Needs a reliable lower bound on the fraction $\lambda$ of the initial state made up of target states, rather than $\lambda$ itself. Grover's earlier $\pi/3$ construction achieves fixed-point convergence but, as Yoder, Low and Chuang put it, such algorithms "lose the very quadratic advantage that makes Grover's algorithm so appealing". Their own construction is described by them as "the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup", and it carries an adjustable bound on the failure probability.

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 routineset each reflection phase from Chebyshev

   Grover's fixed $\pm\pi$ reflections are replaced by $l = (L-1)/2$ tuned generalized iterates — the paper constructs $\mathcal{S}_{L}$ "for any odd integer $L \ge 1$", with $L = 2l+1$, so an even $L$ has no construction at all — $\mathcal{S}_{L} = \prod_{j=1}^{l}G(\alpha_{j}, \beta_{j})$ with $G(\alpha, \beta) = -S_{s}(\alpha)S_{t}(\beta)$, whose phases are read off Chebyshev polynomials: $\alpha_{j} = -\beta_{l-j+1} = 2\cot^{-1}\!\left(\tan(2\pi j/L)\sqrt{1-\gamma^{2}}\right)$ with $\gamma^{-1} = T_{1/L}(1/\delta)$. assumption: the whole construction lives in "the two-dimensional subspace $\mathcal{T}$ spanned by $|s\rangle$ and $|T\rangle$", so the generalized reflections built from $A$, $A^{\dagger}$ and $U$ must preserve it. The resulting overlap is exactly $P_{L} = |\langle T|\mathcal{S}_{L}|s\rangle|^{2} = 1 - \delta^{2}T_{L}\!\left(T_{1/L}(1/\delta)\sqrt{1-\lambda}\right)^{2}$, so the failure probability never exceeds $\delta^{2}$ however large $\lambda$ turns out to be. approximation: the exact threshold $w = 1 - T_{1/L}(1/\delta)^{-2}$ is approximated for large $L$ and small $\delta$ as $w \approx (\log(2/\delta)/L)^{2}$, and that is what yields the clean sufficient query count $L \ge \log(2/\delta)/\sqrt{\lambda}$.

   approximationassumption

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

• Prepare an input state

  Map $|0\ldots0⟩$ to a state whose amplitudes are proportional to a specified vector $b$, to within $\varepsilon$. The cost is set by which description of $b$ you hold, not by the algorithm that consumes it.

  This method asks the slot for "the state preparation $A$ and oracle $U$", and runs "a quantum circuit $\mathcal{S}_{L}$ consisting of $A, A^{\dagger}, U$, and efficiently implementable $n$-qubit gates". $A$ enters only through the reflection about the prepared state, $S_s(\alpha) = I - (1-e^{-i\alpha})|s\rangle\langle s|$, whose circuit (Fig. 2) conjugates a multiply-controlled phase on one reusable ancilla with $A^{\dagger}$ then $A$: forward once, inverse once, per iterate $G(\alpha_j,\beta_j) = -S_s(\alpha_j)S_t(\beta_j)$. The target-side reflection $S_t(\beta) = I - (1-e^{i\beta})|t\rangle\langle t|$ needs no $A$ — it is two calls to $U$ alone. So a length-$L$ sequence $\mathcal{S}_{L} = \prod_{j=1}^{l}G(\alpha_j,\beta_j)$ of $l = (L-1)/2$ iterates, which the paper prices at $L-1 = 2l$ queries to $U$, spends $l$ forward and $l$ inverse calls to $A$. assumption: $A$ must be supplied as a circuit that can be run in reverse; a source that only emits $|s\rangle$ cannot build $S_s(\alpha)$.

  assumption

given  the preparation unitary A and its inverse, a reflection S_0 about |0>,
       and a reflection S_chi marking the good subspace, so that
       Q = - A S_0 A^{-1} S_chi can be applied at arbitrary powers;
       and a reliable lower bound on the fraction lambda of the initial
       state made up of target states

# a lower bound, not lambda itself -- that is all this variant of the layer
# asks for. A is the state-preparation step below this method.

choose the bound on the failure probability -- it is adjustable

replace the +/-1 reflections of textbook amplification with tuned phase
    shifts, and run the resulting iteration for the number of oracle
    queries fixed by the lower bound on lambda and the failure tolerance

# so the iteration converges on the target instead of rotating past it;
# rotating past it is what happens when the success amplitude is known only
# as a lower bound

