Circuit-depth optimization of quantum partial-search algorithms

Kun Zhang; Vladimir Korepin; Xiao-Hui Wang; Yan-Bo Jiang

REVIEW

Circuit-depth optimization of quantum partial-search algorithms

Yan-Bo Jiang, Xiao-Hui Wang, Kun Zhang, Vladimir Korepin

Quantum Physics · Submitted 23 Sep 2026 · 59 references

Pith's verdict: The alternating-sequence result is real and analytically derived, not fitted; the headline numbers just depend more on the linear-depth diffusion model than the abstract lets on.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 24, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2609.28147 (version 1)

Pith IDUV2T46LR, Pith's permanent record of this paper and its review

CitationKun Zhang, Vladimir Korepin, Xiao-Hui Wang, Yan-Bo Jiang. "Circuit-depth optimization of quantum partial-search algorithms." arXiv:2609.28147 (2026). https://pith.science/paper/UV2T46LR

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Read the paper; the central claim holds up. The alternating sequence SA = (Gn G~k_m)^k genuinely beats depth-optimized GRK below a critical depth ratio, and the 20-25% reduction is derived, not fitted. That is the thing to know.

What's actually new: this is the first depth-optimal design of partial search, and the claim is well supported. Lemma 1's exact success probability comes from a clean O(3) spectral decomposition and an amplitude recurrence; Corollary 2 tracks the error orders honestly; and the three-regime optimization handles the integer boundary exactly via q_bd rather than rounding a continuous expression. The consistency check that GRK re-emerges at alpha=100 is good evidence the framework is sound. I also found the edge-correction appendix honest: it says outright that the SD family asymptotics stay open near the Case I-II boundary, which is the right thing to disclose.

The soft spot is exactly where the reader put it: every quantitative number—the ~20% finite-size reduction, the 25% asymptote, the critical ratio alpha_c,A—sits on the linear-depth diffusion model (ell_n = xi n, ell_m = xi m) and on the single-oracle-depth parameterization alpha = ell_Ot/ell_n. If real local diffusion doesn't scale linearly in m, or if the local oracle isn't the same depth as the global one, q_A shifts and both the reduction ratio and the crossing move. That's not an internal error—the assumption is disclosed early and the math is coherent within the stated model—but the abstract's "more than 20%" slightly overstates even the paper's own finite-size value of 19.85% at the representative point. Minor, but worth fixing. Also: no code shipped for the scans, which is a small reproducibility hit.

Who this is for: people working on depth-constrained search implementations and on the query-versus-depth distinction generally. The metric choice (expected depth per successful sample) is itself a useful discussion point.

Recommendation: send it to a serious referee. The internal mathematics is careful, the claim is novel within the subfield, and the cost-model sensitivity is disclosed rather than hidden. The referee's job is to pin down the model dependence, not to chase an error in the derivation—I didn't find one.

Referee report

Significance

The contribution is substantial if the model assumptions are granted. The analytical core is careful and internally consistent: I independently re-derived Lemma 1's success probability (via the trace identity tr(T̃k) = 1 − 4 sin²γ sin²λ and the amplitude recurrence a_j = cosγ cos((2j+1)Ω/2)/cos(Ω/2)), the subleading expansion in Theorem 4 (z* = (3qAθ₂)^{1/3} − (2/5)qAθ₂), Eq. (47), and Corollary 14's arithmetic (C_A = 0.0744, C_G = 0.0517, ηA = 19.19% against the numerical 19.85%), and all agree. The constants κ₀–κ₆ come from closed-form optimizations (stationarity of tan y = 2y, envelope theorem), not from fitting, so the central claim is not circular. A genuine strength is that the structural results (Lemma 1, Theorems 4, 6, 8, 11) are stated in generic operator depths (ℓΔ, ℓGm, qA) and reduce to the linear-depth model only at the final quantitative stage; the paper would be strengthened by using this generality to make the reported reduction numbers robust. The work is a natural and well-positioned extension of the recent query-optimality analyses of the GRK algorithm.

Major objections (2)

[Sec. III A; Eqs. (21), (43), (47); Corollary 14; Abstract] The quantitative headline — the ~20% finite-size depth reduction (ηA = 19.85% for SA and ηD = 20.73% for SD at n = 20, m = 10, α = 1), the asymptotic 25% limit along m = n/2, and all quoted αc,A values — is computed exclusively under the linear-depth diffusion model ℓn = ξn, ℓm = ξm (Sec. III A) with the single-oracle parametrization α = ℓ_Ot/ℓn (Eq. 21). The engine of the claimed advantage is qA = (n − m)/(αn + m) (Eq. 43); if a real device's local diffusion is not exactly proportional to m (ancilla-assisted sublinear constructions, or a hardware-specific constant offset), both ηA and αc,A shift, and the operative premise behind the 25% limit is precisely the assumption ℓm/ℓn = 1/2 at m = n/2. I verified that the structural theorems survive because they are parameterized in generic depths, which is a strength, but the Abstract presents 'more than 20%' without this qualification. I request (i) a short sensitivity analysis, for example ℓm = ξ·m^p with p ∈ (0,1] or ℓm/ℓn = ρ·(m/n) for ρ ∈ [0.5, 2], quantifying the movement of ηA and αc,A; and (ii) a qualifying clause in the Abstract and Conclusions stating that the quantitative reductions are obtained under the linear-depth ancilla-free diffusion model.
[Sec. III B, Table II; Abstract; Conclusions] The global-optimality narrative and the finite-size threshold wording need tightening. The identification of the three-parameter family SD as the depth-optimal operator family rests on exhaustive enumeration at a single size, N = 2^10 with a ktot ≤ 24 cutoff (Table II, Sec. III B), and the analytical theory covers SA and GRK but not the optimality of SD (the text itself states 'A closed-form analytical optimization of SD is not available here,' Sec. V B). Relatedly, the Abstract attributes 'more than 20%' to 'our proposed alternating partial-search sequence' (SA), but at the displayed finite size SA alone gives 19.85%, which is below 20%, and only the three-parameter family SD reaches 20.73%; the asymptotic statement is 25%. The threshold claim should be attached to the specific family and size regime, or reworded as 'approximately 20% at n = 20, up to 25% asymptotically,' so that readers are not misled about which construction delivers the stated reduction at finite size.

