Grover's search with an oracle distinguishing between solutions

Hristo Tonchev; Rosen Bahtev

REVIEW 4 major objections 5 minor 46 references

Grover's search with an oracle distinguishing between solutions

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Grover oracle that phases each solution differently can keep the search near certainty beyond the usual stopping point.

desk verdict Qualitative robustness effect is plausible, but the headline scaling law is an unvalidated extrapolation and the paper needs data, cleanup, and honest scoping before it should be cited. read the letter →

arxiv 2508.19793 v2 pith:MZ7RYNN4 submitted 2025-08-27 quant-ph

Hristo Tonchev , Rosen Bahtev This is my paper · ORCID

classification quant-ph MSC 81P68 PACS 03.67.Ac

keywords GroversearchmultiphaseoraclegeneralizedHouseholderreflectionsuperellipsefitMonteCarlosimulationrobustnesswidthsemiempiricalscalingquantum

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that Grover's search algorithm can be made more forgiving: when there are two solutions, an oracle that marks each solution with a different phase can keep the success probability high even after the number of iterations that would make the standard algorithm start losing probability. The authors derive the Grover operator for a general multiphase oracle, then focus on two solutions and use Monte Carlo simulations to map the pairs of oracle phases that still reach high success probability on time. Those phase pairs form stripes that fit a quarter-superellipse, and the paper uses that fit, together with an asymmetric Hill-type curve for the probability plateau, to give formulas for the best phases and for how the plateau width grows with register size. The central quantitative claim is that the width K_max(N) grows roughly linearly in ln N, about -82.658 + 15.978 * ln(250.867 + N).

What carries the argument

The load-bearing construction is the multiphase oracle (Eq. 24), which attaches a distinct phase e^{i phi_j} to each solution state, combined with a generalized Householder reflection as the diffusion operator. With two solutions, the set of phase pairs that meet the success criteria is mapped by a quarter-superellipse fit (Eq. 32), and the width of the probability plateau is quantified by an asymmetric modified Hill function (Eq. 34). The superellipse parameter p_phi(N) and the plateau width K_max(N) are then fitted semiempirically, producing the extrapolation formulas that carry the paper's central claim.

What would settle it

Simulate the two-solution algorithm for a register size well beyond the fitted range, say N = 1600 or N = 3200, using phases from Eq. (42), and measure the plateau width K_max; if it does not match Eq. (43), or if the success probability does not stay above about 0.92 for the predicted number of iterations, the claimed logarithmic robustness is falsified for large registers.

Watch

Extended reading notes

Core claim

The paper claims that a multi-phase oracle—one that applies a distinct phase to each solution state—can be used to maintain a high probability of finding a solution for a number of iterations equal to or greater than the one required by the deterministic Grover's algorithm. For the two-solution case, the acceptable phase pairs form two symmetric stripes, one of which is fitted by the top-left quadrant of a translated superellipse. Using this fit, the authors identify the phase pair that maximizes robustness and estimate the width of the high-probability plateau. Their semiempirical formulas, Eq. (42) for the optimal phases and Eq. (43) for the plateau width, predict that robustness increases

Load-bearing premise

The load-bearing premise is that the Monte Carlo fits for registers of about 20 to 775 states, and the fitted equations for the superellipse parameter and plateau width, continue to describe much larger registers; no structural proof or out-of-sample test is given for that extrapolation.

Editorial extensions

If this is right

The two-phase oracle with optimal phases keeps a high probability of finding a solution for at least as many iterations as the deterministic Grover algorithm, and for a window that widens logarithmically with register size.
Equations (42) and (43) give concrete, ready-to-use phase values and expected plateau widths for two-solution searches across register sizes.
Because the acceptable phase stripes become thinner as N grows, the method's usefulness depends on precise phase control rather than on more iterations.
With three or more oracle phases, the simulations indicate even wider plateaus, but finding the right phases becomes a high-dimensional fitting problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the logarithmic extrapolation holds, doubling the register size adds only about 15.978 * ln 2, or roughly 11, additional iterations of plateau width, so robustness grows slowly with N.
A natural next check is to simulate registers well beyond the fitted range, e.g. N > 1000, and also to test whether the plateau width formula survives when the number of solutions M grows, since the fits here are for M=2 only.
The phase-stripe narrowing implies a precision threshold: imperfection in setting phi_0 and phi_1 will eventually dominate, so an error-analysis extension could quantify the required phase accuracy.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the qualitative observation is probably real, but the paper's headline scaling law is a curve fit extrapolated far beyond the data, so treat Eq. (43) as a guess, not a result.

