## Quantum search of spatial regions (2005)

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Venue: | THEORY OF COMPUTING |

Citations: | 61 - 9 self |

### BibTeX

@INPROCEEDINGS{Aaronson05quantumsearch,

author = {Scott Aaronson and Andris Ambainis},

title = {Quantum search of spatial regions},

booktitle = {THEORY OF COMPUTING},

year = {2005},

pages = {200--209},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Can Grover’s algorithm speed up search of a physical region—for example a 2-D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional hypercube in time O ( √ n) for d ≥ 3, or O ( √ nlog 5/2 n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almost-tight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of ‘locality’ for unitary matrices acting on graphs. As an application of our results, we give an O (√ n)-qubit communication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov.

### Citations

1512 |
Quantum computation and quantum information
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- 2000
(Show Context)
Citation Context ... not, then does a problem’s complexity in our model ever depend on which criterion is chosen? Let us emphasize 6that these questions are not answered by, for example, the Solovay-Kitaev theorem (see =-=[22]-=-), that an n × n unitary matrix can be approximated using a number of gates polynomial in n. For recall that the definition of C-locality requires the edgewise operations to commute—indeed, without th... |

941 | Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
- Shor
- 1996
(Show Context)
Citation Context ...are needed. It is sometimes asserted that, although the speedup of Grover’s algorithm is only quadratic, this speedup is provable, in contrast to the exponential speedup of Shor’s factoring algorithm =-=[26]-=-. But is that really true? Grover’s algorithm is typically imagined as speeding up combinatorial search—and we do not know whether NP is contained in BPTIME ( 2 n/2) , any more than we know whether fa... |

902 | A fast quantum mechanical algorithm for database search” quant-ph/9605043
- Grover
(Show Context)
Citation Context ...unication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov. 1 Introduction The goal of Grover’s quantum search algorithm =-=[16]-=- is to search an ‘unsorted database’ of size n in a number of queries proportional to √ n. Classically, of course, order n queries are needed. It is sometimes asserted that, although the speedup of Gr... |

551 |
Microscopic Origin of the Bekenstein/Hawking Entropy,” Phys
- See, Strominger, et al.
- 1996
(Show Context)
Citation Context ... linearity, which is what produces the “other” Ω (√ n ) bound. 4 In the case of a black hole horizon, waiting for the bits to be emitted as Hawking radiation—as recent evidence suggests that they are =-=[28]-=-—takes time proportional to r 3 , which is much too long. 4have in mind systems far from the Schwarzschild limit, for which any time dilation is by at most a constant factor independent of n. (3) How... |

392 |
Gaussian elimination is not optimal
- Strassen
- 1969
(Show Context)
Citation Context ... n time. To us this illustrates why one should not assume an algorithm is optimal on heuristic grounds. Painful experience—for example, the “obviously optimal” O ( n3) matrix multiplication algorithm =-=[30]-=-—is what taught computer scientists to see the proving of lower bounds as more than a formality. Our setting is related to that of quantum random walks on graphs [1, 13, 14, 28]. In an earlier version... |

323 | Quantum mechanics helps in searching for a needle in a haystack
- Grover
- 1997
(Show Context)
Citation Context ...unication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov. 1 Introduction The goal of Grover’s quantum search algorithm =-=[17, 18]-=- is to search an ‘unsorted database’ of size n in a number of queries proportional to √ n. Classically, of course, order n queries are needed. It is sometimes asserted that, although the speedup of Gr... |

321 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...0. In Section 4 we prove general facts about our model, including an upper bound of O for the time needed to search any graph with diameter δ, and a proof (using the hybrid argument of Bennett et al. =-=[8]-=-) that this upper bound is tight for certain graphs. We conclude in Section 8 with some open problems. 21.2 Relation to Previous Work In a paper on ‘Space searches with a quantum robot,’ Benioff [7] ... |

279 | Quantum lower bounds by polynomials
- Beals, Buhrman, et al.
(Show Context)
Citation Context ... be safely skipped by readers unconcerned with the physical universe. In Section 3 we define quantum query algorithms on graphs, a model similar to quantum query algorithms as defined by Beals et al. =-=[4]-=-, but with the added requirement that unitary operations be ‘local’ with respect to some graph. In Section 3.1 we address the difficult question, which also arises in work on quantum random walks [1] ... |

