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REVIEW 3 major objections 5 minor 29 references

Nested Grover's Algorithm for Tree Search

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A nested Grover loop cuts quantum tree search from O(2^{m/2}) to O(m·2^{m/4}).

desk verdict Useful framing for nested Grover search, but the claimed speedup rests on an unanalyzed guess that the correct lower path lies in the candidate set. read the letter →

arxiv 2509.07041 v1 pith:N7DJDPHG submitted 2025-09-08 quant-ph

Andreas Wichert This is my paper · ORCID
classification quant-ph
keywords quantumtreesearchnestedGroverconcatenatedoraclepartialcandidatesolutioncomplexityGrover'salgorithmAISAT
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

Quantum tree search normally pays Grover's O(2^{m/2}) to find a goal leaf in a binary tree of depth m. This paper tries to beat that by expanding the tree to a fixed depth g, listing v nodes at that depth as 'partial candidate solutions,' and running Grover twice: once on the lower g bits to pick a candidate, once on the upper m−g bits to check the rest. The paper's key move is a concatenated oracle that tests the global solution only for states whose lower part matches the chosen candidate; because each circuit contains exactly one candidate, the lower and upper parts do not entangle, so a valid solution is measured with certainty. With about m candidates, the paper derives worst-case cost O(m·2^{m/4}), a quadratic improvement over the plain Grover tree search, provided the true lower part of the solution is among the candidates. If correct, this gives quantum tree search a structured speedup and replaces heuristic functions—which Grover's uniform-amplitude requirement rules out—with partial candidate solutions.

What carries the argument

The concatenated oracle is the load-bearing object. For x=z q y, the global oracle is written o(x)=u(z) q l(y), and the lower factor l(y) is replaced by a partial candidate solution h_k(y), a function that returns 1 only for one local path of length g. The algorithm prepares the equal superposition over all m bits, applies Grover amplification to the g lower bits to concentrate on h_k, then applies the concatenated oracle and Grover amplification to the upper m−g bits. Because exactly one candidate is in the circuit, the upper and lower registers do not entangle, and the global solution ξ=u q l is recovered with certainty when l=h_k. The set ℵ of v partial candidate solutions carries all the

What would settle it

Take a uniform binary tree of depth m=6 with a single goal leaf, choose g=3, and let ℵ be any set of v=6 nodes at depth 3 that does not contain the true lower path. Run the paper's iterative Algorithm 4.1 on the corresponding circuits; step 5 can never report success, and every measured state fails the global oracle. A single such run is a direct counterexample to an unconditional O(m·2^{m/4}) claim, confirming that the speedup is conditional on l∈ℵ.

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Extended reading notes

Core claim

The paper claims that a nested Grover search over a uniform binary tree of depth m can beat plain Grover's O(2^{m/2}). The search tree is developed to a fixed depth g; each node at that depth has a local path descriptor y, and a partial candidate solution h_k(y) marks exactly one such node. A concatenated oracle o(x)=u(z) q h_k(y) applies the global solution check only to states whose lower g bits match h_k; the upper part u(z) is checked by a second Grover amplification. The iterative algorithm builds one circuit per h_k in a set ℵ of v candidates. In such a circuit only one lower path is present, so the subspaces U and L are not entangled, and the solution, if it exists, is measured with p

Load-bearing premise

The load-bearing premise is that the set ℵ of about m partial candidate solutions contains the true lower path descriptor l; the paper's Section 7.1 admits that building such a set is 'highly probabilistic' and can produce non-existent candidates, and if l is missing, the loop in Algorithm 4.1 can never succeed.

Editorial extensions

If this is right

  • For a binary tree of depth m with a known branching bound, the iterative nested search replaces one Grover search over 2^m leaves with about m Grover searches over 2^{m/2} subspaces, lowering the worst-case cost from O(2^{m/2}) to O(m·2^{m/4}).
  • Because each single-candidate circuit avoids entanglement, the solution is measured with probability 1, so the algorithm does not need repeated measurements to overcome the 1/v dilution that limits the original nested search.
  • If the upper oracle u(z) is known, the disentangled construction H=U_{h_k≠h_1}⊗...⊗L lets the algorithm identify the correct lower candidate by measuring |u⟩ and then recover l with a Grover search on L.
  • The permutation approach, if the permutation can be synthesized, reduces the cost to O((log n)^{1/2}·n^{1/4}), improving the iterative bound by a factor √log n.
  • The partial-candidate formulation gives quantum tree search a way to use problem structure without violating the uniform-amplitude requirement that blocks heuristic functions.