What's genuinely new: the authors notice that an oracle applying different phases to each solution can keep Grover's success probability high over a wider window of iteration counts than the standard equal-phase version. That is a practical nuisance worth addressing, and their Monte Carlo results for N up to a few hundred make a decent case that the effect exists. The superellipse fit and the asymmetric Hill function are reasonable descriptive tools, and the authors are upfront that the fits are semiempirical and that the superellipse parameter has significant error.

Soft spots, in rough order of importance. First, the quantitative claims — Eq. (42) for optimal phases and Eq. (43) for robustness width K_max — are fits to the same Monte Carlo data they then use to make predictions. There is no out-of-sample test, no error bars, and the extrapolation from N=775 to large N is an act of faith. The paper itself says the fits are for extrapolation but gives no reason the logarithmic form should hold. Second, the thresholds defining 'useful' phases (P_max > 0.92, t_iter < t_opt + 2) are post hoc; changing them would change the fitted curves. Third, the analytical section in 3.1 is hard to follow and the equations look garbled, with normalization errors that suggest it hasn't been checked carefully. Fourth, the originating multiphase oracle work [35] is cited but missing from the bibliography, which is sloppy. There is also no code or data, so the Monte Carlo results cannot be independently checked.

None of this kills the basic qualitative finding. But the abstract's general claim that the method 'maintains a high probability ... for a number of iterations equal to or more than' the standard algorithm is only demonstrated for the small registers they simulated with hand-picked phases. The useful phase region is also a thin stripe, as they note, so the practical value is limited even if the scaling holds.

Who is this for? People working on robust Grover variants or on empirical methods for quantum algorithm design. It is not a definitive paper; it is a workshop-grade exploration, but a serious referee could help the authors fix the analytics, add data, and scope the claims properly. I'd send it to review rather than desk reject, with the expectation of major revision. I wouldn't cite it as it stands.

Referee Report

4 major / 5 minor

Summary. The paper studies a modification of Grover's search in which a multi-solution oracle applies a different phase to each marked state. It derives the general Grover operator and a recursion relation for the amplitudes in the relevant basis, then specializes to the case of two solutions. Using Monte Carlo simulations for register sizes roughly N = 20 to 775, it identifies oracle-phase pairs whose success probability exceeds 0.92 within about the standard optimal iteration count, fits the boundary of this acceptable region to a quarter-superellipse (Eq. 32), and introduces an asymmetric modified Hill function (Eq. 34) to characterize the probability curve P(t). From these fits the paper proposes formulas for the 'optimal' oracle phases (Eq. 42) and claims that the robustness plateau width grows logarithmically with register size: K_max(N) ≈ −82.658 + 15.978 |ln(250.867 + N)| (Eq. 43). The central claim is that a suitably phase-chosen two-solution Grover search maintains high success probability for at least as many iterations as the deterministic Grover algorithm, with a robustness window that widens logarithmically.

Significance. The qualitative phenomenon—that distinct oracle phases can broaden the plateau of high success probability in multi-solution Grover search—is an interesting and potentially useful robustness effect, and the paper's analytical setup in Section 3.1 (recursion relation and explicit matrix form) is a reasonable starting point. The Monte Carlo evidence for moderate N supports the existence of such an effect. However, the quantitative content goes beyond what the data establish: the fitted superellipse parameter p_phi(N), the optimal-phase formula (Eq. 42), and the logarithmic growth law (Eq. 43) are all post-hoc fits to the same simulations, with no out-of-sample validation, no error bars, and no structural argument for extrapolation. The paper also does not provide code or data to make the fits reproducible. The significance therefore depends on a revision that either substantially validates the scaling claim or explicitly restricts the conclusions to the simulated range.

major comments (4)