Minor objections (4)

[Sec. III A, Eq. (25)] The expected-depth metric Eℓ = ℓ(S)/Pr(S) excludes the classical cost of verifying the measured block label, which is disclosed; a one-sentence estimate of this cost relative to ℓ(S) would help readers judge when the model applies.
[Sec. III B, Fig. 3] The caption notes that for m = 7, 8, 9 the operator Gn occupies the entire scan, but the legend does not key the green region clearly; state explicitly in the caption that single-iteration Gn is depth-optimal for these m across the whole scanned α range.
[Sec. IV A, before Lemma 1] The domain restriction 0 < λ ≤ π/2 is asserted before Lemma 1 and justified only later via the folding argument in Sec. IV B and Appendix C; a forward reference or a one-line justification at the point of first use would remove an apparent arbitrariness.
[Sec. III B] State explicitly that the SD-optimality observation is a finite-size (N = 2^10) exhaustive-scan finding and not a proven structural theorem, so that later readers do not over-generalize the claim beyond the enumerated regime; the current wording is honest but could be made unambiguous in one sentence.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

At depth ratios α below a critical value αc,A, an alternating sequence of global and local Grover operators, (Gn Gm^k~)^k, achieves strictly lower expected circuit depth than the depth-optimized Grover–Radhakrishnan–Korepin partial-search…

The reading

Quantum partial search aims to identify the block containing the target rather than the target itself, trading accuracy for speed. The standard algorithm, the global–local–global GRK sequence, is optimal in the number of oracle queries, but query optimality does not imply circuit-depth optimality. The paper claims that the depth-optimal order of the two kinds of Grover operators is instead the alternating sequence (Gn Gm^k~)^k — repeatedly applying the same global-plus-local pair — whenever the oracle is not much deeper than the global diffusion operator. In that regime the alternating sequence is strictly shallower than the depth-optimized GRK form, with a reduction in minimum expected circuit depth of about 20 percent numerically at n=20, m=10, α=1 that approaches 25 percent as the database grows along m=n/2. The boundary between the two regimes is a critical depth ratio αc,A that depends on block size, and the advantage disappears again for large α and for m soon above n/2.

What carries the argument

The central object is the alternating operator T_{k~} = Gn Gm^k~ acting on the three-dimensional O(3) invariant subspace spanned by the target item |t⟩, the uniform non-target state inside the target block |b̄_t⟩, and the uniform state over all non-target blocks |b̄⟩. Because T_{k~} has determinant −1 and its (−1) eigenspace is orthogonal to the initial uniform state, the search dynamics lives entirely in a two-dimensional invariant rotation plane, which yields the exact success probability of Lemma 1. That exact form separates the depth optimization into a Grover stopping-point part (minimizing y/sin²y with the universal constant y0 ≈ 1.1656) and a cost-effectiveness part controlled by the ratio qA = (n−m)/(αn+m) under the linear-depth model ℓn = n, ℓm = m, which encodes how much global-diffusion depth is saved per local iteration relative to the local operator's cost.

Extended reading notes

Core claim

The paper derives an exact success probability for the alternating sequence from O(3) geometry: the product Gn Gm^k~ is an improper orthogonal transformation with one (−1) eigenspace and a two-dimensional invariant rotation plane that contains the initial uniform state, so the block-search amplitude follows a sinusoid with rotation angle Ω = 2 arcsin(sin γ sin λ). From this exact formula (Lemma 1), the paper obtains exact asymptotic expected depths, optimal iteration counts in three parameter regimes, and a complete comparison with the depth-optimized GRK sequence. For a depth ratio α below a critical value αc,A, the alternating sequence is strictly shallower than the depth-optimized GRK sequence; under the linear-depth diffusion model the reduction in minimum expected depth is about 19.9 percent numerically at n=20, m=10, α=1 (20.7 percent for the full three-parameter family SD), tending to 25 percent along m=n/2 as n grows. The critical ratio is non-monotonic in m, with a maximum αc,A^max ≈ 0.2976 × $2^{{n/3}}$ near m ≈ n/3 + 1.2175, and for m > n/2 + c*(α) the partial-search advantage collapses to a single global Grover iteration.

Load-bearing premise

The quantitative depth reduction is computed under the assumption that diffusion depth scales linearly with register size (ℓn = ξn, ℓm = ξm, ancilla-free multi-controlled Toffoli) with a single fixed oracle-depth ratio α; if real diffusion depths are not linear in register size or local diffusion is not substantially cheaper than global, the size of the claimed advantage and the locations of the critical ratios change.

Editorial extensions

If this is right

For 0 < α < αc,A the alternating sequence SA = (Gn Gm^k~)^k is strictly shallower than the depth-optimized GRK sequence, so depth-constrained partial-search implementations should switch to the alternating form in exactly this window.
At α=1 along m=n/2, the reduction in minimum expected circuit depth approaches 25 percent as n grows, with about 19.9 percent numerically at n=20 (ηD ≈ 20.7 percent for the full family SD), so replacing the global-local-global order is cheaper per unit of amplitude amplification at comparable oracle and diffusion costs.
The critical ratio αc,A is non-monotonic in the block-size parameter m, with a maximum ≈ 0.2976 × 2^{n/3} near m ≈ n/3 + 1.2175; the alternating advantage is largest at intermediate partition sizes, not at the most balanced ones.
For m > n/2 + c*(α) with c*(α) ≈ 0.950 at α=1, the partial-search advantage collapses to a single global Grover iteration (Case III), consistent with the known query-complexity behavior of partial search.
Above αc,A the depth-optimal and query-optimal optima coincide again — the GRK form is restored — so query optimality implies depth optimality only when the oracle depth dominates the diffusion depth.