Reading between the lines

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

  • The paper assumes but does not construct the set ℵ; a direct extension would measure how often classically generated candidate sets contain the true lower path, since the O(m·2^{m/4}) regime only appears when inclusion holds.
  • The disentangled construction suggests a candidate-filtering primitive: if the upper oracle u(z) is only partially known, measuring the |u⟩ register could still identify which candidates are consistent with the global solution, pruning ℵ before the upper Grover search.
  • If the permutation on L can be learned from known candidate paths, the compressed cost O((log n)^{1/2} n^{1/4}) becomes reachable; otherwise the basis-preparation overhead the paper cites cancels the advantage.
  • Combining iterative deepening with the concatenated-oracle construction could remove the need to know the tree depth m in advance, at the cost of repeating each amplifier for every depth limit.
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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

3 major / 5 minor

Summary. The paper proposes a nested Grover algorithm for binary tree search. It splits an m-bit path descriptor into lower and upper parts, introduces 'partial candidate solutions' h(y) acting on the lower g qubits, and defines a concatenated oracle o(x)=u(z) q l(y). The central claim is that, with m partial candidate solutions, the iterative nested search costs O(m·2^{m/4}) = O(log n · n^{1/4}), improving on standard Grover's O(2^{m/2}). The manuscript also discusses a disentanglement construction H=U_{h_k≠h1}⊗...⊗L and a permutation-based variant. The abstract, §4.3, §5.2, and conclusion all assert the same asymptotic improvement.

Significance. If the claimed O(m·2^{m/4}) complexity were established, it would be a notable advance over the unstructured Grover bound for tree search: for depth m, it would reduce the exponent from m/2 to m/4+log m. The paper introduces a useful vocabulary (partial candidate solutions, concatenated oracles) and provides small Qiskit demonstrations that are reproducible from the linked notebooks. However, the central complexity claim is conditional on an unproved and, by the paper's own admission, unresolved ingredient: the reliable construction of a set ℵ containing the correct lower path descriptor l. The paper does not analyze P(l∈ℵ), and §7.1 concedes that the proposed construction is 'highly probabilistic' and may yield non-existent candidates. Under the standard reading of the algorithm, failure to include l causes deterministic failure of every iteration. Thus the headline result is not established; the paper currently offers a promising structure rather than a proven algorithm.