151 | Quantum lower bounds by quantum arguments
- Ambainis
(Show Context)
Citation Context ...eater than 1/ √ Λ remains unclear, but if such a database existed, it could be searched! 3 The Model Much of what is known about the power of quantum computing comes from the black-box or query model =-=[4, 5, 8, 16, 26]-=-, in which one counts only the number of queries to an oracle, not the number of computational steps. We will take this model as the starting point for a formal definition of quantum robots. Doing so ... |

135 | Quantum vs. classical communication and computation
- Buhrman, Cleve, et al.
- 1990
(Show Context)
Citation Context ...he disjointness problem, Alice and Bob must decide with high probability whether there exists an i such that xi = yi = 1, using as few bits of communication as possible. Buhrman, Cleve, and Wigderson =-=[13]-=- observed that in the quantum setting, Alice and Bob can solve this problem using only O ( √ n log n) qubits of communication. This was subsequently improved by Høyer and de Wolf [17] to O (√ nc log∗ ... |

131 | Quantum amplitude amplification and estimation
- Brassard, Høyer, et al.
(Show Context)
Citation Context ...e idea is pointed out, an upper bound of O ( n1/2+ε) follows readily. However, to obtain the tighter bounds is more difficult; for that we use the amplitude-amplification framework of Brassard et al. =-=[12]-=-. Section 5 presents the main results; Section 5.4 shows further that, if the number of marked items is at least k, then the search time decreases to O (√ nk−1/2+1/d) for d ≥ 3, and that this upper bo... |

115 |
Tight bounds on quantum searching. Fortschritte der Physik
- Boyer, Brassard, et al.
- 1998
(Show Context)
Citation Context ...’s with xi = 1? If there are k marked items (where k need ( not be known in advance), then Grover’s algorithm can find a marked item with high probability √n/k) in O queries, as shown by Boyer et al. =-=[11]-=-. In our setting, however, this is too much to hope for—since even if there are many marked items, they might all be in a faraway part of the hypercube. Then Ω ( n 1/d) steps are needed, even if √ n/k... |

106 | D.A.(2003), Exponential Algorithmic Speedup by Quantum Walk
- Childs, Cleve, et al.
(Show Context)
Citation Context ...) matrix multiplication algorithm [27]—is what taught computer scientists to see the proving of lower bounds as more than a formality. Our setting is related to that of quantum random walks on graphs =-=[1, 14, 15]-=-. Most work on quantum walks is concerned with preparing a (near-) uniform distribution over vertices, a problem quite different from that of finding a particular marked vertex. However, Shenvi, Kempe... |

95 | Quantum communication complexity of symmetric predicates
- Razborov
(Show Context)
Citation Context ... communication complexity of the well-known disjointness problem is O ( √ n). This improves an O (√ nc log∗ n ) upper bound of Høyer and de Wolf [17], and matches the Ω ( √ n) lower bound of Razborov =-=[22]-=-. The rest of the paper is about the formal model that underlies our results. Section 2 sets the stage for this model, by exploring the ultimate limits on information storage imposed by properties of ... |

94 | On an estimate of the chromatic class of a p-graph - Vizing - 1964 |

90 | A Framework For Fast Quantum Mechanical Algorithms
- Grover
- 1998
(Show Context)
Citation Context ..., once the idea is pointed out, an upper bound of O ( n1/2+ε) follows readily. However, to obtain the tighter bounds is more difficult; for that we use the amplitude-amplification framework of Grover =-=[19]-=- and Brassard et al. [11]. Section 5 presents the main results; ( Section 5.4 shows further that, when there are k or more marked √nlog ) 5/2 vertices, the search time becomes O n when d = 2, or Θ (√ ... |

89 | Quantum walks on graphs
- Aharonov, Ambainis, et al.
- 2001
(Show Context)
Citation Context ... [4], but with the added requirement that unitary operations be ‘local’ with respect to some graph. In Section 3.1 we address the difficult question, which also arises in work on quantum random walks =-=[1]-=- and quantum cellular automata ([31], of what ‘local’ means. √nδ) Section 4 proves general facts about our model, including an upper bound of O for the time needed to search any graph with diameter δ,... |