Reading between the lines

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

My extension: the ≈25 percent asymptotic bound looks structural — it comes from replacing one global rotation with a cheaper local-plus-global unit that still delivers a full effective rotation — so a similar depth-reduction bound may hold for other amplitude-amplification settings beyond partial search, including multiple-target versions.
My extension: the analytic machinery (the O(3) rotation identity of Lemma 1 and the squared-sine success probability) is independent of the depth-ratio model, so it survives even where the linear-depth assumption fails; only the location of αc,A and the size of the advantage would shift.
My extension: a testable next step is whether the advantage persists for block-dependent oracles, since the paper's parameterization uses a single fixed ratio α for both global and local moves, and for nonuniform initial distributions, which the authors list as open problems.

Original abstract

Grover's algorithm is optimal in terms of oracle queries. We can trade accuracy for speed, which gives rise to the quantum partial-search algorithm. The partial-search algorithm is implemented using two kinds of Grover operators, global and local, where the former is the standard Grover operator for full search and the latter has a diffusion operator acting on the search subspace. The global-local-global sequence, also known as the Grover-Radhakrishnan-Korepin (GRK) algorithm, has been proved optimal in the oracle-query metric. In this work, we show that the alternating sequence of global and local Grover operators, formed by repeatedly applying a fixed product of global and local Grover operators, can achieve a lower expected circuit depth than the GRK algorithm. Through systematic analysis, we obtain its exact success probability, expected depth, and asymptotically optimal parameters. We derive the boundary, characterized by the ratio between the depths of the oracle and the global diffusion operator, separating the depth-optimal and oracle-optimal partial-search algorithms. When the depths of the oracle and the global diffusion operator are comparable, our proposed alternating partial-search sequence can reduce the minimum expected circuit depth by more than 20\% compared with the GRK algorithm.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-24: 1 request, 6,285 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.83 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Grover's algorithm is optimal in terms of oracle queries. We can trade accuracy for speed, which gives rise to the quantum partial-search algorithm. The partial-search algorithm is implemented using two kinds of Grover operators, global and local, where the former is the standard Grover operator for full search and the latter has a diffusion operator acting on the search subspace. The global-local-global sequence, also known as the Grover-Radhakrishnan-Korepin (GRK) algorithm, has been proved optimal in the oracle-query metric. In this work, we show that the alternating sequence of global and local Grover operators, formed by repeatedly applying a fixed product of global and local Grover operators, can achieve a lower expected circuit depth than the GRK algorithm. Through systematic analysis, we obtain its exact success probability, expected depth, and asymptotically optimal parameters. We derive the boundary, characterized by the ratio between the depths of the oracle and the global diffusion operator, separating the depth-optimal and oracle-optimal partial-search algorithms. When the depths of the oracle and the global diffusion operator are comparable, our proposed alternating partial-search sequence can reduce the minimum expected circuit depth by more than 20\% compared with the GRK algorithm.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Grover's algorithm is optimal in terms of oracle queries.	background 1.00	0.04	0.02	asserted 1.00
s1	We can trade accuracy for speed, which gives rise to the quantum partial-search algorithm.	background 1.00	0.09	0.03	asserted 0.64
s2	The partial-search algorithm is implemented using two kinds of Grover operators, global and local, where the former is the standard Grover operator for full search and the latter has a diffusion operator acting on the search subspace.	background 0.62	0.09	0.02	asserted 1.00
s3	The global-local-global sequence, also known as the Grover-Radhakrishnan-Korepin (GRK) algorithm, has been proved optimal in the oracle-query metric.	background 1.00	0.04	0.04	asserted 1.00
s4	In this work, we show that the alternating sequence of global and local Grover operators, formed by repeatedly applying a fixed product of global and local Grover operators, can achieve a lower expected circuit depth than the GRK algorithm.	result 0.62	0.95	0.89	asserted 0.97
s5	Through systematic analysis, we obtain its exact success probability, expected depth, and asymptotically optimal parameters.	contribution 0.46	0.97	0.09	asserted 1.00
s6	We derive the boundary, characterized by the ratio between the depths of the oracle and the global diffusion operator, separating the depth-optimal and oracle-optimal partial-search algorithms.	result 0.81	0.93	0.08	asserted 1.00
s7	When the depths of the oracle and the global diffusion operator are comparable, our proposed alternating partial-search sequence can reduce the minimum expected circuit depth by more than 20\% compared with the GRK algorithm.	result 1.00	0.93	0.47	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.63 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Geometric illustration of the two families of partial-search

**Figure 3.** FIG. 3. Phase diagram of depth-optimal partial-search operators

**Figure 5.** FIG. 5. Minimum expected circuit depth

**Figure 6.** FIG. 6. Critical depth ratios as functions of

Showing 4 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.28147 by the authors.

References

Works cited (59)

[1]

Case I: Cost-effective local-iteration regime Figure 3 shows that whenαis below a threshold andm≤ ⌊n/2⌋, the alternating sequenceSA is the depth-optimal oper- ator. We call this Case I. In this regime, local Grover operators are sufficiently cost-effective that additional local iterations are beneficial. Mathematically, minimizingh(x)in Eq. (41) produces ...

View on Pith
[2]

Case II: Local-iteration cost-effectiveness boundary AsqA decreases to orderb −1, the optimalx= ˜k+ 1ap- proaches the first few admissible integers. The integer com- parison must then be performed before taking a continuous expansion. Analytically, we find the boundary value ofq A, below which inserting a local diffusion operator no longer re- duces the e...

View on Pith
[3]

Case III: Large-msingle-iteration solution Figure 3 shows that asmapproachesn/2from above, the quantum partial-search algorithm becomes depth-inefficient. The optimization then gives a single-iteration solution. In this regime, the number of blocksK= 2 n−m becomes too small for repeated amplitude amplification to retain a quantum ad- vantage. Theorem 8.Un...

