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REVIEW 4 major objections 5 minor 47 references

An Optimized Quantum Maximum or Minimum Searching Algorithm and its Circuits

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Optimized quantum min–max search replaces its probabilistic inner search with Grover–Long exact search, pushing success probability near 100 percent when the solution ratio is known.

desk verdict Useful engineering on top of a known combination, but the near-100% success claim needs the estimator and the success condition fixed before it is published as-is. read the letter →

arxiv 1908.07943 v1 pith:VYY64XG4 submitted 2019-08-21 quant-ph

Yanhu Chen , Shijie Wei , Xiong Gao , Cen Wang , Jian Wu , Hongxiang Guo This is my paper · ORCID
classification quant-ph MSC 68Q1281P68 PACS 03.67.Ac03.67.Lx
keywords quantumsearchmaximumandminimumfindingGrover-Longalgorithmamplitudeamplificationcircuitsoracleconstructionfailureratedatabase
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 claims that the known quantum algorithm for finding a maximum or minimum can be strengthened by replacing its probabilistic inner search step with the Grover–Long exact search, provided the fraction of marked database items can be estimated. The authors give a concrete algorithm, QUMMSA, along with general circuits for the oracle and phase-shift operators, and three equivalent simplifications that reduce gate counts. When the estimated ratio $\tilde{M}/\tilde{N}$ is accurate, the success probability approaches 100 percent; for large databases the paper also claims an advantage in total complexity because it prepares far fewer initial states than the earlier method. The key step is demonstrated on a two-qubit superconducting processor, and a six-qubit simulation on a real dataset shows lower failure rates than the earlier approach.

What carries the argument

The load-bearing object is the Grover–Long operator: a single Grover iteration in which the conditional phase shifts in the oracle and the $I_0$ operator are replaced by rotations through an adjustable angle $\phi = 2\arcsin(\sin(\pi/(4J+2))/\sin\beta)$, with $\sin\beta = \sqrt{M/N}$. When $J$ satisfies the lower bound from the paper, measuring the register yields one of the marked states with zero theoretical failure. The paper derives $J$ and $\phi$ from estimated values $\tilde{M}$ and $\tilde{N}$, constructs the diagonal phase-shift operator $I_0 = \mathrm{diag}[e^{i\phi},1,\ldots,1]$, builds oracles that mark individual or threshold-marked basis states, and then applies three simplification principles to lower the cost of controlled-phase gates. This machinery is what turns a probabilistic amplitude-amplification subroutine into a deterministic one while preserving the quadratic speedup.

What would settle it

Run QUMMSA on a database whose values occupy only a small fraction of the $2^n$ code space (e.g., 6 qubits with 10 distinct values spanning the range) and count how often the Grover–Long measurement returns a value above the current threshold; the paper's own contour map predicts an elevated failure rate in that regime, which would contradict the near-100% success claim if the method is used without the sampling refinement.

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Extended reading notes

Core claim

The paper's central claim is that the earlier quantum min–max search DHA can be rebuilt on the Grover–Long operator, a version of Grover's iteration in which the usual phase inversion is replaced by an adjustable rotation through an angle determined by the estimated ratio of marked values to database size. With an iteration count $J$ satisfying $J \geq \lfloor (\pi/2 - \beta)/\beta \rfloor + 1$, where $\sin\beta = \sqrt{M/N}$, the search returns a marked value with zero theoretical failure rate. The paper then computes these parameters from estimates $\tilde{M} = d_0 + 1$ and $\tilde{N} = 2^n$, provides explicit quantum circuits for the phase-shift operator and for oracles marking one or many values, and gives three simplification rules that reduce the number of multi-qubit controlled gates. Compared with the repetition-based inner search of DHA, the QUMMSA has a deterministic iteration count and a worst-case success probability in the main loop of $1 - (1/2)^c$ for a constant $c$, which approaches 1 as $c$ grows.

Load-bearing premise

The central assumption is that the estimated ratio of marked values to database size, taken from a uniform filling of the code space, is close to the true ratio; if the database is sparse or non-uniform, the Grover–Long parameters are wrong and the near-100% success claim relies on sampling refinements that are not part of the main algorithm.

Editorial extensions

If this is right

  • If the ratio $\tilde{M}/\tilde{N}$ is close to the true ratio, the inner Grover–Long search succeeds with probability close to 1, so the full min–max algorithm inherits a near-deterministic success behavior.
  • Replacing the random stopping rule of DHA with a fixed constant $c$ gives an exponential improvement in the worst-case failure rate of the outer loop, $1/2^c$.
  • The simplified oracle circuits reduce the number of two-qubit controlled gates for marking $2^m$ consecutive states from $2^{n+m-1}$ to $2^{n-m-1}$, which directly lowers the gate load for threshold oracles used in min/max search.
  • The complexity analysis implies that QUMMSA prepares about $\log_2 N$ initial states instead of $\log_2^2 N$, so the advantage over DHA grows as the database size increases.

