## Quantum complexities of ordered searching, sorting, and element distinctness (2001)

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Citations: | 32 - 4 self |

### BibTeX

@MISC{Høyer01quantumcomplexities,

author = {Peter Høyer and Jan Neerbek and Yaoyun Shi},

title = {Quantum complexities of ordered searching, sorting, and element distinctness },

year = {2001}

}

### Years of Citing Articles

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### Citations

1789 | Bi-orthogonal bases of compactly supported wavelets
- Cohen, Daubechies, et al.
- 1992
(Show Context)
Citation Context ...neralizations of operator U2 that can be applied to any rooted tree. Such a generalization might be of use in other search problems. Secondly, the operator U2 is related to the Haar wavelet transform =-=[13]-=-. Applying operator U2 is equivalent to applying the inverse of the Haar transform on each of the perfect subtrees rooted at the boundary vertices. Operator U2 as applied in the fourth and final step ... |

952 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...00 ˚Arhus C, Denmark. neerbek@daimi.au.dk. email: ¶ Department of Computer Science, Princeton University, Princeton, NJ 08544, USA. email: shiyy@cs.princeton.edu. 1spossible, as in the case of Shor’s =-=[23]-=- algorithms for factoring and finding discrete logarithms, super-polynomial speedups are so far proven only in restricted models such as the black box model. In the black box model, the input is given... |

329 | Quantum mechanics helps in searching for a needle in a haystack. Phys
- Grover
- 1997
(Show Context)
Citation Context ...the complexity measure is the number of queries. Many problems that allow provable quantum speedups can be formulated in this model, an example being the unordered search problem considered by Grover =-=[18]-=-. Several tight lower bounds are now known for this model, most of them being based on techniques introduced in [6, 4, 2]. We study the quantum complexities of the following three problems. Ordered se... |

321 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...d in this model, an example being the unordered search problem considered by Grover [18]. Several tight lower bounds are now known for this model, most of them being based on techniques introduced in =-=[6, 4, 2]-=-. We study the quantum complexities of the following three problems. Ordered searching Given a list of numbers x = (x0, x1, . . . , xN−1) in non-decreasing order and some number y, find the minimal i ... |

280 | Quantum lower bounds by polynomials
- Beals, Buhrman, et al.
(Show Context)
Citation Context ...d in this model, an example being the unordered search problem considered by Grover [18]. Several tight lower bounds are now known for this model, most of them being based on techniques introduced in =-=[6, 4, 2]-=-. We study the quantum complexities of the following three problems. Ordered searching Given a list of numbers x = (x0, x1, . . . , xN−1) in non-decreasing order and some number y, find the minimal i ... |

213 | Zur Theorie der orthogonalen Funktionensysteme - Haar - 1910 |

153 | Quantum lower bounds by quantum arguments
- Ambainis
- 2002
(Show Context)
Citation Context ...d in this model, an example being the unordered search problem considered by Grover [18]. Several tight lower bounds are now known for this model, most of them being based on techniques introduced in =-=[6, 4, 2]-=-. We study the quantum complexities of the following three problems. Ordered searching Given a list of numbers x = (x0, x1, . . . , xN−1) in non-decreasing order and some number y, find the minimal i ... |

133 | Quantum amplitude amplification and estimation
- Brassard, Høyer, et al.
(Show Context)
Citation Context ...wn leading to quantum algorithms using at most c log 2(N) queries for some constant c strictly less than 1. Whereas most quantum algorithms are based on Fourier transforms and amplitude amplification =-=[9]-=-, our algorithm is based on binary search trees. We initiate several applications of the binary search algorithm in quantum parallel and let them find the element we are searching for in teamwork. By ... |

97 | Grover’s quantum searching algorithm is optimal
- Zalka
- 1999
(Show Context)
Citation Context ... inner products which is only one of the many studied measures for distinguishability of states. A striking example of the limitations of using this measure is given by Jozsa and Schlienz in [21]. In =-=[25]-=-, Zalka uses a non-linear measure to prove the optimality of Grover’s algorithm [18]. Similarly, it might well be that utilizing some other (possibly non-linear) measure of distinguishability could be... |

61 | Quantum Algorithms for Element Distinctness
- Buhrman, Dürr, et al.
(Show Context)
Citation Context ...decision tree complexity of element distinctness has the same Ω(N log N) lower bound as sorting. Interestingly, their quantum complexities differ dramatically: the quantum algorithm by Buhrman et al. =-=[8]-=- uses only O(N 3/4 log N) comparisons. There is still a big gap between this upper bound and our lower bound of Ω(N 1/2 log N). One way of closing this gab might be to consider quantum time-space trad... |