View on Pith
[4]

Large-mregime (Case III) The preceding cases assume thaty= (2k+

View on Pith
[5]

Amplitude recurrence OnV 2, the restriction ofT ˜k is a planar rotation. Therefore it satisfies the quadratic identity T 2 ˜k−2 cos ΩT ˜k +I= 0,(B4) when acting on vectors inV 2. Equivalently, letµdenote an eigenvalue ofT ˜k. The full characteristic equation is det µI−T ˜k = (µ+ 1)(µ 2−2 cos Ωµ+ 1) = 0,(B5) but the−1eigenspace component is absent from the...

View on Pith
[6]

Continuous optimization We first treatyas a continuous variable. ForK≫1, the optimalkscales asΘ( √ K/|sin(xθ 2)|), consistent with Corollary 2, so the continuous approximation is accurate at leading order. Taking the derivative ofy/sin 2ywith re- spect toyand setting it to zero gives the stationary condition 15 tany= 2y. The smallest positive root of this...

View on Pith
[7]

Cost-effective local-iteration regime (Case I) The optimal value of the inner iteration count ˜kis deter- mined by the functionh(x)given by Eq. (41) withx= ˜k+ 1. For any admissiblex >0outside the first positive quarter- period, we reducexθ 2 moduloπand, if necessary, reflect the remainder aboutπ/2. The resulting angle lies in(0,π/2]and has the same absol...

View on Pith
[8]

Boundary solution (Case II) The preceding asymptotic optimization, which treatsxas a continuous variable, cannot determine the first integer switch, sincex= ˜k+1is a positive integer. The continuous stationary point equals2at qstat(b) = tan(2θ2) θ2 −2 = 8 3b +O(b−2).(C16) However, the exact integer transition betweenx= 1andx= 2occurs at a smaller value of...

View on Pith
[9]

I. Kerenidis, J. Landman, A. Luongo, and A. Prakash, q-means: A quantum algorithm for unsupervised machine learning, inAd- vances in Neural Information Processing Systems 32 (NeurIPS 2019)(2019) pp. 4136–4146, arXiv:1812.03584 [quant-ph]

View on Pith Pith review arXiv 2019
[10]

and have since progressed across trapped ions [11], su- perconducting circuits [12–15], silicon spin qubits [16], and error-suppressed Grover search on up to five qubits [17]. De- spite this progress, implementations of Grover search remain small-scale demonstrations, reflecting the gap between an asymptotic query advantage and the resource limitations of...

View on Pith Pith review arXiv 2026
[11]

sin(xθ2)/ √ Kcan reachy 0, withk=O( √ K/sinλ)as specified in Corollary 2. To locate the loss of this asymptotic advantage, setm=n/2 +cwithc=O(1). The success probability of a single application ofG n isK−1 1 +o(1) , so Eℓ(Gn)∼n(α+ 1)2 n/2−c.(C21) For the competing Case I solution,m/n= 1/2 +o(1)and Eq. (47) gives Emin,asy ℓ,A ∼κ 0n(α+ 1/2)2n/2.(C22) Equati...

View on Pith
[12]

Depth optimization of the GRK algorithm Proof of Lemma 9.Using √ b= √ N/ √ Kandn−m= (α+ 1)nqG, Eqs. (57) and (59) give, through order √ b, Eℓ,G(vtot,v 2) = (α+ 1)nv tot √ N sin2(2vtot) − (α+ 1)n √ b sin2(2vtot) qGv2 + 2vtot(sin(2v2)−v 2) sin(4vtot) sin2(2vtot) +o((α+ 1)n √ b).(D1) Herev tot andv 2 are the total and local GRK iteration counts rescaled by √...

View on Pith
[13]

Proof of Theorem 11 Normalize Eqs. (63) and (51) byκ 0n2n/2 and retain the integer minimum in the alternating depth. The one-to-one change of variables α+ m n = n−m nqA , α+ 1 = n−m n 1 +qA qA (D3) also givesqG =qA/(1 +qA). Equating the two depths and cancelling the common factor(n−m)/(nq A)yields Eq. (66) atqA =qA,c. For uniqueness, letA j(α)be the norma...

View on Pith
[14]

Proof of Corollary 12 Consider the interior regime of the Case I windowb −1≪ qasy A,c≪b 1/2 specified in Theorem 4. In this regime, substitut- ingα+m/n= (n−m)/(nq asy A,c)andq G =q asy A,c/(1 +q asy A,c) 18 into Eqs. (47) and (63), then cancelling(n−m)/n, gives Eq. (68). The depth difference is strictly decreasing inα, since its derivative consists of neg...

View on Pith
[15]

Proof of Corollary 13 In the large-nlimit,q G = (n−m)/[n(α+ 1)]→0, so Eq. (D6) controls the leading scaling. Sinceαasy c,A =u 3 c−m/n grows exponentially, its stationarity condition isduc/dm= 0 to leading order. Solving Eq. (D6) together with itsmderiva- tive gives uc≃ 3 5κ−1 6 n−m n 1/3 2m/3,(D9a) 2(3m−n)/2≃ 50 27 κ0κ3 6 κ4 .(D9b) The second relation yie...

View on Pith
[16]

=O |δ|/(qAb)1/3 , which is small under the stated as- sumption. The corresponding change within the rotation sub- space isO(|δ|/ √ N). Equation (B1) also gives⟨ ¯b|v−⟩= O(sinγ), so the component injected into the(−1)eigenspace has the same order, while the quadratic remainder isO(δ2/b). These estimates prove Eq. (E4). Equation (E3) is unconditional, where...