Reading between the lines

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

  • The real bottleneck of the method is estimation, not search: if a user supplies a good rank estimate for the current threshold from a small sample, the same Grover–Long circuit becomes near-deterministic; the paper's own sampling formula suggests such a pre-processing step, though it is not integrated into the pseudocode.
  • Because the estimated fraction $\tilde{M}/\tilde{N}$ is derived from a uniform filling of the code space, the method's success on sparse real-world tables will depend strongly on how well the empirical cumulative distribution matches that assumption; the paper's contour map shows the failure rate climbing away from the diagonal.
  • The three circuit simplifications are not specific to min/max finding and could be reused for any oracle that marks values below or above a threshold, such as range queries or top-$k$ selection.
  • A fully practical version of the algorithm would need a certified procedure that estimates $M/N$ with bounded error before running the quantum search; the paper leaves this as a separate step, so a combined classical-sampling-plus-quantum-search pipeline is the natural next development.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes QUMMSA, a variant of Dürr–Høyer quantum minimum/maximum search. QUMMSA replaces the probabilistic quantum exponential searching subroutine with the Grover–Long exact search algorithm, sets the Grover–Long phase and iteration parameters from estimates M̃ and Ñ of the number of marked items and database size, terminates the outer loop after a constant c repetitions instead of DHA's threshold, and gives explicit circuit constructions for the phase operator I0 and oracles together with three simplification rules. The authors report a two-qubit experiment on IBMQ Yorktown comparing Grover–Long with QESA, a six-qubit numerical simulation for a 36-item excerpt of Titanic passenger ages, a failure-rate analysis for both subroutines, and a complexity comparison with DHA. The central advertised advantage is that the success probability can approach 100% when the ratio M/N is known accurately, while the circuit costs are lower than DHA for large databases.

Significance. Conditional on its assumptions, the paper's central idea is sound: replacing QESA with Grover–Long is a natural way to make the inner search deterministic, and the reported circuit constructions, simplification rules, and hardware demonstration are useful and transparent. The paper includes detailed circuit diagrams for I0 and oracles, three simplification principles, and full data tables for the Titanic example, which facilitate reproduction. I found no sign of circular reasoning or outcome-dependent parameter fitting. However, the strongest advertised claims—near-100% success and complexity advantage—currently rest on an estimator whose accuracy is not proved, and the 6-qubit validation is compromised by a wrong success condition. These are fixable, but they need to be corrected before the claims can be accepted.

major comments (4)
  1. [Section 4.2, text before Fig. 11] The success condition for the minimum-search simulation is stated as "If d1 ≥ d0, the algorithm is thought to operate successfully", but the oracle constructed for minimum search marks all values ≤ d0. A correct minimum-search success condition is d1 ≤ d0. As written, the 6-qubit validation may count measurements above d0 as successes, so the reported comparison with QESA in Section 4.2 does not support the claimed advantage for minimum search.
  2. [Section 5.1, Eq. (14) and Fig. 13] The near-100% success claim depends on M̃/Ñ closely approximating M/N, but the implemented estimator M̃ = d0 + 1, Ñ = 2^n (Section 4.2) is exact only when all code states in [0, 2^n − 1] hold data values and the data are uniform. Hypothesis (6) explicitly permits zero-amplitude basis states, and the Titanic excerpt stores only 36 values in a 64-state 6-qubit code space, so M̃ can systematically exceed the true number of marked items. The paper gives no quantitative bound on the Grover-Long failure rate as a function of the estimation error, and the sampling estimator in Eq. (14) is not inserted into Algorithm 1; hence the abstract's "close to 100%" claim and the complexity comparison in Eq. (21) are not established.
  3. [Section 5.2, Eq. (21) and Fig. 15] The total failure probability ε enters the final complexity formula, but it is never derived from the outer-loop parameter c and the Grover-Long parameter error; Fig. 15 fixes ε = 0.1 without justification. The two failure contributions in Section 5.1 are analyzed separately, so the paper does not prove the overall QUMMSA failure probability or the claimed complexity advantage over DHA for large databases.
  4. [Section 3.1, Eq. (1)] For the general case N < 2^n admitted by Hypothesis (6), the preparation operator W that creates the uniform superposition over the N occupied basis states is not constructed and no circuit for it is given, despite Hypothesis (5) assigning it cost log2(N). Without such a construction, the claim in Section 3 of a general design method for QUMMSA circuits and the complexity accounting in Section 5.2 are incomplete.
minor comments (5)
  1. [Section 5.1] The sentence "Therefore, we assume that each orthonormal basis state stores a data value" appears twice verbatim in the paragraph before Eq. (13); the duplicate should be removed.
  2. [Section 4.2] The text refers to "Fig 10(a)" when discussing the theoretical failure rate of Grover-Long algorithm, but the relevant panels appear in Fig. 12; the figure cross-references should be renumbered.
  3. [References] References [25] and [26] are identical entries for Vandersypen et al.; one should be removed or replaced with a distinct citation.
  4. [Appendix B, Table 2] The column header "Name Age After Encoding" repeats inside the table, and the column alignment is inconsistent; the table should be reformatted so each data column has a single clear header.
  5. [Throughout] There are several typographical errors: "donates" in the Fig. 14 caption should be "denotes", "combing" in Section 6 should be "combining", and "Durr, Hoyer's" should be "Dürr–Høyer's".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algorithm's parameters come from an explicit estimation assumption and external Grover-Long theory, not from fitting the claimed success.

full rationale

The paper's central derivation is self-contained in the relevant sense. QUMMSA replaces DHA's QESA with the Grover-Long algorithm, whose phase and iteration parameters are taken from the external Refs. [16,17] (Eqs. (3)-(4)); the paper does not redefine those parameters in terms of its own success metric. The estimator M̃ = d0+1, Ñ = 2^n (Section 4.2, before Fig. 11, and Section 5.1) is explicitly introduced as an assumption for databases with no prior knowledge, and the paper then quantifies the resulting failure rate as a function of the gap between M̃/Ñ and M/N (Fig. 13 and Eqs. (C5)-(C7)). This is a robustness analysis, not a fitted parameter relabeled as a prediction. The sampling-based improvement in Section 5.1 is conditional ('As long as P̃(x) ≈ P(x), we can calculate the parameters of Grover-Long algorithm, accurately'), and the complexity comparison in Section 5.2 uses the same openly stated M0 ≈ N/2 and ε = 0.1 assumptions. No equation reduces to itself by construction, and the paper's load-bearing references (Grover-Long, DHA, Boyer et al.) are external prior work, not self-citations. The paper's own caveats about rising failure rates for large databases are limitations rather than circularity; the possibly inverted success criterion d1 ≥ d0 in Section 4.2 and the unintegrated sample-size estimator are correctness/soundness gaps, not circular reasoning.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central claim rests on two free parameters (M̃, Ñ) and the stopping constant c, plus assumptions about the database distribution and the exactness of Grover-Long.

free parameters (3)
  • c = positive integer, user-chosen
    Stopping rule constant in Algorithm 1; controls the trade-off between failure rate and runtime.
  • M̃ = d0 + 1
    Estimated number of marked items for the Grover-Long parameters; assumes uniform database over the code space.
  • Ñ = 2^n
    Estimated database size; overestimates N when the database does not fill every code word.
assumptions (5)
  • domain assumption Grover-Long search has zero failure rate when M/N is known exactly and the phase is chosen by formula (3).
    Background result from Ref [16] that the paper uses as its key replacement for QESA.
  • domain assumption Each data value is stored in a distinct orthonormal basis state with amplitude 1/sqrt(N) and unused states have amplitude 0.
    Hypothesis (6) in Section 2; needed so that the initial state matches the Grover framework.
  • ad hoc to paper The estimated database distribution is uniform over [0, 2^n - 1], so M̃/Ñ approximates M/N.
    Used to set Grover-Long parameters in Sections 4.1 and 4.2; not true for sparse or skewed databases.
  • domain assumption The output of Grover-Long is uniformly distributed over marked states, so the expected number of main loops is log2 N and the stopping rule analysis holds.
    Used in Sections 2 and 5.1 for the expectation of main loops and the (1/r0)^c stopping probability.
  • ad hoc to paper The classical sample size formula h = Z^2 sigma^2 / E^2 applies to quantum database estimation.
    Invoked in Section 5.1 to justify sample-based estimation, but not integrated into Algorithm 1 and no proof connects it to the quantum failure rate.

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Cite this review

Pith. "Pith review of An Optimized Quantum Maximum or Minimum Searching Algorithm and its Circuits." pith.science (2026). https://pith.science/paper/VYY64XG4

@misc{pith2026190807943,
  author       = {Pith},
  title        = {Pith review of: An Optimized Quantum Maximum or Minimum Searching Algorithm and its Circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYY64XG4}},
  note         = {Machine review of arXiv:1908.07943}
}
read the original abstract

Finding a maximum or minimum is a fundamental building block in many mathematical models. Compared with classical algorithms, Durr, Hoyer's quantum algorithm (DHA) achieves quadratic speed. However, its key step, the quantum exponential searching algorithm (QESA), which is based on Grover algorithm, is not a sure-success algorithm. Meanwhile, quantum circuits encounter the gate decomposition problem due to variation of the scale of data. In this paper, we propose an optimized quantum algorithm for searching maximum and minimum, based on DHA and the optimal quantum exact search algorithm. Furthermore, we provide the corresponding quantum circuits, together with three equivalent simplifications. In circumstances when we can exactly estimate the ratio of the number of solutions M and the searched space N, our method can improve the successful probability close to 100%. Furthermore, compared with DHA, our algorithm shows an advantage in complexity with large databases and in the gate complexity of constructing oracles. Experiments have been executed on an IBM superconducting processor with two qubits, and a practical problem of finding the minimum from Titanic passengers' age was numerically simulated. Both showed that our optimized maximum or minimum performs more efficiently compared with DHA. Our algorithm can serve as an important subroutine in various quantum algorithms which involves searching maximum or minimum.

Figures

Figures reproduced from arXiv: 1908.07943 by the authors.

Figure 1
Figure 1. The general circuit for 𝐼0 operator, where q[0] denotes the lowest qubit, q[n-1] denotes the highest qubit. 3.3 The design of oracle O Oracle can recognize the solutions of a searching problem. If one orthonormal basis state is one of solutions, it will receive a rotation phase. Here, we elaborate on the construction of oracle from two parts: the searching problem has a unique solution or multiple solutions. Firstly… view at source ↗
Figure 2
Figure 2. General circuits for different oracles which mark an odd state [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. General circuits for different oracles which mark an even state. Secondly, we should discuss that the oracle can mark 𝑀 (0 < 𝑀 ≤ 2 𝑛) quantum states. Namely, the number of solutions is 𝑀. The oracle can be described as a diagonal matrix. 𝑂 = 𝑒 𝑖𝜙 ∑|𝑣𝜏 ⟩⟨𝑣𝜏 | 𝑀 𝜏=1 + ∑ |𝜏⟩⟨𝜏| 2 𝑛−1 𝜏=0,𝜏∉𝑉 (7) Where the number of 𝑒 𝑖𝜙 is M and the set of 𝑒 𝑖𝜙 position is 𝑉 = {𝑣1 , 𝑣2 , … , 𝑣𝑀}. The oracle marking multiple quantum sta… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: A general circuit for marking states |0⟩ and |1⟩. 3.4 Three equivalent simplified principles If there are 2 𝑚 (1 ≤ 𝑚 ≤ 𝑛 − 1) solutions, the oracle will become very complex. Besides, it’s difficult to execute too many entanglement gates on current quantum computers. He…
Figure 5
Figure 5. Figure 5: A schematic diagram of the first equivalent simplified principle. (a) The circuit for marking two continuous odd states by an (n − 1)-qubit controlled phase gate(it has n − 2 control qubits and a single target qubit). (b) The circuit for marking 2 𝑚 continuous odd stat…
Figure 6
Figure 6. Figure 6: A schematic diagram of the second equivalent simplified principle. (a) The original circuit. (b) The simplified circuit. Thirdly, to simplify the oracles marking even and odd states, we proposed the third principle, through elementary algebraic transformation [36] . Af…
Figure 7
Figure 7. Figure 7: A schematic diagram of the third equivalent simplified principle. (a) The original circuit. (b) The simplified circuit [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: 5-qubit superconducting processor: (a)schematic; (b) topology. The experimental device is IBMQ Yorktown which consists of five coupled superconducting transmons. Limited by the accuracy of the experimental device, two qubits 𝑄0 , 𝑄2 were used for the experiment of Grov…
Figure 9
Figure 9. Figure 9: (a) A 2-qubit circuit of Grover-Long algorithm. (b) A 2-qubit circuit of the initial state preparation. (c) A 2-qubit and an ancilla qubit circuit of Grover algorithm [36] . (d) The experimental result and the theoretical result. Among them, |Ψ⟩ is the initial state; |…
Figure 15
Figure 15. Figure 15: Complexity comparison for two algorithms. 6. Conclusions In summary, we demonstrate the advantage of the quantum algorithm for finding maximum or minimum value to alleviate some of the challenges brought by the rapidly increasing amount of data. Based on Grover-Long a…

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