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

# neither the tuned phase shifts nor the query count as a function of
# lambda's lower bound and the failure tolerance is quoted on this record:
# the query count is in the paper's full text, and the record stops at the
# abstract on purpose

# the optimality in their title has exactly this shape: for a given number
# of oracle queries, the adjustable failure bound is guaranteed over "the
# broadest possible range" of lambda

# do not build this: Grover's earlier pi/3 construction also achieves
# fixed-point convergence, but such algorithms "lose the very quadratic
# advantage that makes Grover's algorithm so appealing" -- the construction
# above is described by its authors as the first to reach fixed-point
# behavior without sacrificing the quantum speedup

The abstract states no query-count formula; its cost claims are comparative. The construction keeps the quadratic quantum speedup that fixed-point predecessors gave up, and the optimality named in the title is of exactly this shape: for a given number of oracle queries, the adjustable failure-probability bound is guaranteed, as Yoder, Low and Chuang put it, over "the broadest possible range" of the target fraction $\lambda$. The explicit query count as a function of $\lambda$'s lower bound and the failure tolerance is in the paper's full text and is not quoted here.

• PennyLane's `qml.AmplitudeAmplification` template, `fixed_point=True`

  • Fixed-point quantum search with an optimal number of queries

    Theodore J. Yoder, Guang Hao Low, Isaac L. Chuang · 2014

  About

  PennyLane's single `AmplitudeAmplification` operation implements textbook amplitude amplification by default, and its class docstring states "the implementation of the algorithm is based on" Brassard, Hoyer, Mosca and Tapp, then adds that "the template also unlocks advanced techniques such as fixed-point quantum search [arXiv:1409.3305] ... by reflecting on a subset of wires" -- the same Yoder-Low-Chuang paper this method's own citations name. The fixed-point branch is a mode of the one class rather than a separate one: the constructor takes a boolean `fixed_point` (default `False`) and a `work_wire` the docstring calls required "if fixed_point == True", which the code enforces by raising `WireError("work_wire must be specified if fixed_point == True.")` when it is missing.

  Methods

  `_get_fixed_point_angles(iters, p_min)` sets `delta = np.sqrt(1 - p_min)` and `gamma = np.cos(np.arccos(1 / delta, dtype=np.complex128) / iters, dtype=np.complex128) ** -1`, a complex-valued route to the same fractional-Chebyshev quantity this record's own hop calls $\gamma^{-1} = T_{1/L}(1/\delta)$: because $\cos(ix) = \cosh(x)$, taking `arccos` of the complex `1/delta` and dividing by `iters` computes what `arccosh(1/delta)/iters` would compute directly. It returns `alphas = [np.real(2 * np.arctan(1 / (np.tan(2 * np.pi * j / iters) * np.sqrt(1 - gamma**2)))) for j in range(1, iters // 2 + 1)]` and `betas = [-alphas[-j] for j in range(1, iters // 2 + 1)]` -- the palindromic pairing this record states as $\alpha_j = -\beta_{l-j+1}$. `compute_decomposition` then builds, for each of `iters // 2` generalized iterates, a Hadamard-controlled(`O`)-Hadamard-`PhaseShift(betas[it])`-Hadamard-controlled(`O`)-Hadamard sandwich on the caller-supplied `work_wire` -- an ancilla construction of the target-side reflection from calls to the oracle `O` alone -- followed by `Reflection(U, -alphas[it], reflection_wires=reflection_wires)` for the state-side reflection built from the preparation operator `U`. The `fixed_point=False` branch it falls back to instead applies `O` directly and `Reflection(U, np.pi, ...)`, the textbook $\pm\pi$ reflections this record's `conditions` contrasts the construction against.

  Data

  None found yet.

  Code

  `pennylane/templates/subroutines/amplitude_amplification.py` in https://github.com/PennyLaneAI/pennylane -- Python, Apache License 2.0, header "Copyright 2018-2024 Xanadu Quantum Technologies Inc.", 294 lines on `main` at commit `45622b173b686ee0ea37915f4cc7bf4291af2e6f`, read 2026-08-27. The class is `AmplitudeAmplification(Operation)`, exported from `pennylane/templates/subroutines/__init__.py` as `from .amplitude_amplification import AmplitudeAmplification`. Constructor: `__init__(self, U, O, iters=1, fixed_point=False, work_wire=None, p_min=0.9, reflection_wires=None)`. Neither `__init__` nor `_get_fixed_point_angles` checks that `iters` is odd, though this record's own hop states the paper constructs the sequence "for any odd integer $L \ge 1$ ... so an even $L$ has no construction at all" -- an even `iters` is accepted and silently halved by `iters // 2` rather than rejected.