View on Pith
[17]

L. K. Grover, A fast quantum mechanical algorithm for database search, inProceedings of the 28th ACM Symposium on Theory of Computing (STOC)(1996) pp. 212–219

View on Pith 1996
[18]

L. K. Grover, Quantum mechanics helps in searching for a nee- dle in a haystack, Physical Review Letters79, 325 (1997)

View on Pith 1997
[19]

C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM Jour- nal on Computing26, 1510 (1997)

View on Pith 1997
[20]

C. Zalka, Grover’s quantum searching algorithm is optimal, Physical Review A60, 2746 (1999)

View on Pith 1999
[21]

G. Brassard, P. Høyer, M. Mosca, and A. Tapp, Quantum am- plitude amplification and estimation, inQuantum Computation and Information, Contemporary Mathematics, V ol. 305, edited by S. J. Lomonaco, Jr. and H. E. Brandt (American Mathemat- ical Society, 2002) pp. 53–74, arXiv:quant-ph/0005055

View on Pith Pith review arXiv 2002
[22]

W. P. Baritompa, D. W. Bulger, and G. R. Wood, Grover’s quan- tum algorithm applied to global optimization, SIAM Journal on Optimization15, 1170 (2005)

View on Pith 2005
[23]

M. Grassl, B. Langenberg, M. Roetteler, and R. Steinwandt, Applying Grover’s algorithm to AES: Quantum resource es- timates, inPost-Quantum Cryptography (PQCrypto 2016) (Springer, 2016) pp. 29–43

View on Pith 2016
[24]

E. Dantsin, V . Kreinovich, and A. Wolpert, On quantum ver- sions of record-breaking algorithms for SAT, ACM SIGACT News36, 103 (2005)

View on Pith 2005
[25]

I. L. Chuang, N. Gershenfeld, and M. Kubinec, Experimen- tal implementation of fast quantum searching, Physical Review Letters80, 3408 (1998)

View on Pith 1998
[26]

C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Deb- nath, and C. Monroe, Complete 3-qubit Grover search on a programmable quantum computer, Nature Communications8, 1918 (2017)

View on Pith 2017
[27]

A. Dewes, R. Lauro, F. R. Ong, V . Schmitt, P. Milman, P. Bertet, D. Vion, and D. Esteve, Quantum speeding-up of computation demonstrated in a superconducting two-qubit processor, Physi- cal Review B85, 140503(R) (2012)

View on Pith 2012
[28]

K. Zhang, P. Rao, K. Yu, H. Lim, and V . Korepin, Implemen- tation of efficient quantum search algorithms on NISQ comput- ers, Quantum Information Processing20, 233 (2021)

View on Pith 2021
[29]

F. Yan, Z. Song, H. Li, K. Zhang, Y . Li, Z. Wang, X. Sun, and D. Yu, Scalable algorithm simplification using quantum AND logic, Nature Physics19, 126 (2023)

View on Pith 2023
[30]

M. AbuGhanem, Characterizing Grover search algorithm on large-scale superconducting quantum computers, Scientific Re- ports15, 1281 (2025)

View on Pith 2025
[31]

I. Thorvaldson, D. Poulos, C. M. Moehle, S. H. Misha, H. Edl- bauer, J. Reiner, H. Geng, B. V oisin, M. T. Jones, M. B. Don- nelly,et al., Grover’s algorithm in a four-qubit silicon processor above the fault-tolerant threshold, Nature Nanotechnology20, 472 (2025)

View on Pith 2025
[32]

B. Pokharel and D. A. Lidar, Better-than-classical Grover search via quantum error detection and suppression, npj Quan- tum Information10, 23 (2024)

View on Pith 2024
[33]

J. Preskill, Quantum computing in the NISQ era and beyond, Quantum2, 79 (2018)

View on Pith 2018
[34]

F. Leymann and J. Barzen, The bitter truth about gate-based quantum algorithms in the NISQ era, Quantum Science and Technology5, 044007 (2020)

View on Pith 2020
[35]

L. K. Grover and J. Radhakrishnan, Is partial quantum search of a database any easier?, inProceedings of the 17th Annual ACM Symposium on Parallelism in Algorithms and Architec- tures (SPAA’05)(Association for Computing Machinery, 2005) pp. 186–194

View on Pith 2005
[36]

V . E. Korepin and Y . Xu, Hierarchical quantum search, Interna- tional Journal of Modern Physics B21, 5187 (2007)

View on Pith 2007
[37]

V . E. Korepin and Y . Xu, Quantum search algorithms, Interna- tional Journal of Modern Physics B23, 5727 (2009)

View on Pith 2009
[38]

V . E. Korepin and L. K. Grover, Simple algorithm for partial quantum search, Quantum Information Processing5, 5 (2006)

View on Pith 2006
[39]

V . E. Korepin, Optimization of partial search, Journal of Physics A: Mathematical and General38, L731 (2005)

View on Pith 2005
[40]

V . E. Korepin and J. Liao, Quest for fast partial search algo- rithm, Quantum Information Processing5, 209 (2006)

View on Pith 2006
[41]

Y .-B. Jiang, X.-H. Wang, K. Zhang, and V . Korepin, Asymp- totic bounds on quantum partial search algorithm and its ap- plications to parallel search, Physical Review A114, 012412 (2026)

View on Pith 2026
[42]

K. Zhang, K.-Y . Chen, X.-H. Wang, and V . Korepin, Asymp- totic optimality of Grover–Radhakrishnan–Korepin algorithm, Journal of Physics A: Mathematical and Theoretical59, 315301 (2026), arXiv:2604.15886 [quant-ph]

View on Pith Pith review arXiv 2026
[43]

K. Zhang and V . E. Korepin, Depth optimization of quantum search algorithms beyond Grover’s algorithm, Physical Review A101, 032346 (2020)

View on Pith 2020
[44]

M. Bria ´nski, J. Gwinner, V . Hlembotskyi, W. Jarnicki, S. Pli´s, and A. Szady, Introducing structure to expedite quantum searching, Physical Review A103, 062425 (2021)

View on Pith 2021
[45]

J. Liu and H. Zhou, Hardware efficient quantum search algo- rithm (2021), arXiv:2103.14196 [quant-ph]

View on Pith Pith review arXiv 2021
[46]

K. Zhang, H.-L. Shi, X.-H. Wang, and V . Korepin, Palindromic structure of depth-efficient quantum search algorithms (2026), arXiv:2606.00960 [quant-ph]

View on Pith Pith review arXiv 2026
[47]

J. Burke and C. McGoldrick, Quantum search without global diffusion (2026), arXiv:2604.15435 [quant-ph]

View on Pith Pith review arXiv 2026
[48]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cambridge 20 University Press, 2010)