75 | The holographic principle
- Bousso
- 2002
(Show Context)
Citation Context ...osed by physics, then we should acknowledge that the speed of light is finite, and that a bounded region of space can store only a finite amount of information, according to the holographic principle =-=[10]-=-. We discuss the latter constraint in detail in Section 2; for now, we say only that it suggests a model in which a ‘quantum robot’ occupies a superposition over finitely many locations, and moving th... |

66 |
An example of the difference between quantum and classical random walks
- Childs, Farhi, et al.
- 103
(Show Context)
Citation Context ...) matrix multiplication algorithm [27]—is what taught computer scientists to see the proving of lower bounds as more than a formality. Our setting is related to that of quantum random walks on graphs =-=[1, 14, 15]-=-. Most work on quantum walks is concerned with preparing a (near-) uniform distribution over vertices, a problem quite different from that of finding a particular marked vertex. However, Shenvi, Kempe... |

50 | Positive Vacuum Energy and the N-bound
- Bousso
(Show Context)
Citation Context ...have in mind systems far from the Schwarzschild limit, for which any time dilation is by at most a constant factor independent of n. (3) How do cosmological considerations affect our analysis? Bousso =-=[9]-=- argues that, in a spacetime with positive cosmological constant Λ > 0, the total number of bits accessible to any one experiment is at most 3π/ (Λ ln 2), or roughly 10122 given current experimental b... |

50 |
Gaussian elimination is not optimal. Numerische Mathematik
- Strassen
- 1969
(Show Context)
Citation Context ...) steps. To us this illustrates why one should not assume an algorithm is optimal on heuristic grounds. Painful experience—for example, the “obviously optimal” O ( n3) matrix multiplication algorithm =-=[27]-=-—is what taught computer scientists to see the proving of lower bounds as more than a formality. Our setting is related to that of quantum random walks on graphs [1, 14, 15]. Most work on quantum walk... |

48 |
A universal upper bound on the entropy to energy ratio for bounded systems
- Bekenstein
- 1981
(Show Context)
Citation Context ...ht. The answer seems to be that it is not tight, since (i) the entropy on a black hole horizon is not efficiently accessible4 , and (ii) weakly-gravitating systems are subject to the Bekenstein bound =-=[6]-=-, an even stronger entropy constraint than the holographic bound. Yet it is still of basic interest to know whether n bits in a radius-r ball can be searched in time o(min {n, r √ n})—that is, whether... |

46 |
Quantum random walk search algorithm, Phys. Rev. A 67
- Shenvi, Kempe, et al.
- 2003
(Show Context)
Citation Context ...n quantum walks is concerned with preparing a (near-) uniform distribution over vertices, a problem quite different from that of finding a particular marked vertex. However, Shenvi, Kempe, and Whaley =-=[25]-=- recently showed how to use a quantum walk to find a marked vertex on a hypercube, with efficiency matching that of Grover’s algorithm. Moreover, based on numerical evidence, N. Shenvi (personal commu... |

43 | Adiabatic Quantum State Generation and Statistical Zero Knowledge
- Aharonov, Ta-Shma
- 2003
(Show Context)
Citation Context ...ms quite difficult; a related problem has stymied research on quantum cellular automata for years (see Watrous [30]). On the other hand, perhaps the ‘sparse Hamiltonian lemma’ of Aharonov and Ta-Shma =-=[2]-=- could be modified to show any H-local unitary is approximable by a small product of C-local unitaries. A second problem is to prove significant lower bounds in our model, perhaps by using the adversa... |

38 | A complete promise problem for statistical zero-knowledge
- Sahai, Vadhan
- 1997
(Show Context)
Citation Context ... linearity, which is what produces the “other” Ω (√ n ) bound. 2 In the case of a black hole horizon, waiting for the bits to be emitted as Hawking radiation—as recent evidence suggests that they are =-=[27]-=-—takes time proportional to r 3 , which is much too long. 4the algorithms of this paper are unnecessary. Our response is that, while there might be n ‘passive’ computing elements (capable of storing ... |

37 | On one-dimensional quantum cellular automata
- Watrous
- 1995
(Show Context)
Citation Context ...nt that unitary operations be ‘local’ with respect to some graph. In Section 3.1 we address the difficult question, which also arises in work on quantum random walks [1] and quantum cellular automata =-=[30]-=-, of what ‘local’ means. We offer several definitions; some results about the robustness of these definitions are proved ( in Appendix √nδ) 10. In Section 4 we prove general facts about our model, inc... |

36 | Computational capacity of the universe
- Lloyd
(Show Context)
Citation Context ...thus sidestep a major difficulty for quantum walks [1], which is how to ensure that a process on an unknown graph is unitary. At any time, the robot’s state has the form ∑ αi,z |vi, z〉. 5 Also, Lloyd =-=[18]-=- argues that the total number of bits accessible up till now is at most the square of the number of Planck times elapsed so far, or about ( 10 61) 2 = 10 122 . Lloyd’s bound, unlike Bousso’s, does not... |

30 | Coins make quantum walks faster
- Ambainis, Kempe, et al.
- 2005
(Show Context)
Citation Context ...r certain graphs. We conclude in Section 8 with some open problems. ) ) 2This paper ( d = 2 d = 3 d = 4 d ≥ 5 √nlog ) 3/2 O n O ( √ n) O ( √ n) O ( √ n) [16] O (n) O ( n 5/6) O ( √ n log n) O ( √ n) =-=[3, 15]-=- O ( √ n log n) O ( √ n) O ( √ n) O ( √ n) Table 2: Time needed to find a unique marked item in a d-dimensional hypercube, using the divide-andconquer algorithms of this paper, the original quantum wa... |

29 | Improved quantum communication complexity bounds for disjointness and equality
- Høyer, Wolf
- 2002
(Show Context)
Citation Context ...d application of our search algorithm, that the quantum communication complexity of the well-known disjointness problem is O ( √ n). This improves an O (√ nc log∗ n ) upper bound of Høyer and de Wolf =-=[17]-=-, and matches the Ω ( √ n) lower bound of Razborov [22]. The rest of the paper is about the formal model that underlies our results. Section 2 sets the stage for this model, by exploring the ultimate ... |

19 |
AND AVI WIGDERSON: Quantum vs. classical communication and computation
- BUHRMAN, CLEVE
- 1998
(Show Context)
Citation Context ...he disjointness problem, Alice and Bob must decide with high probability whether there exists an i such that xi = yi = 1, using as few bits of communication as possible. Buhrman, Cleve, and Wigderson =-=[12]-=- observed that in the quantum setting, Alice and Bob can solve this problem using only O( √ nlogn) qubits of communication. This was subsequently improved by Høyer and de Wolf [20] to O (√ nclog∗ n ) ... |

15 |
Spatial search by quantum walk
- Childs, Goldstone
(Show Context)
Citation Context ... et al. [7]) that this upper bound is tight for certain graphs. We conclude in Section 8 with some open problems. ) ) 2This paper ( d = 2 d = 3 d = 4 d ≥ 5 √nlog ) 3/2 O n O ( √ n) O ( √ n) O ( √ n) =-=[16]-=- O (n) O ( n 5/6) O ( √ n log n) O ( √ n) [3, 15] O ( √ n log n) O ( √ n) O ( √ n) O ( √ n) Table 2: Time needed to find a unique marked item in a d-dimensional hypercube, using the divide-andconquer ... |

13 |
BEKENSTEIN: A universal upper bound on the entropy to energy ratio for bounded systems
- D
- 1981
(Show Context)
Citation Context ...ht. The answer seems to be that it is not tight, since (i) the entropy on a black hole horizon is not efficiently accessible2 , and (ii) weakly-gravitating systems are subject to the Bekenstein bound =-=[5]-=-, an even stronger entropy constraint than the holographic bound. 1 Admittedly, the holographic principle is part of quantum gravity and not general relativity per se. All that matters for us, though,... |

12 |
Spatial search and the Dirac equation
- Childs, Goldstone
(Show Context)
Citation Context ...r certain graphs. We conclude in Section 8 with some open problems. ) ) 2This paper ( d = 2 d = 3 d = 4 d ≥ 5 √nlog ) 3/2 O n O ( √ n) O ( √ n) O ( √ n) [16] O (n) O ( n 5/6) O ( √ n log n) O ( √ n) =-=[3, 15]-=- O ( √ n log n) O ( √ n) O ( √ n) O ( √ n) Table 2: Time needed to find a unique marked item in a d-dimensional hypercube, using the divide-andconquer algorithms of this paper, the original quantum wa... |

