The DOI matches its Crossref record; 24 of the 24 references it cites were found in a registry or an index; none of the 8 sources its claims cite is checked for support yet.
A fast quantum mechanical algorithm for database search
Open accessAbstract
An unsorted database contains N records, of which just one satisfies a particular property.The problem is to identify that one record.Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records.Quantum mechanical systems can do several operations simultaneously due to their wave like properties.This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record.It is within a constant factor of the fastest possible quantum mechanical algorithm.
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19 No basis given · 8 Sources not found · 4 Own interpretation
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“This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record.”
§ A fast quantum mechanical algorithm for database search > Abstract
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- The authors state this as an assumption: There is a unique state that satisfies the condition C(S)=1.
- The authors state this as an assumption: For any state S, the condition C(S) can be evaluated in unit time.
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“Corollary 3.1-Let the state vector be as followsfor any one state the amplitude is k, for each of the remaining (N-1) states the amplitude is 1.Let k and 1 be real numbers (in general the amplitudes can be complex).Let k be negative and 1 be positive such that ~< ~N.Then after applying the diffusion transform both k and 1are positive numbers.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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“First, it is proved that the system is a valid quantum mechanical system (theorems 1 & 2)”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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“As mentioned in the introduction, the algorithm of this paper does not use any Imowledge about the problem.”
§ A fast quantum mechanical algorithm for database search > 114
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In the paper
“Theorem 1-The matix D is unitary.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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“The following sections prove that the system discussed in section 3 is indeed a valid quantum mechanical system and that it converges to the desired state with a probability O(l).”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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- The authors state this as an assumption: There is a unique state that satisfies the condition C(S)=1.
- The authors state this as an assumption: For any state S, the condition C(S) can be evaluated in unit time.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Theorem 2 -D can be expressed as D = FRF, where F, the Fourier Transform Matrix and R, the rotation matrix are defined as follows”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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In the paper
“Since the proba-& bility of the system being found in any particular state is proportional to the square of the amplitude, it follows that the probability of the system being in tlhe desired state when k is ~, 1 J2 is k2 = ~.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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“In case there is a unique state SV such that C(SV) = 1, the final state is SV with a probability of at least ~. 2”
§ A fast quantum mechanical algorithm for database search > T. Acknowledgments
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- The authors state this as an assumption: There is a unique state that satisfies the condition C(S)=1.
- The authors state this as an assumption: For any state S, the condition C(S) can be evaluated in unit time.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It can be easily modified to take care of the case when there are multiple states satisfying the condition C(S) = 1 and it is required to return one of these.”
§ A fast quantum mechanical algorithm for database search > 114
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- Probably not a substantial claim. 8
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In the paper
“Theorem 5-Let the probability of a particular outcome in an experiment be &.Then if the experiment be repeated ~times, the probability that the outcome does not happen even once is less than or equal to e-K.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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In the paper
“The change in k ( Ak) after steps (a) and (b) of the algorithm is lower bounded by Ak>~.Also after steps (a) and (b), 1>0.2fi”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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In the paper
“Using this it immediately follows that there exists a number M less than EN, such that in M repetitions of the loop in step (ii), k wifl become ~.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
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In the paper
“The most efficient classical algorithm for this is to examine the items in the database one by one.”
§ A fast quantum mechanical algorithm for database search > ------------------------------------------------
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In the paper
“Very recent results [Shor95] have shown the existence of quantum mechanical error corr~ting cod~which enable transmission of data even in the presence of noise”
§ A fast quantum mechanical algorithm for database search > 114
What the sources say
- Not matchedcites [Shor95]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This algorithm, by cleverly making use of some features of the structure of the problem, is the only known algorithm that solves the problem in O (logIV) steps.”
§ A fast quantum mechanical algorithm for database search > 114
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The phase does not have an analog in classical probabilistic algorithms.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This illustrates the point that, in quantum mechanics, the complete description of the system requires the amplitudes over all states, just the probabilities is not enough.”
§ A fast quantum mechanical algorithm for database search > 1.1 Search Problems in Computer Science
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“For example Shor's primality testing algorithm [Shor94] for a number N can be viewed as a search of the (N-l) numbers smaller than N to see if any of them is a factor of N.”