View on Pith 2010
[49]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Mar- golus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, El- ementary gates for quantum computation, Physical Review A 52, 3457 (1995), arXiv:quant-ph/9503016

View on Pith Pith review arXiv 1995
[50]

M. Saeedi and M. Pedram, Linear-depth quantum circuits for n-qubit Toffoli gates with no ancilla, Physical Review A87, 062318 (2013)

View on Pith 2013
[51]

Y . He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, De- compositions of n-qubit Toffoli gates with linear circuit com- plexity, International Journal of Theoretical Physics56, 2350 (2017)

View on Pith 2017
[52]

B. Zindorf and S. Bose, Efficient implementation of multicon- trolled quantum gates, Physical Review Applied24, 044030 (2025)

View on Pith 2025
[53]

M. Boyer, G. Brassard, P. Høyer, and A. Tapp, Tight bounds on quantum searching, Fortschritte der Physik46, 493 (1998)

View on Pith 1998
[54]

R. M. Gingrich, C. P. Williams, and N. J. Cerf, General- ized quantum search with parallelism, Physical Review A61, 052313 (2000)

View on Pith 2000
[55]

V . E. Korepin and B. C. Vallilo, Group theoretical formulation of a quantum partial search algorithm, Progress of Theoretical Physics116, 783 (2006)

View on Pith 2006
[56]

B.-S. Choi and V . E. Korepin, Quantum partial search of a database with several target items, Quantum Information Pro- cessing6, 243 (2007)

View on Pith 2007
[57]

P.-C. Zhong, W.-S. Bao, and Y . Wei, Quantum partial search- ing algorithm of a database with several target items, Chinese Physics Letters26, 020301 (2009)

View on Pith 2009
[58]

K. Zhang and V . E. Korepin, Quantum partial search for uneven distribution of multiple target items, Quantum Information Pro- cessing17, 143 (2018)

View on Pith 2018
[59]

E. Biham, O. Biham, D. Biron, M. Grassl, and D. A. Lidar, Grover’s quantum search algorithm for an arbitrary initial am- plitude distribution, Physical Review A60, 2742 (1999)

View on Pith 1999

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Case I: Cost-effective local-iteration regime Figure 3 shows that whenαis below a threshold andm≤ ⌊n/2⌋, the alternating sequenceSA is the depth-optimal oper- ator. We call this Case I. In this regime, local Grover operators are sufficiently cost-effective that additional local iterations are beneficial. Mathematically, minimizingh(x)in Eq. (41) produces ...

View on Pith

[2] [2]

Case II: Local-iteration cost-effectiveness boundary AsqA decreases to orderb −1, the optimalx= ˜k+ 1ap- proaches the first few admissible integers. The integer com- parison must then be performed before taking a continuous expansion. Analytically, we find the boundary value ofq A, below which inserting a local diffusion operator no longer re- duces the e...

View on Pith

[3] [3]

Case III: Large-msingle-iteration solution Figure 3 shows that asmapproachesn/2from above, the quantum partial-search algorithm becomes depth-inefficient. The optimization then gives a single-iteration solution. In this regime, the number of blocksK= 2 n−m becomes too small for repeated amplitude amplification to retain a quantum ad- vantage. Theorem 8.Un...

View on Pith

[4] [4]

Large-mregime (Case III) The preceding cases assume thaty= (2k+

View on Pith

[5] [5]

Amplitude recurrence OnV 2, the restriction ofT ˜k is a planar rotation. Therefore it satisfies the quadratic identity T 2 ˜k−2 cos ΩT ˜k +I= 0,(B4) when acting on vectors inV 2. Equivalently, letµdenote an eigenvalue ofT ˜k. The full characteristic equation is det µI−T ˜k = (µ+ 1)(µ 2−2 cos Ωµ+ 1) = 0,(B5) but the−1eigenspace component is absent from the...

View on Pith

[6] [6]

Continuous optimization We first treatyas a continuous variable. ForK≫1, the optimalkscales asΘ( √ K/|sin(xθ 2)|), consistent with Corollary 2, so the continuous approximation is accurate at leading order. Taking the derivative ofy/sin 2ywith re- spect toyand setting it to zero gives the stationary condition 15 tany= 2y. The smallest positive root of this...

View on Pith

[7] [7]

Cost-effective local-iteration regime (Case I) The optimal value of the inner iteration count ˜kis deter- mined by the functionh(x)given by Eq. (41) withx= ˜k+ 1. For any admissiblex >0outside the first positive quarter- period, we reducexθ 2 moduloπand, if necessary, reflect the remainder aboutπ/2. The resulting angle lies in(0,π/2]and has the same absol...

View on Pith

[8] [8]

Boundary solution (Case II) The preceding asymptotic optimization, which treatsxas a continuous variable, cannot determine the first integer switch, sincex= ˜k+1is a positive integer. The continuous stationary point equals2at qstat(b) = tan(2θ2) θ2 −2 = 8 3b +O(b−2).(C16) However, the exact integer transition betweenx= 1andx= 2occurs at a smaller value of...

View on Pith

[9] [9]

I. Kerenidis, J. Landman, A. Luongo, and A. Prakash, q-means: A quantum algorithm for unsupervised machine learning, inAd- vances in Neural Information Processing Systems 32 (NeurIPS 2019)(2019) pp. 4136–4146, arXiv:1812.03584 [quant-ph]

View on Pith Pith review arXiv 2019

[10] [10]

and have since progressed across trapped ions [11], su- perconducting circuits [12–15], silicon spin qubits [16], and error-suppressed Grover search on up to five qubits [17]. De- spite this progress, implementations of Grover search remain small-scale demonstrations, reflecting the gap between an asymptotic query advantage and the resource limitations of...

View on Pith Pith review arXiv 2026

[11] [11]

sin(xθ2)/ √ Kcan reachy 0, withk=O( √ K/sinλ)as specified in Corollary 2. To locate the loss of this asymptotic advantage, setm=n/2 +cwithc=O(1). The success probability of a single application ofG n isK−1 1 +o(1) , so Eℓ(Gn)∼n(α+ 1)2 n/2−c.(C21) For the competing Case I solution,m/n= 1/2 +o(1)and Eq. (47) gives Emin,asy ℓ,A ∼κ 0n(α+ 1/2)2n/2.(C22) Equati...

