Requires the amplified algorithm to decompose into stages carrying a per-branch stopping flag; a routine with a single uniform stopping time gains nothing from it. Its natural position is above a solver whose branches differ in cost — which is where the $\kappa$ dependence of a linear-system solve actually lives — rather than above a bare preparation.

The preparation unitary $A$ and its inverse, a reflection about $\lvert 0\rangle$, and a reflection marking the good subspace — the Grover operator $Q = -A S_0 A^{-1} S_\chi$ must be applicable at arbitrary powers. Individual variants additionally require a lower bound on $a$, or a per-branch stopping flag.

1. Routine with a good branch → Reliable routineamplify stage by stage, skip the stopped

   The routine is allowed to stop at different times on different branches, and the amplification follows it. At each stopping time the state decomposes as $|\psi_{i}\rangle = \alpha_{i,0}|0\rangle|\psi_{i,0}\rangle + \alpha_{i,1}|1\rangle|\psi_{i,1}\rangle + \alpha_{i,2}|2\rangle|\psi_{i,2}\rangle$, flagging each branch stopped-bad, stopped-good, or still running. assumption: a branch that has stopped must not move again — $P_{H_{i}}|\psi_{i+1,0}\rangle = |\psi_{i,0}\rangle$ and $P_{H_{i}}|\psi_{i+1,1}\rangle = |\psi_{i,1}\rangle$, so that "the part of the state where the computation stopped at time $t_{i}$ should not change after that". A nested sequence then amplifies each stage separately, approximation: only far enough to reach $r_{i} \ge 1/9m$ rather than $\Omega(1)$, which is what keeps $m$ nested amplifications from compounding, giving $T_{i} \le (1 + \frac{1}{3m-1})\sqrt{r_{i}/r_{i}'}(T_{i-1} + 2^{i-1})$ and a total $O\!\left(T_{max}\sqrt{\log T_{max}} + \frac{T_{av}}{\sqrt{p_{succ}}}\log^{1.5}T_{max}\right)$ against $O(T_{max}/\sqrt{p_{succ}})$ for amplifying the routine as one block, where $T_{av} = \sqrt{\sum_{i}p_{i}t_{i}^{2}}$.

   approximationassumption

A routine that produces the wanted branch with a stated failure probability, together with the query count and the maximum sequential depth consumed.

given  the routine to be amplified, decomposed into stages that carry a
       per-branch stopping flag; and A, its inverse, the reflection S_0
       about |0> and the reflection S_chi marking the good subspace

# a routine with a single uniform stopping time gains nothing from this

amplify in nested stages, so that branches which finish early are not
    charged at the worst-case depth

# how the stages nest is not stated on this record and is not invented here

return the wanted branch with a stated failure probability, together with
    the query count and the maximum sequential depth consumed

# the depth is the line to read: it is the resource a coherence-limited
# device actually runs out of first

# its natural position is above a solver whose branches differ in cost --
# which is where the kappa dependence of a linear-system solve actually
# lives -- rather than above a bare preparation

# Ambainis generalizes amplitude amplification to the case when parts of the
# algorithm being amplified stop at different times, improving the Harrow
# et al. linear-systems running time from O(kappa^2 log N) to
# O(kappa log^3 kappa log N), kappa the condition number of the system

# Chakraborty, Gilyen and Jeffery build variable-time amplitude estimation
# directly on this technique

Ambainis states his generalization of amplitude amplification "to the case when parts of the quantum algorithm that is being amplified stop at different times" improves the running time of the Harrow et al. linear-systems algorithm "from O(kappa^2 log N) to O(kappa log^3 kappa log N) where kappa is the condition number of the system of equations". Chakraborty, Gilyén and Jeffery build on it directly: they "develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique".