REVIEW 2 major objections 4 minor 1 cited by
Depth optimization of quantum search algorithms beyond Grover's algorithm
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Grover's algorithm is not optimal in circuit depth: mixing global and local Grover operators reduces expected depth whenever the oracle-to-diffusion depth ratio is finite.
desk verdict Finite-size depth reduction is real and worth a look; the asymptotic critical-ratio theorems are not established as stated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the mixed Grover sequence $S_{n,m}(j_1,\ldots,j_q) = G_n^{j_1} G_m^{j_2} \cdots$, built from the global Grover operator $G_n = D_n U_t$ and the local Grover operator $G_m = D_m U_t$, where $D_m$ is a partial diffusion operator acting on the same $m$-qubit block in every occurrence. In the three-state basis $\{|t\rangle, |ntt\rangle, |u\rangle\}$ borrowed from quantum partial search, both operators become $O(3)$ matrices, so the success probability of any sequence is obtained by multiplying $3 \times 3$ matrices. The optimization is the minimization of expected depth $d(S)/P(S)$ over the block size $m$ and the non-negative exponents; the critical ratio $\alpha_c$ is the value of $\alpha$ at which this minimum crosses Grover's expected depth $d_G(\alpha)$.
What would settle it
Compute the exact $3\times 3$ matrix for the sandwich sequence $S_{n,n-1}(1,1,1)$ without small-angle approximations, use the paper's Toffoli depth table for $d(D_n)$, and check inequality (45) for a range of $\alpha$ and $n$; if no positive $\alpha$ satisfies it, the claimed $O(n^{-1}2^{n/2})$ critical ratio fails. Alternatively, enumerate all sequences with at most a fixed number of oracle calls for $n=4,\dots,20$ and compare the numerical $\alpha_{c,1}$ to the asymptotic scaling.
Extended reading notes
Core claim
The central discovery is a depth-optimality result: for finite $\alpha$, Grover's algorithm is never the depth-optimal search strategy, and the improvement is quantified by critical ratios. For the one-stage algorithm, $\alpha_{c,1} = O(n^{-1}2^{n/2})$, meaning even an oracle considerably deeper than the diffusion operator still admits a shallower sequence for large database size $N = 2^n$. For the two-stage algorithm, $\alpha_{c,2}$ tends to $1 + \sqrt{3} \approx 2.732$, so a two-stage search wins only when the oracle depth is comparable to the global diffusion depth; its practical payoff is error mitigation and parallelization. The depth saving comes from replacing some global diffusion operators $D_n$ with local ones $D_m$: the local operator is cheaper, and the small loss in success probability is more than compensated. The argument represents $G_n$ and $G_m$ as $O(3)$ matrices on the three-dimensional subspace spanned by the target state, the sum of nontarget states in the target block, and the sum of states in all other blocks, then compares expected depths of candidate sequences.
Load-bearing premise
The whole analysis assumes every local diffusion operator in a sequence acts on the same set of qubits, so the amplitude dynamics stays in the three-state subspace $\{|t\rangle, |ntt\rangle, |u\rangle\}$; if a sequence mixes different local subspaces, that three-state description and every critical-ratio formula derived from it break down.
Editorial extensions
If this is right
- For any finite oracle-to-diffusion depth ratio, the minimal expected depth is strictly below Grover's, so depth-constrained implementations should replace a fraction of global diffusion operators with local ones.
- The one-stage critical ratio grows as $O(n^{-1}2^{n/2})$, so for large databases the depth advantage survives even when the oracle is much deeper than the diffusion operator.
- The two-stage critical ratio tends to $1+\sqrt{3}$, which delimits when the two-stage rescaling strategy beats Grover's expected depth; beyond that, the two-stage design remains useful for error reduction and parallelization.
- Dividing the search into stages, each finding part of the target address, gives a natural way to run search in parallel on multiple quantum computers.
- The depth reduction does not change the oracle-complexity benchmark: the optimized sequences still use $O(\sqrt{N})$ oracle calls and preserve Grover's quadratic query speedup.
Reading between the lines
- The paper fixes one block size per stage; allowing several block sizes within a single sequence is a natural extension it leaves open, and the same matrix technique could optimize that larger family.
- If local diffusion operators are allowed to act on different qubit subsets from one application to the next, the three-state truncation fails; a testable extension is to map the resulting multi-subspace dynamics and see whether even lower depths exist.
- The same ratio-based depth comparison could be applied to fixed-point or amplitude-amplification variants of Grover search, using the generalized phase oracle the paper mentions but does not optimize.
- Hardware with limited qubit connectivity may make local diffusion on a connected block cheaper than the Toffoli depth table used here, which would widen the depth gap; poor connectivity could erase it, so the critical-ratio framework doubles as a hardware-calibration tool.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes optimizing the circuit depth of Grover search by mixing global diffusion operators D_n with local diffusion operators D_m, measuring efficiency through the expected depth d(α)=depth/success probability for a fixed oracle-to-diffusion depth ratio α. It claims that Grover's algorithm is not optimal in depth, presents critical ratios α_c,1=O(n^{-1}2^{n/2}) for one-stage and lim α_c,2=1+√3 for two-stage search, and supports the proposal with numerical tables and an explicit n=6 example in which S_{6,4}(1,1,2) has success probability ≈0.755, single-run depth 360, and expected depth ≈477 versus Grover's ≈617 at α=1. The paper also discusses multistage and parallel search implementations. The central existence claim for a finite-depth advantage is credible from the explicit example, but the asymptotic critical-ratio theorems are not established as stated.
Significance. If the finite-depth advantage is accepted under the stated depth model, the paper provides a useful NISQ-oriented observation: replacing some global diffusion operators by same-subspace local diffusion operators can reduce expected depth. The explicit n=6 calculation and the numerical Tables I-III give reproducible evidence for this advantage. The multistage and parallel-running discussions are plausible extensions. However, the headline scaling claims in the abstract and in Section IV are not supported by the proofs: Theorem 1 supplies a lower bound on the critical ratio, not the claimed upper bound, and Theorem 2 contains a sign/factor inconsistency. With those theorems corrected or reframed as sufficient conditions, the core finite-n contribution would remain valuable.
major comments (2)
- [Sec. IV A, Theorem 1] The proof analyzes only the specific sequence S_{n,n-1}(1,1,1). Inequality (45)-(46) gives a sufficient condition under which this sequence beats Grover, namely α below a quantity of order Ω(n^{-1}2^{n/2}). This is a lower bound on α_c,1, not an upper bound. The statement α_c,1=O(n^{-1}2^{n/2}) is therefore not proved: other sequences could still beat Grover for larger α, and the proof contains no argument excluding that possibility. The abstract's scaling claim should be revised to a lower-bound statement unless an upper bound is supplied.
- [Sec. IV B, Theorem 2] The displayed inequality (55) and its solution (56) are inconsistent. With j=√3 j̃, the two-stage expected depth is approximately j̃[(2α+1)d(D_n)+d(D_2)]/(P_n(j)+δ), so the correct comparison against Grover is (α+1)d(D_n)/P_n(j) > [(2α+1)d(D_n)+d(D_2)]/[√3(P_n(j)+δ)]. In the large-N limit this gives α<1+√3 as a sufficient condition for the constructed sequence to win, not α>1+√3 as written in Eq. (56). Consequently the proof does not establish lim α_c,2=1+√3; it establishes at most a lower-bound direction for the critical ratio. The theorem statement and the text following Eq. (56) need correction.
minor comments (4)
- [Eq. (14) and Table II/Appendix A] The sequence notation is inconsistent: Eq. (14) defines S_{n,m}(j1,...,jq) with G_n first and G_m last, but Table II and Appendix A use the reverse convention, e.g. S_{6,4}(1,1,2)=G4G6G4^2. Please standardize the convention in both the definition and the examples.
- [Tables I-III and Appendix B] The depth values used in the tables do not match the stated formula. Appendix B defines d(D_n)=d(Λ_{n-1}(X))+2, which gives d(D_6)=63 and d(D_4)=15, but Table II's header lists d(D_6)=64; the single-run depth 360 for n=6 corresponds to 63 and 15, not 64 and 16. Please reconcile the tables, headers, and appendix.
- [Eq. (45)-(46)] The exact consequence of inequality (45) is α < [2(d(D_n)-d(D_{n-1}))P_n(j)-3d(D_n)δ]/[3d(D_n)δ]; the expression in Eq. (46) omits the -1 term that comes from moving α+1 across the inequality.
- [Throughout] There are a few typographical issues, including 'expect depth' immediately after Eq. (45), and the garbled formatting of Eq. (55) in the text. These should be cleaned up together with the substantive fixes.
Circularity Check
No circular derivation: the depth advantage is computed from explicit gate decompositions and algebraic probabilities; reused QPSA material is a tool, not the target claim.