[Sections 3.3 and 3.6; Eqs. (33), (42), (43)] The headline quantitative claim—that the robustness width K_max(N) grows as 15.978 ln(250.867 + N)—is an unvalidated extrapolation. The formula is fitted to Monte Carlo results for N between about 20 and 775, and the paper explicitly states in Section 3.3 that simulations for large register sizes could not be performed and that there is 'significant error' in fitting p_phi. No out-of-sample test, cross-validation, or independent benchmark is given, and no analytic or structural argument supports the logarithmic form beyond the fitted range. Since Eqs. (42) and (43) both depend on the extrapolated p_phi(N), the abstract's claim that the modification maintains high probability 'for a number of iterations equal to or more than the one required by the deterministic Grover's algorithm' is not established for large registers.
[Eq. (42) and Section 3.6] Eq. (42) is not a closed formula for the optimal oracle phases: it contains z_max(N), which is only constrained to lie in the interval [Φ−(N), Φ+(N)] by Eq. (41). The text and Fig. 11 show z_max(N) as a curve, but no equation, algorithm, or selection rule is provided to determine z_max(N) for a given N. Without such a rule, Eq. (42) cannot be implemented, and the claim that these are the phases giving maximal robustness is not computable from the manuscript.
[Section 3.2; Fig. 4] The acceptance criteria for the Monte Carlo phase samples are post hoc and not tied to the stated requirements. The text says the maximal probability 'must not be lower than in the original Grover's algorithm' and must need the same number of iterations, but the actual filter is P_max > 0.92 and t_iter < t_opt + 2. No comparison is made between 0.92 and the actual success probability of the standard algorithm, and the '+2' iteration tolerance is an arbitrary threshold. All subsequent fits—the superellipse, p_phi(N), and K_max(N)—inherit this choice, so the quantitative conclusions are conditioned on it.
[Section 3.5; Eq. (36)] The quantity Ω(z) = (b(z)/b_max)(k(z)/k_max) is presented as identifying the phase choice with maximal robustness, but no independent definition of 'robustness' is given, and no evidence is provided that maximizing Ω is equivalent to maximizing any directly measured property of the probability curve, such as the width of the plateau or the area under P(t). The text merely asserts that 'the result shows that this quantity always gives good results.' Since b and k are themselves fitted parameters of the modified Hill function, this step is circular unless Ω is validated against a concrete, separately defined robustness metric.

minor comments (5)

[Section 2.1 and Eq. (28)] Several equations contain garbled or missing notation due to font/encoding issues, notably Eq. (28) and the definition of the oracle in Eq. (24). The recursion relation is hard to verify as printed; please rewrite in clean LaTeX.
[Section 3.4; Eq. (34)] The notation '⟧±' is not defined clearly. It appears to mean a sign-dependent branch, but the sign convention should be stated explicitly, and the formula should be checked for typographical errors.
[Section 3.4; Eq. (35)] The standard deviation expression is incomplete: the summation variable and the index range are not fully specified, and the dependence of P_j on t and the phases is omitted. Please restate precisely.
[Figures 4, 5, and 13] Several figures are hard to read in grayscale and lack sufficient axis labels or legends. In particular, Fig. 13's solid red and dotted purple lines are difficult to distinguish; please use more distinct markers or styles and include a caption listing the plotted quantities.
[General] There are duplicated 'Acknowledgments' headings and a reference formatting mismatch (e.g., the second acknowledgment block before the references). Please also define 'K_max' before first use in the abstract/introduction, since it appears later without a formal definition in Section 3.6.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytic part follows from definitions, and the quantitative formulas are openly semiempirical fits rather than disguised predictions.

full rationale

The paper's analytic content (Section 3.1, Eqs. 28–31) is a direct matrix derivation from the definitions of the multiphase oracle and the generalized Householder reflection; it does not assume the robustness conclusion. The numerical part is explicitly semiempirical: Eq. (33) for p_phi(N) and Eq. (43) for K_max(N) are regressions on Monte Carlo data, not first-principles derivations of the target claim. The abstract's qualitative claim is supported by simulations for the studied register sizes; extrapolation to larger N is flagged by the authors themselves as an extrapolation ('we will fit the results for p_phi and use the resulting equation to make an extrapolation'). The self-citations [27] and [28] supply a fitting function (the modified Hill function) that is redefined in this paper (Eq. 34), so no load-bearing theorem is imported. The absence of out-of-sample validation is a limitation and predictive-risk concern, not a circular reduction: the fitted formulas are used to summarize the data they were fit to, and the paper does not claim to predict a distinct quantity from the same data. No step in the derivation reduces to its own input by construction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on Monte Carlo simulation and on several ad hoc modeling choices: the superellipse shape for the valid-phase region, the six-parameter asymmetric Hill function used to summarize P(t), and the definition of the robustness metric Omega(z). The quantitative outputs (Eqs 33, 42, 43) are fits to the same data they describe. No code, dataset, or error analysis is provided.