major comments (3)
  1. [§4, §4.1, §4.3, §7.1] The iterative algorithm succeeds only if the correct lower descriptor l belongs to the precomputed set ℵ. Section 4.1 steps 1–5 simply try each h_k and verify; if h_k≠l for all k, every verification fails. Section 4.3 computes the cost v·(2·2^{m/4}+1) under the constraint v<2^{m/4}, but never bounds P(l∈ℵ). For a uniformly chosen ℵ of size v from a lower subspace of dimension 2^g, with g=m/2, P(l∈ℵ)=v/2^g ≤ 2^{m/4}/2^{m/2}=2^{-m/4}; making the success probability constant requires about 2^{m/4} repetitions, which multiplies the claimed cost to Ω(2^{m/2}). This is exactly the gap identified by the stress-test note, and it is not repaired anywhere in the text. Section 7.1 explicitly says the construction of partial candidate solutions 'is highly probabilistic and may occasionally yield non-existent partial candidate solutions or values exceeding one.' The central complexity claim in Eq. (2
  2. [§4.3, Eq. (22)–(24)] The derivation substitutes v≈m while the preceding constraint is v<2^{m/4}. For large m, m<2^{m/4} is true asymptotically, so this part is not internally inconsistent for the asymptotic claim. The real problem is that the constraint is a cost constraint, not a correctness constraint. The statement that v is 'approximately equal to the number of nodes located at one-fourth of the depth' provides no mechanism for selecting the v candidates so that l is among them. Without a probabilistic or deterministic guarantee on ℵ, Eq. (24) is a conditional bound on a sequential guessing strategy, not a bound on an algorithm that finds the solution.
  3. [§5, Eq. (31)–(35)] The disentanglement construction is also missing a success-probability analysis. The tensor product H=U_{h_k≠h1}⊗...⊗U_{h_k≠hv}⊗L in Eq. (33) is not a decomposition of the original m-qubit Hilbert space into independent registers; it represents v parallel circuits, and the cost in Eq. (35) counts v Grover searches on the upper subspace. Section 5.2 then observes that for v>4 'a substantial number of measurements is necessary' and proposes the practical choice v*=4 with v/v* iterations. This reintroduces the same issue: if only four candidates are used, the paper gives no argument that the correct l is among them, and the claimed factor of 1.45 over the iterative approach is not a guaranteed worst-case speedup. The numbers in Eq. (36)–(39) compare costs only under the assumption that one of the tested h_k equals l.
minor comments (5)
  1. [§1, §3.6] Typos: 'contented oracles' should be 'concatenated oracles' and 'disentanglement sunspaces' should be 'disentanglement subspaces'.
  2. [§3.5, Eqs. (9)–(12)] The normalizations 1/√(m−g) and 1/√g appear to be dimension errors; they should be 1/√(2^{m−g}) and 1/√(2^g). As printed, the uniform superpositions are not normalized.
  3. [§5, Eqs. (25)–(28)] The sums are written over z∈B^m, but the upper subspace U has dimension 2^{m−g}; the notation should be z∈B^{m−g}. The normalization 1/√(2^m) is also inconsistent with the 1/√v factor and the subsequent equations.
  4. [§4.2, Eq. (20)] The notation √(2^m·(1−a)) is ambiguous: the intended expression is presumably √(2^{m(1−a)}) + √(2^{ma}), whose minimum is at a=1/2. The typesetting should be corrected for clarity.
  5. [§6.2] The discussion of the permutation operator P acknowledges that 'there appears to be no straightforward method to ascertain the mapping represented by P.' This is a significant practical limitation, but the section presents it as a direction for future work; the paper should more clearly state that the permutation-based variant is not a complete algorithm.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the central cost bound is conditional on an unproved candidate-set assumption, not on circular reasoning.

full rationale

The paper's main complexity claim is derived internally as a function of the input parameter v. Equation (22) gives the cost as v*(2*2^{m/4}+1) by summing two Grover searches plus verification and multiplying by the worst-case number of candidate trials; Equation (23) imposes the constraint v < 2^{m/4}; Equation (24) then substitutes v ≈ m to write O(v·n^{1/4}) = O(log n·n^{1/4}) = O(m·2^{m/4}). This is a conditional complexity expression in v, not a fitted parameter renamed as a prediction, and it does not assume the target result. The serious gap is that the iterative algorithm succeeds only if the correct lower path descriptor l belongs to the precomputed candidate set ℵ, and the paper never bounds P(l∈ℵ) or shows how to construct such a set efficiently; Section 7.1 concedes the construction "is highly probabilistic and may occasionally yield non-existent partial candidate solutions or values exceeding one." That is an unproven completeness/success assumption and a correctness risk, but it is not circular reasoning: the cost formula is not equivalent to the candidate-set assumption by construction. The paper's self-citations ([18], [19], [20], [26]–[28]) support background material such as quantum tree search, iterative deepening, and Qiskit examples; none of them carries the O(m·2^{m/4}) claim, and no load-bearing equation reduces to a self-cited theorem. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The central claim rests on the unexamined assumption that a small candidate set containing the solution's prefix can be constructed. The parameters v and g are chosen to make the cost formula work, but the probability of success is not modeled. The paper supplies no independent evidence for the key construct (partial candidate solution).