§ A fast quantum mechanical algorithm for database search > 114
What the sources say
- Not matchedcites [Shor94]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“There is also a scheme discussed in [VV86] by which it is possible to modify any algorithm that solves an NPsearch problem with a unique solution and use it to solve an NP-search problem in general.”
§ A fast quantum mechanical algorithm for database search > 114
What the sources say
- Not matchedcites [VV86]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“This operation is one of the things that makes quantum mechanical algorithms more powerful than classical algorithms and forms the basis for most significant quantum mechanical algorithms.”
§ A fast quantum mechanical algorithm for database search > T. Acknowledgments
What the sources say
- Not matchedcites [DJ92]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
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The authors state this as an assumption.
There is a unique state that satisfies the condition C(S)=1.
“Let there be a unique state, say Sv, that satisfies the condition C(SV) = 1, whereas for all other states S, C(S) = 0”
“The algorithm as discussed here assumes a unique state that satisfies the desired condition.”
The authors state this as an assumption.
For any state S, the condition C(S) can be evaluated in unit time.
“(assume that for any state S, the condition C(S) can be evaluated in unit time)”
The paper relies on this without support.
There does not exist any sorting on the database that would aid selection of the desired item.
“However, there does not exist any sorting on the database that would aid its selection.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Symmetric cryptography remains comparatively resilient, yet Grover's algorithm offers a quadratic speed-up for brute-force key search, effectively halving the security margin of symmetric ciphers [13].”
From Harvest-Now, Decrypt-Later: A Temporal Cybersecurity Risk in the Quantum Transition (2025) · cites this paper as [13]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Similarly, for hash functions, Grover's algorithm [185] can facilitate quantum birthday attack to find collision, thereby necessitating an output of larger size.”
From Zero Trust Architecture (ZTA): A Comprehensive Survey (2022) · cites this paper as [185]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Similarly, Grover’s algorithm concedes a quadratic advantage for mining blocks in certain consensus protocols such as proof of work.”
From Quantum-resistance in blockchain networks (2023) · cites this paper as [71]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Quantum computers will be able to find these nonces quadratically faster using Grover’s algorithm71.”
From Quantum-resistance in blockchain networks (2023) · cites this paper as [71]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“This model is realistic given the falling cost of storage, the rise of large-scale surveillance infrastructures, and the expected disruption of quantum algorithms such as Shor's and Grover's [1,2,13].”
From Harvest-Now, Decrypt-Later: A Temporal Cybersecurity Risk in the Quantum Transition (2025) · cites this paper as [13]
Experimental — accuracy not yet measured
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Grover's algorithm provides only a quadratic speedup, effectively halving brute-force resistance. Comparable security margins can be restored by doubling key sizes (for example, AES-128 to AES-256) [13].”
From Harvest-Now, Decrypt-Later: A Temporal Cybersecurity Risk in the Quantum Transition (2025) · cites this paper as [13]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“This explains why AES-256 maintains post-quantum security margins, whereas AES-128 requires upgrading to maintain equivalent protection levels [13,14].”
From Harvest-Now, Decrypt-Later: A Temporal Cybersecurity Risk in the Quantum Transition (2025) · cites this paper as [13]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“At the same time, Grover's algorithm shows that symmetric cryptography, though more resilient, also demands doubled key sizes to maintain equivalent margins [13].”
From Harvest-Now, Decrypt-Later: A Temporal Cybersecurity Risk in the Quantum Transition (2025) · cites this paper as [13]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“In this sense, Grover-based optimization algorithms [184] such as Grover Adaptive Search [161] have been proposed.”
From Quantum Combinatorial Optimization in the NISQ Era: A Systematic Mapping Study (2024) · cites this paper as [184]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The former bound is tight since recent work of Grover [13] shows how to accept the class NP relative to any oracle on a quantum computer in time O (2 n/ 2 ).”
From Strengths and Weaknesses of Quantum Computing (1997) · cites this paper as [13]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“In view of Grover's algorithm [13], we know that the constant '1 / 2' in the statement of Theorem 3.5 cannot be improved.”
From Strengths and Weaknesses of Quantum Computing (1997) · cites this paper as [13]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The former bound is tight since recent work of Grover [13] shows how to accept the class NP relative to any oracle on a quantum computer in time O (2 n/ 2 ).”