full rationale
The paper's central claim—that Grover's algorithm is not optimal in depth—is supported by explicit comparisons of expected depth between Grover's algorithm and alternative sequences of global and local Grover operators. The definitions dG(α) in Eq. (12), d1(α) in Eq. (17), and d2(α) in Eq. (26) are genuine minima over search sequences; the inequality d1(α) ≤ dG(α) in Eq. (21) follows by construction because Grover's algorithm is included as the special case with no local operators, but this is a bounding inequality, not the paper's conclusion that d1(α) < dG(α) for finite α. The explicit n = 6 example (S6,4(1,1,2), depth 360, success probability 0.755, expected depth ≈477 versus Grover's ≈617 at α = 1) is an internally consistent computation from the algebraic success probabilities and the gate depths taken from the external Barenco et al. decomposition [27]. No parameter is fitted to data and then renamed as a prediction; the numerical optimizations for n = 4,...,10 are direct minimizations, not fits. The reuse of the QPSA formalism and O(3) representations from the authors' earlier work [22,23,24,31] is a tool: the target assertion that Grover's algorithm is not depth-optimal is not asserted by those citations, and the specific matrices and probabilities are computed in the present paper. Theorems 1 and 2 contain apparent proof gaps—Theorem 1 establishes only a sufficient condition of order Ω(n^{-1}2^{n/2}) rather than the claimed upper bound O(n^{-1}2^{n/2}), and Theorem 2's inequality (55) appears to have a sign/substitution inconsistency—but these are correctness concerns, not circularity. They do not make the derivation equivalent to its inputs. The self-citations are minor and not load-bearing for the central existence claim, so the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (4)
- domain assumption Depth is counted with a universal gate set and is additive for concatenated unitaries.
- domain assumption Oracle depth scales polynomially with n and alpha >= 1 for the one-target search.
- domain assumption The three-dimensional subspace spanned by |t>, |ntt>, |u> is invariant for any global/local Grover sequence when all local diffusions act on the same qubits.
- standard math The n-qubit Toffoli gate depths from Barenco et al. [27] are correct and apply to the diffusion operator.
Cite this review
Pith. "Pith review of Depth optimization of quantum search algorithms beyond Grover's algorithm." pith.science (2026). https://pith.science/paper/B7YNYHR6
@misc{pith2026190804171,
author = {Pith},
title = {Pith review of: Depth optimization of quantum search algorithms beyond Grover's algorithm},
year = {2026},
howpublished = {\url{https://pith.science/paper/B7YNYHR6}},
note = {Machine review of arXiv:1908.04171}
}
read the original abstract
Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a new depth optimization method for quantum search algorithms. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for parallel running of the quantum search algorithm.
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Reference graph
Works this paper leans on
-
[1]
We also study the depth optimization in the multi-stage quantum search algorithm
In examples, our algorithm has around 20% lower depth than Grover’s algorithm. We also study the depth optimization in the multi-stage quantum search algorithm. In each stage, the circuit has lower depth than in Grover’s algorithm. The multistage quantum search algorithm gives a natural way for parallel running of the quantum search algorithm. Ideas in th...
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[2]
Then the oracle gives a sign flip acting on the target state: Uf(1 12n⊗H)|x⟩⊗| 1⟩ = (−1)f (x)(1 12n⊗H)|x⟩⊗| 1⟩ (2) Here 1 12n is the identity operator on the 2n dimensional Hilbert space. For convenience, we denote the oracleUf as Ut = 1 12n− 2|t⟩⟨t| (3) if the ancillary qubitH|1⟩ is prepared. The general phase flip can be constructed as follows:Ut,φ = 1 12...
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[3]
Because N = 2n is a large num- ber, the angleθ2 is a small number. We can expand: γ = 3 √ 2θ2 +O ( θ2 2 ) , (41a) ⟨v±|sn⟩ = 1√ 2 +O (θ2) (41b) We substitute the above relations into Eq. (40). After some al- gebra, we can get the success probability of finding the target state: |⟨t|S ˜j n,n−1(1, 1, 1)|sn⟩|2 = sin2 ( 3 √ 2˜jθ2 ) +O(θ2) (42) Because the sandw...
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[4]
One-oracle Algorithm • Grover’s algorithm. The one-iteration Grover’s algo- rithm gives |0⟩ H Ut D6 |0⟩ H |0⟩ H |0⟩ H |0⟩ H |0⟩ H Measurements at the end are computational basis mea- surements. The whole circuit has depth d(G6) = 126 (A5) We can incorporate the initial Hadamard gates intoG6. The success probability of finding the target state is P6(1) =|⟨t...
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[5]
We can also design a two-stage quantum search algorithm
Two-oracle Algorithm We can apply same strategy for the two-iteration search al- gorithm: design a circuit with local diffusion operators and find the optimal one with the least expected depth. We can also design a two-stage quantum search algorithm. And for each stage we use two oracles. • Grover’s algorithm. The two-iteration Grover’s algo- rithm gives: ...
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[6]
We divide the target state into two parts: |t1⟩ and|t2⟩
= 252 (A11) The success probability of finding the target state is P6(2, 0) =|⟨t|G2 6|s6⟩|2≈ 0.3439, (A12) and the expected depth is d(G2 6) P6(2)≈ 733 (A13) • Our two-stage search algorithm. We divide the target state into two parts: |t1⟩ and|t2⟩. Here t1 is two bits long and t2 is four bits long. Accordingly, we can design a search algorithm which has tw...