  Results

  The class docstring's own worked example amplifies $|2\rangle$ on 3 qubits, with `U` a Hadamard-built uniform superposition and `O = qp.FlipSign(2, wires=range(3))`, run as `qp.AmplitudeAmplification(U, O, iters=5, fixed_point=True, work_wire=3)` inside a `qp.qnode` on `qp.device("default.qubit")`, PennyLane's state-vector simulator. The printed output, `print(np.round(circuit(),3))`, is `[0.013 0.013 0.91 0.013 0.013 0.013 0.013 0.013]` -- probability 0.91 on the marked index 2.

• `ampamp`'s `FixedPointEngine`

  • Fixed-point quantum search with an optimal number of queries

    Theodore J. Yoder, Guang Hao Low, Isaac L. Chuang · 2014

  • Amplitude Amplification Algorithms

    Mithilesh Kumar, Yusuf Tahir, Varun Daiya · 2026

  About

  `ampamp` is a Python "research library for building quantum amplification circuits, inspecting their mathematical behavior, profiling compilation cost, and validating ideal/noisy backend behavior from one compact API" (its README), maintained by the GitHub org `QuantumAmplification` under the MIT licence and distributed on PyPI. It is organised one engine per amplification variant rather than one generic amplifier: the README's own capability table lists "Fixed-point AA" against `FixedPointEngine` in a row separate from "Variable-time AA" against `VTAAEngine`, and its section headers likewise separate "### Fixed-point amplification" from a later "### Variable-time branch analysis" -- this record's artefact is the former module only. A companion survey article by the same three authors, "Amplitude Amplification Algorithms" (Zenodo, 2026), states in its abstract that it follows "the progression through oblivious, fixed-point, and distributed amplification schemes" and lists "Fixed-Point Amplification" among its own keywords.

  Methods

  `FixedPointEngine.__init__(self, L, delta)` requires `L` odd -- it raises `ValueError("L must be an odd integer for FPAA.")` when `L % 2 == 0` -- and sets `self.num_grover_iterates = (L - 1) // 2`, this record's own $l = (L-1)/2$. `_compute_gamma` returns `1.0 / np.cosh(np.arccosh(1.0 / self.delta) / self.L)`, documented as computing "gamma through gamma^{-1} = T_{1/L}(1 / delta)". `_generate_phase_pairs` then sets, for `j` from 1 to `num_grover_iterates`, `alphas[j-1] = 2.0 * np.arctan2(1.0, np.tan(2.0 * np.pi * j / self.L) * sq_term)` with `sq_term = sqrt(1 - gamma**2)`, and `betas = -alphas[::-1]` -- this record's $\alpha_j = -\beta_{l-j+1}$. `success_probability(lambda_val)` evaluates `1 - delta**2 * T_L((1/gamma) * sqrt(1-lambda_val))**2` through its own `_chebyshev_polynomial`, this record's closed form $P_L = 1-\delta^2 T_L(\cdot)^2$. `build_fixed_point_circuit(num_qubits, marked_indices)` synthesizes the circuit directly in Qiskit: it Hadamards every qubit, then for each `(a_j, b_j)` pair applies a `DiagonalGate` placing phase $e^{i\beta_j}$ on the marked computational-basis indices (the target-side reflection, built from the oracle alone), Hadamards, a `DiagonalGate` placing phase $e^{-i\alpha_j}$ on index 0 (the reflection about the Hadamard-prepared source state), and Hadamards again to close the sandwich.

  Data

  None found yet.

  Code

  `src/ampamp/fixed_point.py` in https://github.com/QuantumAmplification/ampamp -- Python, MIT licence, 176 lines on `main` at commit `a482ee2eea80a53db02ff33d6bbace7f529fcee2`, read 2026-08-27. Distributed on PyPI as `ampamp`; `pyproject.toml` at that commit records `version = "0.1.4"`. The README's own usage example imports it at top level, `from ampamp import FixedPointEngine`, and the class depends on `from qiskit import QuantumCircuit` plus, inside `build_fixed_point_circuit`, `from qiskit.circuit.library import DiagonalGate as Diagonal` with a fallback `from qiskit.circuit.library import Diagonal` for older Qiskit releases.