From Strengths and Weaknesses of Quantum Computing (1997) · cites this paper as [13]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Grover’s algorithm [54] is a quantum algorithm for searching in a database, ofering quadratic speedup over classical searching methods.”
From QASMBench: A Low-Level Quantum Benchmark Suite for NISQ Evaluation and Simulation (2023) · cites this paper as [54]
The cited source says what the claim attributes to it.
In the citing paper
“However, quantum machines can recognize this language quadratically faster, in time O ( √ 2 n ), using Grover's algorithm [13].”
From Strengths and Weaknesses of Quantum Computing (1997) · cites this paper as [13]
The cited source says what the claim attributes to it.
In the citing paper
“That this was a real possibility is clear from Grover's [13] result, which gives a black-box approach to solving NP -complete problems in square-root as much time as is required classically.”
From Strengths and Weaknesses of Quantum Computing (1997) · cites this paper as [13]
The cited source says what the claim attributes to it.
In the citing paper
“Various strategies can be imagined, but a good one (called 'inversion about the average' by Grover [13]) is to perform an oracle-independent unitary transformation so as to change the phase difference into an amplitude difference, leaving the y term with the same sign as all the other terms but a magnitude approximately threefold larger.”
From Strengths and Weaknesses of Quantum Computing (1997) · cites this paper as [13]
The check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“The HNDL adversary departs from conventional cryptanalysis in its temporal asymmetry. Instead of attempting immediate decryption, the attacker passively collects encrypted traffic or stored archives and defers decryption until sufficient computational resources become available.”
From Harvest-Now, Decrypt-Later: A Temporal Cybersecurity Risk in the Quantum Transition (2025) · cites this paper as [13]
Experimental — accuracy not yet measuredIn the citing paper
“This algorithm uses a single function evaluation to label the single state which satisfies the function, followed by a series of gates which drive the system into this particular state.”
From Implementation of a quantum search algorithm on a quantum computer (1998) · cites this paper as [9]
In the citing paper
“Here we show how such a computer can be used to implement a fast quantum search algorithm initially developed by Grover. 9, 10”
From Implementation of a quantum search algorithm on a quantum computer (1998) · cites this paper as [9]
In the citing paper
“Among other applications Grover's algorithms enable an extremely rapid search over the domain of a binary function to find elements for which this function is satisfied (that is, the function has the value 1).”
From Implementation of a quantum search algorithm on a quantum computer (1998) · cites this paper as [9]
In the citing paper
“the ability to efficiently perform Hamiltonian simulation is used in conjunction with another technique known as the quantum phase estimation algorithm (QPEA) 42 for efficiently obtaining the eigen-energies and eigenstates of a quantum system 38 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [38]
In the citing paper
“The Quantum Phase Estimation (QPE) algorithm 38,99 is the natural translation of the FCI procedure to quantum computers 45 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [38]
In the citing paper
“Abrams and Lloyd 38 focused on state preparation in more detail and described concrete initialization procedures.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [38]
In the citing paper
“They pointed out that state preparation is conceptually easier in second quantization, where a Fock state is represented by a simple product state of qubits in {| 0 〉 , | 1 〉} states.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [38]
Citations
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Metadata
- Authors
- Grover, L. K.
- Affiliations
- AT&T (United States) (US)
- Corresponding author
- Grover, Lov K.
- Venue
- Proceedings of the twenty-eighth annual ACM symposium on Theory of computing - STOC '96
- Publisher
- ACM Press
- Published
- 1996
- Pages
- 212-219
- OpenAlex
- W2084652510
- Semantic Scholar
- 298d799da82395a64a3bda38ef9d2a4646828ccb
- Cited by
- 8,830
- References
- 24
- Citations by year
- 2026: 664 · 2025: 1066 · 2024: 896 · 2023: 841 · 2022: 674 · 2021: 655 · 2020: 529 · 2019: 423 · 2018: 288 · 2017: 283 · 2016: 219 · 2015: 221 · 2014: 184 · 2013: 161 · 2012: 158
- FWCI
- 36.24
- Type
- conference paper
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- ACM Symposium on Theory of Computing (STOC) (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- quantum search, unsorted database search, Grover's algorithm, quantum algorithms, time complexity, quantum parallelism, lower bounds
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