-
[7]
M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (2010)
work page 2010
-
[8]
C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, SIAM journal on Computing 26, 1510 (1997)
work page 1997
Show all 41 references
-
[9]
L. K. Grover, Physical Review Letters 79, 325 (1997)
1997
-
[10]
P. R. Giri and V . E. Korepin, Quantum Information Processing 16, 315 (2017)
2017
-
[11]
Barends, J
R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell,et al., Nature 508, 500 (2014)
2014
-
[12]
Ballance, T
C. Ballance, T. Harty, N. Linke, M. Sepiol, and D. Lucas, Phys- ical Review Letters 117, 060504 (2016)
2016
-
[13]
Figgatt, D
C. Figgatt, D. Maslov, K. Landsman, N. M. Linke, S. Debnath, and C. Monroe, Nature Communications 8, 1918 (2017)
2017
-
[14]
Arute, K
F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, et al., Nature 574, 505 (2019)
2019
-
[15]
Preskill, Quantum 2, 79 (2018)
J. Preskill, Quantum 2, 79 (2018)
2018
-
[16]
A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta, Physical Review A 100, 032328 (2019)
2019
-
[17]
L. K. Grover, Physical Review Letters 80, 4329 (1998)
1998
-
[18]
Brassard, P
G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Contemporary Mathematics 305, 53 (2002)
2002
-
[19]
Tulsi, Physical Review A 86, 042331 (2012)
A. Tulsi, Physical Review A 86, 042331 (2012)
2012
-
[20]
Tulsi, Physical Review A 91, 052307 (2015)
A. Tulsi, Physical Review A 91, 052307 (2015)
2015
-
[21]
P. Kim, D. Han, and K. C. Jeong, Quantum Information Pro- cessing 17, 339 (2018)
2018
-
[22]
Jaques, M
S. Jaques, M. Naehrig, M. Roetteler, and F. Virdia, arXiv preprint arXiv:1910.01700 (2019)
2019 arXiv
-
[23]
Boyer, G
M. Boyer, G. Brassard, P. Høyer, and A. Tapp, Fortschritte der Physik: Progress of Physics 46, 493 (1998)
1998
-
[24]
Zalka, Physical Review A 60, 2746 (1999)
C. Zalka, Physical Review A 60, 2746 (1999)
1999
-
[25]
Kato, Physical Review A 72, 032319 (2005)
G. Kato, Physical Review A 72, 032319 (2005)
2005
-
[26]
Jiang, E
Z. Jiang, E. G. Rieffel, and Z. Wang, Physical Review A 95, 062317 (2017)
2017
-
[27]
L. K. Grover and J. Radhakrishnan, in Proceedings of the sev- enteenth annual ACM symposium on Parallelism in algorithms and architectures (ACM, 2005) pp. 186–194
2005
-
[28]
V . E. Korepin and L. K. Grover, Quantum Information Process- ing 5, 5 (2006)
2006
-
[29]
V . E. Korepin, Journal of Physics A: Mathematical and General 38, L731 (2005)
2005
-
[30]
V . E. Korepin and J. Liao, Quantum Information Processing 5, 209 (2006)
2006
-
[31]
M. E. Morales, T. Tlyachev, and J. Biamonte, Physical Review A 98, 062333 (2018)
2018
-
[32]
T. J. Yoder, G. H. Low, and I. L. Chuang, Physical Review Let- ters 113, 210501 (2014)
2014
-
[33]
Barenco, C
A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Mar- golus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys- ical Review A 52, 3457 (1995)
1995
-
[34]
He, M.-X
Y . He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, In- ternational Journal of Theoretical Physics 56, 2350 (2017)
2017
-
[35]
Gokhale, J
P. Gokhale, J. M. Baker, C. Duckering, N. C. Brown, K. R. Brown, and F. T. Chong, arXiv preprint arXiv:1905.10481 (2019)
2019 arXiv
-
[36]
R. M. Gingrich, C. P. Williams, and N. J. Cerf, Physical Review A 61, 052313 (2000)
2000
-
[37]
V . E. Korepin and B. C. Vallilo, Progress of Theoretical Physics 116, 783 (2006)
2006
-
[38]
Choi and V
B.-S. Choi and V . E. Korepin, Quantum Information Processing 6, 243 (2007)
2007
-
[39]
Zhang and V
K. Zhang and V . Korepin, Quantum Information Processing17, 143 (2018)
2018
-
[40]
V . E. Korepin and Y . Xu, International Journal of Modern Physics B 21, 5187 (2007)
2007
-
[41]
Diao, Physical Review A 82, 044301 (2010)
Z. Diao, Physical Review A 82, 044301 (2010)
2010
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