View on Pith

[12] [12]

Depth optimization of the GRK algorithm Proof of Lemma 9.Using √ b= √ N/ √ Kandn−m= (α+ 1)nqG, Eqs. (57) and (59) give, through order √ b, Eℓ,G(vtot,v 2) = (α+ 1)nv tot √ N sin2(2vtot) − (α+ 1)n √ b sin2(2vtot) qGv2 + 2vtot(sin(2v2)−v 2) sin(4vtot) sin2(2vtot) +o((α+ 1)n √ b).(D1) Herev tot andv 2 are the total and local GRK iteration counts rescaled by √...

View on Pith

[13] [13]

Proof of Theorem 11 Normalize Eqs. (63) and (51) byκ 0n2n/2 and retain the integer minimum in the alternating depth. The one-to-one change of variables α+ m n = n−m nqA , α+ 1 = n−m n 1 +qA qA (D3) also givesqG =qA/(1 +qA). Equating the two depths and cancelling the common factor(n−m)/(nq A)yields Eq. (66) atqA =qA,c. For uniqueness, letA j(α)be the norma...

View on Pith

[14] [14]

Proof of Corollary 12 Consider the interior regime of the Case I windowb −1≪ qasy A,c≪b 1/2 specified in Theorem 4. In this regime, substitut- ingα+m/n= (n−m)/(nq asy A,c)andq G =q asy A,c/(1 +q asy A,c) 18 into Eqs. (47) and (63), then cancelling(n−m)/n, gives Eq. (68). The depth difference is strictly decreasing inα, since its derivative consists of neg...

View on Pith

[15] [15]

Proof of Corollary 13 In the large-nlimit,q G = (n−m)/[n(α+ 1)]→0, so Eq. (D6) controls the leading scaling. Sinceαasy c,A =u 3 c−m/n grows exponentially, its stationarity condition isduc/dm= 0 to leading order. Solving Eq. (D6) together with itsmderiva- tive gives uc≃ 3 5κ−1 6 n−m n 1/3 2m/3,(D9a) 2(3m−n)/2≃ 50 27 κ0κ3 6 κ4 .(D9b) The second relation yie...

View on Pith

[16] [16]

=O |δ|/(qAb)1/3 , which is small under the stated as- sumption. The corresponding change within the rotation sub- space isO(|δ|/ √ N). Equation (B1) also gives⟨ ¯b|v−⟩= O(sinγ), so the component injected into the(−1)eigenspace has the same order, while the quadratic remainder isO(δ2/b). These estimates prove Eq. (E4). Equation (E3) is unconditional, where...

View on Pith

[17] [17]

L. K. Grover, A fast quantum mechanical algorithm for database search, inProceedings of the 28th ACM Symposium on Theory of Computing (STOC)(1996) pp. 212–219

View on Pith 1996

[18] [18]

L. K. Grover, Quantum mechanics helps in searching for a nee- dle in a haystack, Physical Review Letters79, 325 (1997)

View on Pith 1997

[19] [19]

C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and weaknesses of quantum computing, SIAM Jour- nal on Computing26, 1510 (1997)

View on Pith 1997

[20] [20]

C. Zalka, Grover’s quantum searching algorithm is optimal, Physical Review A60, 2746 (1999)

View on Pith 1999

[21] [21]

G. Brassard, P. Høyer, M. Mosca, and A. Tapp, Quantum am- plitude amplification and estimation, inQuantum Computation and Information, Contemporary Mathematics, V ol. 305, edited by S. J. Lomonaco, Jr. and H. E. Brandt (American Mathemat- ical Society, 2002) pp. 53–74, arXiv:quant-ph/0005055

View on Pith Pith review arXiv 2002

[22] [22]

W. P. Baritompa, D. W. Bulger, and G. R. Wood, Grover’s quan- tum algorithm applied to global optimization, SIAM Journal on Optimization15, 1170 (2005)

View on Pith 2005

[23] [23]

M. Grassl, B. Langenberg, M. Roetteler, and R. Steinwandt, Applying Grover’s algorithm to AES: Quantum resource es- timates, inPost-Quantum Cryptography (PQCrypto 2016) (Springer, 2016) pp. 29–43

View on Pith 2016

[24] [24]

E. Dantsin, V . Kreinovich, and A. Wolpert, On quantum ver- sions of record-breaking algorithms for SAT, ACM SIGACT News36, 103 (2005)

View on Pith 2005

[25] [25]

I. L. Chuang, N. Gershenfeld, and M. Kubinec, Experimen- tal implementation of fast quantum searching, Physical Review Letters80, 3408 (1998)

View on Pith 1998

[26] [26]

C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Deb- nath, and C. Monroe, Complete 3-qubit Grover search on a programmable quantum computer, Nature Communications8, 1918 (2017)

View on Pith 2017

[27] [27]

A. Dewes, R. Lauro, F. R. Ong, V . Schmitt, P. Milman, P. Bertet, D. Vion, and D. Esteve, Quantum speeding-up of computation demonstrated in a superconducting two-qubit processor, Physi- cal Review B85, 140503(R) (2012)

View on Pith 2012

[28] [28]

K. Zhang, P. Rao, K. Yu, H. Lim, and V . Korepin, Implemen- tation of efficient quantum search algorithms on NISQ comput- ers, Quantum Information Processing20, 233 (2021)

View on Pith 2021

[29] [29]

F. Yan, Z. Song, H. Li, K. Zhang, Y . Li, Z. Wang, X. Sun, and D. Yu, Scalable algorithm simplification using quantum AND logic, Nature Physics19, 126 (2023)

View on Pith 2023

[30] [30]

M. AbuGhanem, Characterizing Grover search algorithm on large-scale superconducting quantum computers, Scientific Re- ports15, 1281 (2025)

View on Pith 2025

[31] [31]