free parameters (6)

superellipse exponent p_phi(N) = Eq (33): p_phi(N) = -65.7376 + 12.5476 * ln(N + 252.0719)
Fitted per register size from the Monte Carlo stripe in Section 3.3, then regressed on N; the offset 252.0719 is an arbitrary fit constant.
plateau width K_max(N) = Eq (43): K_max(N) ~ -82.658 + 15.978 * |ln(250.867 + N)|
Fitted to the Hill-function width parameter k across the simulated range of N in Section 3.6.
success probability threshold = 0.92
Ad hoc cutoff in Section 3.2 used to define acceptable phase pairs; all subsequent stripe and curve fits depend on it.
iteration tolerance = t_iter < t_opt + 2
Post-hoc requirement in Section 3.2 that the maximum probability is reached no more than one iteration after the deterministic Grover optimum.
optimal superellipse parameter z_max(N) = read from numerical scans (Fig 11)
Positions of the maxima of Omega(z) are obtained from fits to the simulation data, not from a closed-form expression.
Hill function normalizers b_max(N), k_max(N) = not tabulated
Used in Eq (36); they are maxima over z of parameters that are themselves fitted to the probability curves P(t).

assumptions (6)

domain assumption Phase matching between oracle and reflection phases (phi = omega) yields a deterministic Grover rotation with zero failure rate.
Adopted from Long et al. [34] in Sections 2.3 and 3.2; the reflection operator P_N is fixed to the optimal phase phi_max throughout the simulations.
domain assumption The multiphase oracle is a product of phase oracles and acts trivially on all non-solution states.
Eqs (20) and (24); this is the standard phase-oracle model in quantum search.
domain assumption Ideal unitary state-vector evolution in simulation exactly predicts experimental success probabilities; decoherence and gate errors are ignored.
All Monte Carlo simulations in Sections 3.2 through 3.7 use ideal unitary evolution.
ad hoc to paper The boundary of the useful phase region is a quarter of a superellipse (Eq 32) for all register sizes.
Imposed in Section 3.3 after visual inspection of simulated stripes; the axes are fixed by matching single-solution endpoints, but the exponent p_phi is fitted.
ad hoc to paper The six-parameter asymmetric modified Hill function (Eq 34) adequately represents P(t), and its fitted height b and width k capture the robustness plateau.
Introduced in Section 3.4 and used in Section 3.5 to define the robustness metric Omega(z).
ad hoc to paper The quantity Omega(z) = (b(z)/b_max)(k(z)/k_max) identifies the phase choice with maximal robustness.
Defined in Eq (36) without derivation; it is an ad hoc product of normalized fit parameters.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Grover's search with an oracle distinguishing between solutions." pith.science (2026). https://pith.science/paper/MZ7RYNN4

@misc{pith2026250819793,
  author       = {Pith},
  title        = {Pith review of: Grover's search with an oracle distinguishing between solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZ7RYNN4}},
  note         = {Machine review of arXiv:2508.19793}
}

read the original abstract

Here we suggest a modification of Grover's algorithm, based on a multiphase oracle which marks each solution with a different phase when there is more than one solution. Such a modification can be used to maintain a high probability of finding a solution for a number of iterations equal to or more than the one required by the deterministic Grover's algorithm (the one based on generalized Householder reflections). We use various semiempirical methods to show that the interval of number of iterations for which the algorithm keeps the probability of finding solution high depends on the register size and the oracle phases.

Figures

Figures reproduced from arXiv: 2508.19793 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 36 canonical work pages

[35]

D. L. Mohr, W. J. Wilson, and R. J. Freund, Statistical Methods. London, United Kingdom ; San Diego, CA: Academic Press, 2022

work page 2022
[1]

Each of the three algorithms mentioned is quadratically faster than the classical search algorithms and has applications for which it is more suitable than the other two

Introduction There are various quantum algorithms for searching for an element in an unordered database, examples include Grover’s search [1], quantum random walk search [2] and fixed- point quantum search [3]. Each of the three algorithms mentioned is quadratically faster than the classical search algorithms and has applications for which it is more suit...

work page
[2]

These elements are also called solutions

Grover’s algorithm and some of its modifications Grover's search algorithm is a probabilistic quantum algorithm for searc hing in an unordered database and returns one of the elements fulfilling the search condition s. These elements are also called solutions. Invented in 1996 [1], it is one of the oldest quantum algorithms and is quadratically faster tha...

