gives the exact expected oracle cost for every fixed measurement strength and a closed-form optimal strength. Using the strength selected from p_0, the expected number of Grover iterations is at most 1/2√2(1/√p+1/√(p_0)), where the expected cost decreases as the actual p increases, and gives the leading constant 1/√2 at the promise boundary p = p_0. For completely unknown p, we identify measurement strength κ(t)=Θ(1/t) as the critical scale within the regular schedules considered here and analyse κ_b(t)=min{1/2,b/t}. For each fixed b > 2, we rigorously derive a closed-form Gamma-function expression C(b) giving (C(b)/2+o(1))/√p expected iterations. Numerical minimisation of this explicit expression gives b ≈ 5.2 and C(b)/2 ≈ 1.01. The results show that weak measurements preserve the Θ(1/√p) search scale while adding an explicit fixed-point control mechanism and provable oracle savings.
ER -