free parameters (2)
  • v (number of partial candidate solutions) = v approx m
    The headline complexity O(m*2^(m/4)) is obtained by setting v approx m candidates. The constraint (23) requires v < 2^(m/4)/2 to beat Grover, and no method ensures the candidate set contains the solution.
  • g (subspace split) = m/2
    Chosen to minimize the per-iteration Grover cost sqrt(2^g)+sqrt(2^(m-g)). The minimization ignores success probability and the chance that l is in the candidate set.
assumptions (4)
  • domain assumption The search tree is uniform with known depth m and branching factor 2.
    Section 1 states: 'we assume that the tree is uniform and its size is known beforehand.'
  • domain assumption Grover's algorithm requires uniform distributions; non-uniform distributions require adaptation with the same complexity O(sqrt n).
    Section 2.1: 'Uniform distributions are essential for the Grover['s] algorithms...'
  • ad hoc to paper The solution ξ can be decomposed as ξ = u ⊗ l, and the set ℵ contains l.
    Section 5 states: 'We really on the assumption that ξ exists and can be decomposed as ξ = u ⊗ l.' The iterative algorithm only succeeds if some h_k = l.
  • ad hoc to paper A set ℵ of v partial candidate solutions can be identified 'by determining their similarity or distance to the goal state' without a specified efficient procedure.
    Section 7.1 suggests this identification is possible but admits it is 'highly probabilistic and may occasionally yield non-existent partial candidate solutions.' This is the key grounding assumption for the speedup.
invented entities (2)
  • Partial candidate solution h(y)
    purpose: Replaces heuristic functions; marks a single node at depth g as a candidate for the lower path descriptor.
    New construct introduced by the paper. It has no falsifiable handle outside the method itself, and the paper provides no procedure to construct it for a given problem.
  • Concatenated oracle o(x) = u(z) q l(y)
    purpose: Decomposes the global oracle into upper and lower parts to enable nested Grover search.
    A definitional bookkeeping device: it splits the oracle along the subspace decomposition. It is not an independently testable entity.

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Cite this review

Pith. "Pith review of Nested Grover's Algorithm for Tree Search." pith.science (2026). https://pith.science/paper/N7DJDPHG

@misc{pith2026250907041,
  author       = {Pith},
  title        = {Pith review of: Nested Grover's Algorithm for Tree Search},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7DJDPHG}},
  note         = {Machine review of arXiv:2509.07041}
}
read the original abstract

We investigate optimizing quantum tree search algorithms by employing a nested Grover Algorithm. This approach seeks to enhance results compared to previous Grover-based methods by expanding the tree of partial assignments to a specific depth and conducting a quantum search within the subset of remaining assignments. The study explores the implications and constraints of this approach, providing a foundation for quantum artificial intelligence applications. Instead of utilizing conventional heuristic functions that are incompatible with quantum tree search, we introduce the partial candidate solution, which indicates a node at a specific depth of the tree. By employing such a function, we define the concatenated oracle, which enables us to decompose the quantum tree search using Grover algorithm.

Figures

Figures reproduced from arXiv: 2509.07041 by the authors.

Figure 1
Figure 1. Search tree for B = 2 and depth m = 2. Each question can be represented by a bit. Each binary number (11, 10, 01, 00) represents a path descriptor m from the root to the leaf. number represents a path from the root to a leaf. For each goal, a specific binary number 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Search tree for B = 2 and depth m = 5. Each question can be represented by a bit. a constant branching factor B > 2, each question has B possible answers. The m answers can be represented by a base-B number with m digits. In a quantum computation, we can simultaneously represent all possible path descriptors. There’s one path descriptor for each leaf off the tree. Using Grover’s algorithm, we search through all poss… view at source ↗
Figure 3
Figure 3. The decomposition of the Hilbert space H of dimension n = 2m into subspaces L and U of dimension 2 g and 2 m−g . In our example m = 5 and g = 2. since 1 < √ 2 g · √ 2m−g − √ 2 g + √ 2m−g + 1 1 < ( √ 2 g − 1) · (1 − √ 2m−g) because 1 < ( √ 2 g − 1), 1 < (1 − √ 2m−g) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The concept of cost savings is predicated on the inequality that decomposes the Hilbert space [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: (a) Circuit using Grover’s amplification representing oracles [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Search tree for B = 2 and depth m = 5. With h1 = 011, h2 = 000, u = 11, l = 011 and ξ = 01111. values to some values hk with y ∈ {h1, h2, · · · , hv} the constraint oracle can be expressed as l = hk with o(x) = o(z, hk) = u(z) q l(hk). 3.5 Not Entangled Subspaces Imagi…
Figure 7
Figure 7. Figure 7: (a) (a) Circuit employing Grover’s amplification to represent oracles [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Example for v = 4. Determine the solution h1 by Grover algorithm on the subspace L. Verify if the solution indicated by the oracle oξ(x) exists. Repeat the procedure for h2, h3, h4. 4.1 Algorithm Given set ℵ of v partial candidate solutions h (k) (y) = hk with k ∈ {1, …
Figure 9
Figure 9. Figure 9: (a) Circuit using Grover’s amplification representing oracle [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: (a) Circuit employing Grover’s amplification to represent oracles [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: (a) After applying Grover’s algorithm with two solutions indicated by [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: (a) The corresponding circuit, the permutation operator [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: (a) Representation of the permutation operator [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]