work page 1996
[3]

When the required number of iterations to obtain the solution is surpassed, the probability of obtaining it decreases as fast as the algorithm finds it

Oracle applying different phases to each solution The original Grover’s algorithm has a periodic behavior as a Grover iteration can be seen as a rotation in the space spanned by two vectors, one of which is an equal superposition of all solutions and the other one is an equal superposition of all non-solutions. When the required number of iterations to ob...

work page
[4]

An oracle i. e. a reflection against the state that is an equal superposition of all non-solution states. It is based on a function that can recognize the solution states, and marks them by flipping their sign. Let us denote the solution states by ݉௜⟩ே with − 1). The oracle can be expressed as: ܱே݉௜ܫே − 2|ߚ⟩ே⟨ߚ|ே (2) where ܫே is an identity matrix of dime...

work page
[5]

The probability that the outcome of the measurement is a solution state is high and depends on both the register size and the number of solutions

A reflection operator against the equal superposition of all states: ܲே =ܫே − 2|0)⟩𝑁⟨0)|𝑁 (3) Let us denote the whole Grover iteration by ܩே, thus: ܩே =ܲேܱே (4) After the required number of iterations, a measurement is done on the register. The probability that the outcome of the measurement is a solution state is high and depends on both the register siz...

work page
[6]

Grover algorithm with zero theoretical failure rate,

G. L. Long, “Grover algorithm with zero theoretical failure rate,” Phys. Rev. A, vol. 64, no. 2, p. 022307, Jul. 2001, doi: 10.1103/PhysRevA.64.022307

work page doi:10.1103/physreva.64.022307 2001
[7]

remains the same. We use the same oracle as the one used in [35]: ܱே݉௝, ∀߮௝)ܫே − (1 −݁௜ఝೕ)|݉௝⟩ே⟨݉௝|ே) ெ−1 ௜=0 (24) The Grover operator ܩே is constructed using ܱே݉௝, ∀߮௝) and for ܲே is used generalized Householder reflection with phase ߱ : ܲே(߱)ܫ= 𝑁 − (1 −݁݅ఠ)|߰0⟩ 𝑁 ⟨߰0| 𝑁 (25) Let’s define the state |)ݐ(߰⟩ே of the register after the ݐth iteration as: |)ݐ(...

work page

Show all 46 references

[8]

The maximal probability of finding a solution must not be lower than in the original Grover’s algorithm
[9]

The reflection ܲே is constructed using the optimal phase

It must need the same number of iterations to obtain a solution as the original Grover’s algorithm with two marked states We did Monte Carlo simulations of the algorithm, during which the two phases ߮0 and ߮1 are chosen at random. The reflection ܲே is constructed using the opt...
[10]

Find all parameters of the fitting curve and how they change with the increase of the register size
[11]

We also want to obtain the top left quadrant of the superellipse

Find the part of the parametric curve that gives the highest robustness The point where the curve crosses the ߮0 axis ߮0߮௠௔௫߮1 = 0) corresponds to Grover’s algorithm with only one solution. We also want to obtain the top left quadrant of the superellipse. These two restriction...
[12]

We begin with defining the operators, and use their analytical form to obtain the general case for the Grover iteration and some useful recursive relations

Conclusion In this work we study a modification of the Grover’s algorithm, with an oracle that marks each solution with different phase. We begin with defining the operators, and use their analytical form to obtain the general case for the Grover iteration and some useful recu...

2021
[13]

A fast quantum mechanical algorithm for database search,

L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC ’96, Philadelphia, Pennsylvania, United States: ACM Press, 1996, pp. 212–219. doi: 10.1145/237814.237866

1996
[14]

Quantum random-walk search algorithm,

N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A, vol. 67, no. 5, p. 052307, May 2003, doi: 10.1103/PhysRevA.67.052307

2003 doi
[15]

Fixed-Point Quantum Search with an Optimal Number of Queries,

T. J. Yoder, G. H. Low, and I. L. Chuang, “Fixed-Point Quantum Search with an Optimal Number of Queries,” Phys. Rev. Lett., vol. 113, no. 21, p. 210501, Nov. 2014, doi: 10.1103/PhysRevLett.113.210501

2014 doi
[16]