11 |
Introduction to Cosmology
- Ryden
- 2003
(Show Context)
Citation Context ... elapsed so far, or about ( 10 61) 2 = 10 122 . Lloyd’s bound, unlike Bousso’s, does not depend on Λ being positive. The numerical coincidence between the two bounds reflects the experimental finding =-=[21, 24]-=- that we live in a transitional era, when both Λ and “dust” contribute significantly to the Universe’s net energy balance (ΩΛ ≈ 0.7, Ωdust ≈ 0.3). In earlier times dust (and before that radiation) dom... |

7 |
and 31 others (Supernova Cosmology Project). Measurements of Ω and Λ from 42 high-redshift supernovae
- Perlmutter
- 1999
(Show Context)
Citation Context ...s that, in a spacetime with positive cosmological constant Λ > 0, the total number of bits accessible to any one experiment is at most 3π/ (Λ ln 2), or roughly 10122 given current experimental bounds =-=[21]-=- on Λ. 5 More precisely, Bousso assumes that an experiment (or in our setting, computation) has a definite beginning and end, represented by spacetime points p and q respectively. He then defines the ... |

7 | Space searches with a quantum robot
- Benioff
- 2002
(Show Context)
Citation Context ... [8]) that this upper bound is tight for certain graphs. We conclude in Section 8 with some open problems. 21.2 Relation to Previous Work In a paper on ‘Space searches with a quantum robot,’ Benioff =-=[7]-=- asked whether Grover’s algorithm can speed up search of a physical region, as opposed to a combinatorial search space. His answer was discouraging: for a 2-D grid of size √ n × √ n, Grover’s algorith... |

7 |
Umesh Vazirani, Quantum walks on graphs
- Aharonov, Ambainis, et al.
(Show Context)
Citation Context ... [4], but with the added requirement that unitary operations be ‘local’ with respect to some graph. In Section 3.1 we address the difficult question, which also arises in work on quantum random walks =-=[1]-=- and quantum cellular automata [31], of(√ what )‘local’ means. Section 4 proves general facts about our model, including an upper bound of O nδ for the time needed to search any graph with diameter δ,... |

6 |
Quantum searching a classical database (or how we learned to stop worrying and love the bomb
- RUDOLPH, GROVER
(Show Context)
Citation Context ...cosmological bounds are also relevant. Let us address these questions in turn. (1) One could argue that to maintain a ‘quantum database’ of size n requires n computing elements ([32], though see also =-=[24]-=-). So why not just exploit those elements to search the database in parallel? Then it becomes trivial to show that the search time is limited only by the radius of the database, so 1 Admittedly, the h... |

5 |
AMBAINIS: Quantum lower bounds by quantum arguments
- unknown authors
(Show Context)
Citation Context ...ylog 1 √ Λ 1/ √ Λ remains unclear, but if such a database existed, it could be searched! 3 The Model Much of what is known about the power of quantum computing comes from the black-box or query model =-=[2, 4, 7, 17, 29]-=-, in which one counts only the number of queries to an oracle, not the number of computational steps. We will take this model as the starting point for a formal definition of quantum robots. Doing so ... |

4 |
Introduction to Cosmology (Addison-Wesley
- Ryden
- 2003
(Show Context)
Citation Context ... elapsed so far, or about ( 10 61) 2 = 10 122 . Lloyd’s bound, unlike Bousso’s, does not depend on Λ being positive. The numerical coincidence between the two bounds reflects the experimental finding =-=[26, 25]-=- that we live in a transitional era, when both Λ and “dust” contribute significantly to the universe’s net energy balance (ΩΛ ≈ 0.7, Ωdust ≈ 0.3). In earlier times dust (and before that radiation) dom... |

3 |
RAZBOROV: Quantum communication complexity of symmetric predicates
- A
(Show Context)
Citation Context ...tum communication complexity of the well-known disjointness problem is O( √ n). This improves an O (√ nclog∗ n ) upper bound of Høyer and de Wolf [20], and matches the Ω( √ n) lower bound of Razborov =-=[23]-=-. The rest of the paper is about the formal model that underlies our results. Section 2 sets the stage for this model, by exploring the ultimate limits on information storage imposed by properties of ... |