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Works this paper leans on

29 extracted references · 19 canonical work pages

  1. [1]

    PhD thesis, Hebrew University, July 1999

    Dorit Aharonov.Noisy Quantum Computation. PhD thesis, Hebrew University, July 1999

    work page 1999

  2. [2]

    Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani

    Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing, 1997. URLhttp://www.citebase.org/ abstract?id=oai:arXiv.org:quant-ph/9701001

    arXiv 1997

  3. [3]

    Tight bounds on quan- tum searching.Fortschritte der Physik, 46:493, 1998

    Michel Boyer, Gilles Brassard, Peter Hoeyer, and Alain Tapp. Tight bounds on quan- tum searching.Fortschritte der Physik, 46:493, 1998. URLhttp://www.citebase. org/abstract?id=oai:arXiv.org:quant-ph/9605034

    arXiv 1998

  4. [4]

    Brassard, P

    G. Brassard, P. Hoyer, M. Mosca, and A. Tapp. Quantum Amplitude Amplification and Estimation.eprint arXiv:quant-ph/0005055, May 2000. 24

    arXiv 2000

  5. [5]

    Quantum Counting

    Gilles Brassard, Peter Hoyer, and Alain Tapp. Quantum counting, 1998. URLhttp: //www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9805082

    work page Pith review arXiv 1998

  6. [6]

    Cerf, Lov K

    Nicolas J. Cerf, Lov K. Grover, and Colin P. Williams. Nested quantum search and structured problems.Phzsical Review A, 61(3):032303, 2000

    work page 2000

  7. [7]

    M. C. Diamantini and C. A. Trugenberger. Mirror modular cloning and fast quantum associative retrieval.ArXiv Quantum Physics e-prints, abs/2206.01644, 2022

    work page Pith review arXiv 2022

  8. [8]

    Lov K. Grover. A fast quantum mechanical algorithm for database search. InSTOC ’96: Proceedings of the twenty-eighth annual ACM symposium on Theory of comput- ing, pages 212–219, New York, NY, USA, 1996. ACM. ISBN 0-89791-785-5. DOI: http://doi.acm.org/10.1145/237814.237866

    arXiv 1996

Show all 29 references
  1. [9]

    Lov K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters, 79:325, 1997. URLdoi:10.1103/PhysRevLett.79.325

    1997 doi

  2. [10]

    Lov K. Grover. A framework for fast quantum mechanical algorithms. InSTOC ’98: Proceedings of the thirtieth annual ACM symposium on Theory of comput- ing, pages 53–62, New York, NY, USA, 1998. ACM. ISBN 0-89791-962-9. DOI: http://doi.acm.org/10.1145/276698.276712

    1998

  3. [11]

    Lov K. Grover. Quantum computers can search rapidly by using almost any trans- formation.Phys. Rev. Lett., 80(19):4329–4332, May 1998. DOI: 10.1103/Phys- RevLett.80.4329

    1998 doi

  4. [12]

    Kaye, Raymond Laflamme, and Michele Mosca.An Introduction to Quantum Computing

    Philip R. Kaye, Raymond Laflamme, and Michele Mosca.An Introduction to Quantum Computing. Oxford University Press, USA, 2007