Time-efficient implementation of quantum search with qudits,

S. S. Ivanov, H. S. Tonchev, and N. V. Vitanov, “Time-efficient implementation of quantum search with qudits,” Phys. Rev. A, vol. 85, no. 6, p. 062321, Jun. 2012, doi: 10.1103/PhysRevA.85.062321

2012 doi
[17]

A single-interaction step implementation of a quantum search in coupled micro-cavities,

E. S. Kyoseva, D. G. Angelakis, and L. C. Kwek, “A single-interaction step implementation of a quantum search in coupled micro-cavities,” EPL, vol. 89, no. 2, p. 20005, Feb. 2010, doi: 10.1209/0295-5075/89/20005

2010 doi
[18]

Quantum dialogue protocol based on Grover’s search algorithms,

A. Yin, K. He, and P . Fan, “Quantum dialogue protocol based on Grover’s search algorithms,” Mod. Phys. Lett. A, vol. 34, no. 21, p. 1950169, Jul. 2019, doi: 10.1142/S0217732319501694

2019 doi
[19]

Quantum Amplitude Amplification and Estimation,

G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, “Quantum Amplitude Amplification and Estimation,” vol. 305, 2002, pp. 53–74. doi: 10.1090/conm/305/05215

2002 doi
[20]

Arbitrary phases in quantum amplitude amplification,

P. Høyer, “Arbitrary phases in quantum amplitude amplification,” Phys. Rev. A, vol. 62, no. 5, p. 052304, Oct. 2000, doi: 10.1103/PhysRevA.62.052304

2000 doi
[21]

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge: Cambridge Univ. Press, 2007

2007
[22]

Synthesis of Ternary Grover’s Algorithm,

S. B. Mandal, A. Chakrabarti, and S. Sur- Kolay, “Synthesis of Ternary Grover’s Algorithm,” in 2014 IEEE 44th International Symposium on Multiple-Valued Logic, May 2014, pp. 184–189. doi: 10.1109/ISMVL.2014.40

2014 doi
[23]

Grover search revisited: Application to image pattern matching,

H. Tezuka, K. Nakaji, T. Satoh, and N. Yamamoto, “Grover search revisited: Application to image pattern matching,” Phys. Rev. A, vol. 105, no. 3, p. 032440, Mar. 2022, doi: 10.1103/PhysRevA.105.032440

2022 doi
[24]

Quantum lower bounds for the collision and the element distinctness problems,

Y. Shi, “Quantum lower bounds for the collision and the element distinctness problems,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., Nov. 2002, pp. 513–519. doi: 10.1109/SFCS.2002.1181975

2002 arXiv
[25]

Comparing Grover’s Quantum Search Algorithm with Classical Algorithm on Solving Satisfiability Problem,

R. Wang, “Comparing Grover’s Quantum Search Algorithm with Classical Algorithm on Solving Satisfiability Problem,” in 2021 IEEE Integrated STEM Education Conference (ISEC), Mar. 2021, pp. 204–204. doi: 10.1109/ISEC52395.2021.9764017

2021
[26]

Quantum optimization for training support vector machines,

D. Anguita, S. Ridella, F. Rivieccio, and R. Zunino, “Quantum optimization for training support vector machines,” Neural Networks, vol. 16, no. 5, pp. 763–770, Jun. 2003, doi: 10.1016/S0893-6080(03)00087-X

2003 doi
[27]

and quantum random walk search [28] algorithms. A superellipse (known also as a Lame curve) is a parametric curve, that is used for fitting functions in a variety of fields, including the characteristic equation of photovoltaics [29], face recognition in computer vision [30], ...
[28]

Grover’s Algorithm for Neural Networks,

A. Haverly, S. Rahimi, and M. A. Novotny, “Grover’s Algorithm for Neural Networks,” in 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC), Mar. 2025, pp. 667–671. doi: 10.1109/QCNC64685.2025.00111

2025
[29]

Implementing quantum genetic algorithms: a solution based on Grover’s algorithm,

M. Udrescu, L. Prodan, and M. Vlăduţiu, “Implementing quantum genetic algorithms: a solution based on Grover’s algorithm,” in Proceedings of the 3rd conference on Computing frontiers, in CF ’06. New York, NY, USA: Association for Computing Machinery, May 2006, pp. 71–82. doi: ...