3 |
WHALEY: A quantum random walk search algorithm
- SHENVI, KEMPE, et al.
(Show Context)
Citation Context ...) matrix multiplication algorithm [30]—is what taught computer scientists to see the proving of lower bounds as more than a formality. Our setting is related to that of quantum random walks on graphs =-=[1, 13, 14, 28]-=-. In an earlier version of this paper, we asked whether quantum walks might yield an alternative spatial search algorithm, possibly even one that outperforms our divide-and-conquer algorithm. Motivate... |

3 |
Could Grover’s algorithm help in searching an actual database
- Zalka
- 1999
(Show Context)
Citation Context ...its we are imagining, cosmological bounds are also relevant. Let us address these questions in turn. (1) One could argue that to maintain a ‘quantum database’ of size n requires n computing elements (=-=[31]-=-, though see also [23]). So why not just exploit those elements to search the database in parallel? Then it becomes trivial to show that the search time is limited only by the radius of the database, ... |

1 |
BENIOFF: Space searches with a quantum robot
- unknown authors
- 2002
(Show Context)
Citation Context ... Bennett et al. [7]) that this upper bound is tight for certain graphs. We conclude in Section 8 with some open problems. 1.2 Related Work In a paper on ‘Space searches with a quantum robot,’ Benioff =-=[6]-=- asked whether Grover’s algorithm can speed up search of a physical region, as opposed to a combinatorial search space. His answer was discouraging: for a 2-D grid of size √ n× √ n, Grover’s algorithm... |

1 |
RYDEN: Introduction to Cosmology
- S
- 2002
(Show Context)
Citation Context ... elapsed so far, or about ( 10 61) 2 = 10 122 . Lloyd’s bound, unlike Bousso’s, does not depend on Λ being positive. The numerical coincidence between the two bounds reflects the experimental finding =-=[26, 25]-=- that we live in a transitional era, when both Λ and “dust” contribute significantly to the universe’s net energy balance (ΩΛ ≈ 0.7, Ωdust ≈ 0.3). In earlier times dust (and before that radiation) dom... |

1 |
ZALKA: Could Grover’s algorithm help in searching an actual database
- CH
- 1999
(Show Context)
Citation Context ...its we are imagining, cosmological bounds are also relevant. Let us address these questions in turn. (1) One could argue that to maintain a ‘quantum database’ of size n requires n computing elements (=-=[32]-=-, though see also [24]). So why not just exploit those elements to search the database in parallel? Then it becomes trivial to show that the search time is limited only by the radius of the database, ... |

1 |
Quantum walks on graphs
- Kempe, Vazirani
- 2001
(Show Context)
Citation Context ... [5], but with the added requirement that unitary operations be ‘local’ with respect to some graph. In Section 3.1 we address the difficult question, which also arises in work on quantum random walks =-=[1]-=- and quantum cellular automata [30], of what ‘local’ means. We offer several definitions; some results about the robustness of these definitions are proved ( in Appendix √nδ) 10. In Section 4 we prove... |

1 |
On optimal matchings, Combinatorica 4(4):259–264
- Ajtai, Komlós, et al.
- 1984
(Show Context)
Citation Context ...at the same phenomenon—a polylog n multiplicative factor for the 2-D square, which disappears for cubes of dimension 3 and higher—shows up in a different context in work by Ajtai, Komlós, and Tusnády =-=[3]-=-. 5.1 Amplitude Amplification We start by describing amplitude amplification [12], a generalization of Grover search. Let A be a quantum algorithm that, with probability ǫ, outputs a correct answer to... |

1 |
Open problems list from Workshop on Discrete Metric Spaces and their Algorithmic Applications
- Matouˇsek
(Show Context)
Citation Context ...uantum analogue. Can we prove or disprove (√ ) DSQUARE (t (n),t(n)) ⊆ DSQUARE t (n), t (n) , (√ ) QSQUARE(t (n),t(n)) ⊆ QSQUARE t (n), t (n) ? This seems closely related to a problem of Feige (5.5 in =-=[19]-=-) about Lipschitz mapping of n grid points onto a square. Finally, can the O ( √ n polylog n) bound for search on a 2-D grid be improved, perhaps even to O ( √ n)? As mentioned in Section 1.2, Shenvi,... |