    2007

  5. [13]

    Richard E. Korf. Depth-first iterative-deepening : An optimal admissible tree search. Artificial Intelligence, 27(1):97 – 109, 1985. ISSN 0004-3702. DOI: DOI: 10.1016/0004- 3702(85)90084-0

    1985 doi

  6. [14]

    Quantum-walk speedup of backtracking algorithms.Theory of Computing, 14(15):1–24, 2018

    Ashley Montanaro. Quantum-walk speedup of backtracking algorithms.Theory of Computing, 14(15):1–24, 2018

    2018

  7. [15]

    Qiskit: An open-source framework for quantum computing

    Qiskit-Contributors. Qiskit: An open-source framework for quantum computing. 10.5281/zenodo.2573505, 2023

    2023 doi

  8. [16]

    Hybrid divide-and-conquer approach for tree search algorithms.Quantum, 7:959, 2023

    Mathys Rennela, Sebastiaan Brand, Alfons Laarman, and Vedran Dunjko. Hybrid divide-and-conquer approach for tree search algorithms.Quantum, 7:959, 2023

    2023

  9. [17]

    Russell and P

    S.J. Russell and P. Norvig.Artificial intelligence: a modern approach. Prentice Hall series in artificial intelligence. Prentice Hall, 2010. ISBN 9780136042594. URLhttp: //books.google.pt/books?id=8jZBksh-bUMC

    2010

  10. [18]

    Tree search and quantum computation.Quan- tum Information Processing, 10(4):475–500, 2011

    Luís Tarrataca and Andreas Wichert. Tree search and quantum computation.Quan- tum Information Processing, 10(4):475–500, 2011. ISSN 1570-0755. 10.1007/s11128- 010-0212-z

    2011 doi

  11. [19]

    Iterative quantum tree search

    Luís Tarrataca and Andreas Wichert. Iterative quantum tree search. CiE 2012 - How the World Computes, 2012, 2012

    2012

  12. [20]

    Quantum iterative deepening with an applica- tion to the halting problem.PLOS ONE, 8(3), 2013

    Luís Tarrataca and Andreas Wichert. Quantum iterative deepening with an applica- tion to the halting problem.PLOS ONE, 8(3), 2013

    2013

  13. [21]

    Face recognition with quantum associative networks using overcomplete gabor wavelet.Cognitive Computation, pages 1–6,

    Nuo Tay, Chu Loo, and Mitja Perus. Face recognition with quantum associative networks using overcomplete gabor wavelet.Cognitive Computation, pages 1–6,

  14. [22]

    Trugenberger

    Carlo A. Trugenberger. Probabilistic quantum memories.Physical Review Letters, 87 (6):1–4, 2001

    2001

  15. [23]

    Trugenberger

    Carlo A. Trugenberger. Quantum pattern recognition.Quantum Information Pro- cessing, 1(6):471–493, 2003. 25

    2003

  16. [24]

    Ventura and T

    D. Ventura and T. Martinez. Quantum associative memory with exponential capac- ity. InNeural Networks Proceedings, 1998. IEEE World Congress on Computational Intelligence., volume 1, pages 509–513, 1988

    1998

  17. [25]

    Ventura and T

    D. Ventura and T. Martinez. Quantum associative memory.Information Sciences, 124(1):273–296, 2000

    2000

  18. [26]

    World Scientific, 2020

    Andreas Wichert.Principles of Quantum Artificial Intelligence: Quantum Problem Solving and Machine Learning, 2nd Edition. World Scientific, 2020

    2020

  19. [27]

    Quantum tree search with qiskit.Mathematics, 10(17):3013, 2022

    Andreas Wichert. Quantum tree search with qiskit.Mathematics, 10(17):3013, 2022

    2022

  20. [28]

    CRC Press, 2024

    Andreas Wichert.Quantum Artificial Intelligence with Qiskit. CRC Press, 2024. 26

    2024

  21. [2010]

    URLhttp://dx.doi.org/10.1007/s12559-010-9047-2

    ISSN 1866-9956. URLhttp://dx.doi.org/10.1007/s12559-010-9047-2. 10.1007/s12559-010-9047-2

    doi

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