2006
[30]

A Quantum Public Key Cryptographic Scheme Using Entangled States and Grover Operator,

S. Bajpayee, S. Sen, P. Dey, and I. Mukherjee, “A Quantum Public Key Cryptographic Scheme Using Entangled States and Grover Operator,” in Security and Privacy, S. Mesnager, P. Stănică, and S. K. Debnath, Eds., Cham: Springer Nature Switzerland, 2025, pp. 171–181. doi: 10.1007/...

2025 doi
[31]

Quantum secret-sharing protocol based on Grover’s algorithm,

L.- Y. Hsu, “Quantum secret-sharing protocol based on Grover’s algorithm,” Phys. Rev. A, vol. 68, no. 2, p. 022306, Aug. 2003, doi: 10.1103/PhysRevA.68.022306

2003 doi
[32]

The Improved Quantum Secret Sharing Protocol Based on Grover Algorithm,

Z. Yu, “The Improved Quantum Secret Sharing Protocol Based on Grover Algorithm,” J. Phys.: Conf. Ser., vol. 2209, no. 1, p. 012031, Feb. 2022, doi: 10.1088/1742- 6596/2209/1/012031

2022 doi
[33]

The Hill equation: a review of its capabilities in pharmacological modelling,

S. Goutelle et al., “The Hill equation: a review of its capabilities in pharmacological modelling,” Fundamental & Clinical Pharmacology, vol. 22, no. 6, pp. 633– 648, 2008, doi: 10.1111/j.1472-8206.2008.00633.x

2008
[34]

Superellipse fitting to partial data,

X. Zhang and P. L. Rosin, “Superellipse fitting to partial data,” Pattern Recognition, vol. 36, no. 3, pp. 743–752, Mar. 2003, doi: 10.1016/S0031-3203(02)00088-2

2003 doi
[36]

Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits

U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits. Accessed: Dec. 09, 2022. [Online]. Available: https://www.routledge.com/An- Introduction-to-Systems-Biology-Design-Principles-of-Biological- Circuits/Alon/p/book/9781439837177

2022
[37]

The Hill equation and the origin of quantitative pharmacology,

R. Gesztelyi, J. Zsuga, A. Kemeny- Beke, B. Varga, B. Juhasz, and A. Tosaki, “The Hill equation and the origin of quantitative pharmacology,” Arch. Hist. Exact Sci., vol. 66, no. 4, pp. 427–438, Jul. 2012, doi: 10.1007/s00407-012-0098-5

2012 doi
[38]

A method for estimating Hill function-based dynamic models of gene regulatory networks,

F. Ehsan Elahi and A. Hasan, “A method for estimating Hill function-based dynamic models of gene regulatory networks,” Royal Society Open Science, vol. 5, no. 2, p. 171226, Feb. 2018, doi: 10.1098/rsos.171226

2018 doi
[39]

Robustness of different modifications of Grover’s algorithm based on generalized Householder reflections with different phases,

H. Tonchev and P. Danev, “Robustness of different modifications of Grover’s algorithm based on generalized Householder reflections with different phases,” Results in Physics, vol. 59, p. 107595, Apr. 2024, doi: 10.1016/j.rinp.2024.107595

2024
[40]

Studying the robustness of quantum random walk search on Hypercube against phase errors in the traversing coin by semi-empirical methods,

H. Tonchev and P. Danev, “Studying the robustness of quantum random walk search on Hypercube against phase errors in the traversing coin by semi-empirical methods,” in Proceedings of 11th International Conference of the Balkan Physical Union — PoS(BPU11), vol. 427, SISSA Media...

2023 doi
[41]

Theoretical derivation of super-ellipse model as approximation of physics-based implicit model for solar PV,

A. Kumar Das and R. Singh, “Theoretical derivation of super-ellipse model as approximation of physics-based implicit model for solar PV,” Solar Energy, vol. 274, p. 112551, May 2024, doi: 10.1016/j.solener.2024.112551

2024
[42]

An ellipsoidal model for generating realistic 3D facial textures,

Y. Park, S. Jang, J. Lee, Y. Lee, and G. Kim, “An ellipsoidal model for generating realistic 3D facial textures,” International Journal of Computer Applications in Technology, vol. 46, no. 1, pp. 36–44, Jan. 2013, doi: 10.1504/IJCAT.2013.051386

2013
[43]

Lame’ curve-based signature discovery learning technique for network traffic classification,