  Results

  The library's own test suite checks the synthesized circuit against the closed form on a state-vector simulator. `tests/test_core_engines.py`'s `test_fixed_point_engine_generates_phases_and_circuit` builds `FixedPointEngine(L=5, delta=0.1)` -- 2 generalized iterates -- and asserts `engine.success_probability(engine.lambda_min) == pytest.approx(1.0 - engine.delta**2)`, and, for a synthesized one-qubit circuit with `marked_indices=[1]`, that `Statevector.from_instruction(one_qubit_circuit).probabilities()[1] == pytest.approx(engine.success_probability(0.5))` -- Qiskit's `Statevector` state-vector simulator, not hardware. A separate parametrized test, `test_fixed_point_engine_rejects_invalid_parameters`, checks that invalid `(L, delta)` pairs raise `ValueError` rather than silently building a circuit.

• pyqsp's `FPSearch` phase generator (`fpsearch`)

  • Fixed-point quantum search with an optimal number of queries

    Theodore J. Yoder, Guang Hao Low, Isaac L. Chuang · 2014

  About

  pyqsp is a quantum-signal-processing phase-finding package maintained on GitHub under the username `ichuang`; its own `CITATION` file's primary reference, "Grand Unification of Quantum Algorithms", is coauthored by Isaac L. Chuang, the third author of the fixed-point paper this method records. `pyqsp/phases.py`'s module docstring describes the file as "Known QSP phases for specific responses", a library of precomputed sequences rather than a general numerical solver, and one entry in it, the `FPSearch` class, is documented as returning "phases for fixed point quantum search, following https://arxiv.org/abs/1409.3305" -- this method's own citation.

  Methods

  `FPSearch.generate(self, d, delta=None, gamma=None, return_alpha=False)` sets `L = 2 * d + 1`, this record's odd length with `d` its $l$, and, when `gamma` is not supplied, computes it from `delta` (default 0.1) as `gamma = 1 / np.cosh((1 / L) * np.arccosh(1 / delta))` -- the same $\gamma^{-1}=T_{1/L}(1/\delta)$ quantity this record names. With `sg = np.sqrt(1 - gamma**2)`, it sets `avec = 2 * np.arctan2(1, (np.tan(2 * np.pi * kvec / L) * sg))` for `kvec = np.arange(1, d + 1)` -- this record's $\alpha_j = 2\cot^{-1}(\tan(2\pi j/L)\sqrt{1-\gamma^2})$, since `arctan2(1, x)` equals $\cot^{-1}(x)$ for $x>0$ -- and `bvec = - avec[::-1]`, the same palindromic pairing. Unless `return_alpha=True` is passed, `generate` does not return `avec`/`bvec` directly: it repacks them into a length-`2*d` array `phivec`, via `phivec[2*k] = -avec[d-k-1] / 2` and `phivec[2*k+1] = bvec[d-k-1] / 2`, because pyqsp's own output convention is a single QSP phase sequence in what its README calls the "$W_x$ convention" rather than the two per-iterate reflection angles this record lists separately.

  Data

  None found yet.

  Code

  `pyqsp/phases.py` in https://github.com/ichuang/pyqsp -- Python, 109 lines at commit `4436bb63d4aea0f636015ace26bb5453c77b999c`, read 2026-08-27; distributed on PyPI as `pyqsp`, version 0.2.0. `FPSearch(PhaseGenerator)` is registered under the string key `'fpsearch'` in the module-level dict `phase_generators`. `pyqsp/main.py` reaches it from the command line: its `elif args.cmd == "fpsearch":` branch instantiates `pg = pyqsp.phases.FPSearch()` and calls `pg.generate(*args.seqargs)`, and the tool's own help text gives a worked invocation, `pyqsp --plot-npts=4000 --plot-positive-only --plot-magnitude --plot --seqargs=1000,1.0e-20 --seqname fpsearch angles` -- $d=1000$, $\delta=10^{-20}$.

  Results

  The package's own continuous-integration test suite exercises this path end to end rather than only unit-testing the class: `pyqsp/test/test_main.py` includes `"--plot-positive-only --plot-magnitude --plot-npts=400 --seqargs=10,0.01 fpsearch"` in its list of `test_cmds`, run through `pyqsp.main` -- $d=10$, $\delta=0.01$ -- and plotted as a magnitude response. The test asserts only that the command completes; it makes no numeric assertion on the phases `generate` returns.