I. Thorvaldson, D. Poulos, C. M. Moehle, S. H. Misha, H. Edl- bauer, J. Reiner, H. Geng, B. V oisin, M. T. Jones, M. B. Don- nelly,et al., Grover’s algorithm in a four-qubit silicon processor above the fault-tolerant threshold, Nature Nanotechnology20, 472 (2025)

View on Pith 2025

[32] [32]

B. Pokharel and D. A. Lidar, Better-than-classical Grover search via quantum error detection and suppression, npj Quan- tum Information10, 23 (2024)

View on Pith 2024

[33] [33]

J. Preskill, Quantum computing in the NISQ era and beyond, Quantum2, 79 (2018)

View on Pith 2018

[34] [34]

F. Leymann and J. Barzen, The bitter truth about gate-based quantum algorithms in the NISQ era, Quantum Science and Technology5, 044007 (2020)

View on Pith 2020

[35] [35]

L. K. Grover and J. Radhakrishnan, Is partial quantum search of a database any easier?, inProceedings of the 17th Annual ACM Symposium on Parallelism in Algorithms and Architec- tures (SPAA’05)(Association for Computing Machinery, 2005) pp. 186–194

View on Pith 2005

[36] [36]

V . E. Korepin and Y . Xu, Hierarchical quantum search, Interna- tional Journal of Modern Physics B21, 5187 (2007)

View on Pith 2007

[37] [37]

V . E. Korepin and Y . Xu, Quantum search algorithms, Interna- tional Journal of Modern Physics B23, 5727 (2009)

View on Pith 2009

[38] [38]

V . E. Korepin and L. K. Grover, Simple algorithm for partial quantum search, Quantum Information Processing5, 5 (2006)

View on Pith 2006

[39] [39]

V . E. Korepin, Optimization of partial search, Journal of Physics A: Mathematical and General38, L731 (2005)

View on Pith 2005

[40] [40]

V . E. Korepin and J. Liao, Quest for fast partial search algo- rithm, Quantum Information Processing5, 209 (2006)

View on Pith 2006

[41] [41]

Y .-B. Jiang, X.-H. Wang, K. Zhang, and V . Korepin, Asymp- totic bounds on quantum partial search algorithm and its ap- plications to parallel search, Physical Review A114, 012412 (2026)

View on Pith 2026

[42] [42]

K. Zhang, K.-Y . Chen, X.-H. Wang, and V . Korepin, Asymp- totic optimality of Grover–Radhakrishnan–Korepin algorithm, Journal of Physics A: Mathematical and Theoretical59, 315301 (2026), arXiv:2604.15886 [quant-ph]

View on Pith Pith review arXiv 2026

[43] [43]

K. Zhang and V . E. Korepin, Depth optimization of quantum search algorithms beyond Grover’s algorithm, Physical Review A101, 032346 (2020)

View on Pith 2020

[44] [44]

M. Bria ´nski, J. Gwinner, V . Hlembotskyi, W. Jarnicki, S. Pli´s, and A. Szady, Introducing structure to expedite quantum searching, Physical Review A103, 062425 (2021)

View on Pith 2021

[45] [45]

J. Liu and H. Zhou, Hardware efficient quantum search algo- rithm (2021), arXiv:2103.14196 [quant-ph]

View on Pith Pith review arXiv 2021

[46] [46]

K. Zhang, H.-L. Shi, X.-H. Wang, and V . Korepin, Palindromic structure of depth-efficient quantum search algorithms (2026), arXiv:2606.00960 [quant-ph]

View on Pith Pith review arXiv 2026

[47] [47]

J. Burke and C. McGoldrick, Quantum search without global diffusion (2026), arXiv:2604.15435 [quant-ph]

View on Pith Pith review arXiv 2026

[48] [48]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition(Cambridge 20 University Press, 2010)

View on Pith 2010

[49] [49]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Mar- golus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, El- ementary gates for quantum computation, Physical Review A 52, 3457 (1995), arXiv:quant-ph/9503016

View on Pith Pith review arXiv 1995

[50] [50]

M. Saeedi and M. Pedram, Linear-depth quantum circuits for n-qubit Toffoli gates with no ancilla, Physical Review A87, 062318 (2013)

View on Pith 2013

[51] [51]

Y . He, M.-X. Luo, E. Zhang, H.-K. Wang, and X.-F. Wang, De- compositions of n-qubit Toffoli gates with linear circuit com- plexity, International Journal of Theoretical Physics56, 2350 (2017)

View on Pith 2017

[52] [52]

B. Zindorf and S. Bose, Efficient implementation of multicon- trolled quantum gates, Physical Review Applied24, 044030 (2025)

View on Pith 2025

[53] [53]

M. Boyer, G. Brassard, P. Høyer, and A. Tapp, Tight bounds on quantum searching, Fortschritte der Physik46, 493 (1998)

View on Pith 1998

[54] [54]

R. M. Gingrich, C. P. Williams, and N. J. Cerf, General- ized quantum search with parallelism, Physical Review A61, 052313 (2000)

View on Pith 2000

[55] [55]

V . E. Korepin and B. C. Vallilo, Group theoretical formulation of a quantum partial search algorithm, Progress of Theoretical Physics116, 783 (2006)

View on Pith 2006

[56] [56]

B.-S. Choi and V . E. Korepin, Quantum partial search of a database with several target items, Quantum Information Pro- cessing6, 243 (2007)

View on Pith 2007

[57] [57]

P.-C. Zhong, W.-S. Bao, and Y . Wei, Quantum partial search- ing algorithm of a database with several target items, Chinese Physics Letters26, 020301 (2009)

View on Pith 2009

[58] [58]

K. Zhang and V . E. Korepin, Quantum partial search for uneven distribution of multiple target items, Quantum Information Pro- cessing17, 143 (2018)

View on Pith 2018

[59] [59]

E. Biham, O. Biham, D. Biron, M. Grassl, and D. A. Lidar, Grover’s quantum search algorithm for an arbitrary initial am- plitude distribution, Physical Review A60, 2742 (1999)

View on Pith 1999