S. Suthaharan, L. Sunkara, and S. Keshapagu, “Lame’ curve-based signature discovery learning technique for network traffic classification,” in 2013 IEEE International Conference on Intelligence and Security Informatics, Jun. 2013, pp. 321–326. doi: 10.1109/ISI.2013.6578851

2013
[44]

Auto-Generation System Based on Fractal Geometry for Batik Pattern Design,

G. Tian, Q. Yuan, T. Hu, and Y. Shi, “Auto-Generation System Based on Fractal Geometry for Batik Pattern Design,” Applied Sciences, vol. 9, no. 11, Art. no. 11, Jan. 2019, doi: 10.3390/app9112383

2019 doi
[45]

A generic geometric transformation that unifies a wide range of natural and abstract shapes,

J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” American Journal of Botany, vol. 90, no. 3, pp. 333–338, 2003, doi: 10.3732/ajb.90.3.333

2003 doi
[46]

Phase matching in quantum searching,

G. L. Long, Y. S. Li, W. L. Zhang, and L. Niu, “Phase matching in quantum searching,” Physics Letters A, vol. 262, no. 1, pp. 27–34, Oct. 1999, doi: 10.1016/S0375- 9601(99)00631-3

1999 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [35]

D. L. Mohr, W. J. Wilson, and R. J. Freund, Statistical Methods. London, United Kingdom ; San Diego, CA: Academic Press, 2022

work page 2022

[2] [1]

Each of the three algorithms mentioned is quadratically faster than the classical search algorithms and has applications for which it is more suitable than the other two

Introduction There are various quantum algorithms for searching for an element in an unordered database, examples include Grover’s search [1], quantum random walk search [2] and fixed- point quantum search [3]. Each of the three algorithms mentioned is quadratically faster than the classical search algorithms and has applications for which it is more suit...

work page

[3] [2]

These elements are also called solutions

Grover’s algorithm and some of its modifications Grover's search algorithm is a probabilistic quantum algorithm for searc hing in an unordered database and returns one of the elements fulfilling the search condition s. These elements are also called solutions. Invented in 1996 [1], it is one of the oldest quantum algorithms and is quadratically faster tha...

work page 1996

[4] [3]

When the required number of iterations to obtain the solution is surpassed, the probability of obtaining it decreases as fast as the algorithm finds it

Oracle applying different phases to each solution The original Grover’s algorithm has a periodic behavior as a Grover iteration can be seen as a rotation in the space spanned by two vectors, one of which is an equal superposition of all solutions and the other one is an equal superposition of all non-solutions. When the required number of iterations to ob...

work page

[5] [4]

An oracle i. e. a reflection against the state that is an equal superposition of all non-solution states. It is based on a function that can recognize the solution states, and marks them by flipping their sign. Let us denote the solution states by ݉௜⟩ே with − 1). The oracle can be expressed as: ܱே݉௜ܫே − 2|ߚ⟩ே⟨ߚ|ே (2) where ܫே is an identity matrix of dime...

work page

[6] [5]

The probability that the outcome of the measurement is a solution state is high and depends on both the register size and the number of solutions

A reflection operator against the equal superposition of all states: ܲே =ܫே − 2|0)⟩𝑁⟨0)|𝑁 (3) Let us denote the whole Grover iteration by ܩே, thus: ܩே =ܲேܱே (4) After the required number of iterations, a measurement is done on the register. The probability that the outcome of the measurement is a solution state is high and depends on both the register siz...

work page

[7] [6]

Grover algorithm with zero theoretical failure rate,

G. L. Long, “Grover algorithm with zero theoretical failure rate,” Phys. Rev. A, vol. 64, no. 2, p. 022307, Jul. 2001, doi: 10.1103/PhysRevA.64.022307

work page doi:10.1103/physreva.64.022307 2001

[8] [7]

remains the same. We use the same oracle as the one used in [35]: ܱே݉௝, ∀߮௝)ܫே − (1 −݁௜ఝೕ)|݉௝⟩ே⟨݉௝|ே) ெ−1 ௜=0 (24) The Grover operator ܩே is constructed using ܱே݉௝, ∀߮௝) and for ܲே is used generalized Householder reflection with phase ߱ : ܲே(߱)ܫ= 𝑁 − (1 −݁݅ఠ)|߰0⟩ 𝑁 ⟨߰0| 𝑁 (25) Let’s define the state |)ݐ(߰⟩ே of the register after the ݐth iteration as: |)ݐ(...

work page