28 | A general sequential time–space tradeoff for finding unique elements
- Beame
- 1991
(Show Context)
Citation Context ...ted problems have been studied for the classical case. A Time · Space lower bound of Ω(N 2 ) is proved for comparison-based sorting by Borodin et al. [8], and for the R-way branching program by Beame =-=[5]-=-. Formulations and results on the quantum time-space tradeoffs for sorting and other problems such as Element Distinctness would be interesting. Our algorithm for searching an ordered list with comple... |

28 |
Tricks or treats with the Hilbert matrix
- Choi
- 1983
(Show Context)
Citation Context ...ral norm, i.e., for any complex matrix M ∈ C m×m , ⎤ ⎥ . ⎥ ⎦ |M |2 := max v∈C m ,�v�2=1 �Mv�2. Our lower bound proofs make use of the following property of the Hilbert matrix. Lemma 6 |L |2 = π. Choi =-=[12]-=- has an elegant proof of this lemma. 4 Lower bound for ordered searching Searching ordered lists is a non-Boolean promise problem: the list is promised to be sorted, and the answer is an index, not a ... |

25 | A better lower bound for quantum algorithms searching an ordered list. Available at the LANL preprint archive
- Ambainis
- 1999
(Show Context)
Citation Context ...at most ɛ ≥ 0 requires at least � 1 − 2 � �√ N ɛ(1 − ɛ) 2π (HN − 1) (3) comparisons. 2 1 ksThe previously best known quantum lower bound for ordered searching is 1 12 log 2(N) − O(1), due to Ambainis =-=[1]-=-. For comparison-based sorting and element distinctness, the previously best known quantum lower bounds are Ω(N) and Ω( √ N), respectively, both of which can be proven in many ways. We prove our lower... |

24 | A time-space tradeoff for sorting on non-oblivious machines
- Borodin, Fischer, et al.
- 1981
(Show Context)
Citation Context ...arisons. Space-time tradeoffs for sorting and related problems have been studied for the classical case. A Time · Space lower bound of Ω(N 2 ) is proved for comparison-based sorting by Borodin et al. =-=[8]-=-, and for the R-way branching program by Beame [5]. Formulations and results on the quantum time-space tradeoffs for sorting and other problems such as Element Distinctness would be interesting. Our a... |

23 | A lower bound for randomized algebraic decision trees
- GRIGORIEV, KARPINSKI, et al.
- 1996
(Show Context)
Citation Context ...is at least 0.622 then 1 12 log2(N) > (1 − ɛ)0.220 log2(N), in which case the lower bound of 1 12 log2(N) + O(1) is stronger. 18sThe result of Grigoriev, Karpinski, Meyer auf der Heide, and Smolensky =-=[17]-=- implies that if only comparisons are allowed, the randomized decision tree complexity of Element Distinctness has the same Ω(N log N) lower bound as sorting. Interestingly, their quantum complexities... |

22 | Invariant quantum algorithms for insertion into an ordered list
- Farhi, Goldstone, et al.
- 1999
(Show Context)
Citation Context ...ent, or a probabilistic adversary argument. This proof technique is based on the work of Bennett, Bernstein, Brassard, and Vazirani [6] and Ambainis [2]. Farhi, Goldstone, Gutmann, and Sipser give in =-=[15]-=- an exact quantum algorithm for ordered searching using roughly 0.526 log 2(N) queries. We provide an alternative quantum algorithm that is also exact and uses log 3(N) + O(1) ≈ 0.631 log 2(N) queries... |

19 |
The role of relative entropy in quantum information theory
- Vedral
(Show Context)
Citation Context ... for non-symmetric (possibly partial) functions. It could be interesting to consider other measures than inner products, as discussed, for instance, by Zalka [18], Jozsa and Schlienz [15], and Vedral =-=[17]-=-. The result of Grigoriev, Karpinski, Meyer auf der Heide, and Smolensky [13] implies that if only comparisons are allowed, the randomized decision tree complexity of element distinctness has the same... |

18 | On the power of quantum computation
- Vazirani
- 1998
(Show Context)
Citation Context ...g lower bounds for quantum computing was introduced by Bennett, Bernstein, Brassard and Vazirani in their influential paper [6]. Their beautiful technique is nicely described in Vazirani’s exposition =-=[24]-=-. Our technique is a natural generalization of theirs as well as of Ambainis’ powerful entanglement lower bound approach recently proposed in [2]. Here is the basic idea: Consider a quantum algorithm ... |

15 | Quantum computation. in: Complexity theory retrospective - Berthiaume - 1997 |

14 | Distinguishability of states and von Neumann entropy, Phys Rev A 62
- Jozsa, Schlienz
- 1999
(Show Context)
Citation Context ...re to use inner products which is only one of the many studied measures for distinguishability of states. A striking example of the limitations of using this measure is given by Jozsa and Schlienz in =-=[21]-=-. In [25], Zalka uses a non-linear measure to prove the optimality of Grover’s algorithm [18]. Similarly, it might well be that utilizing some other (possibly non-linear) measure of distinguishability... |

13 | Wavelet Transforms: Fast Algorithms and Complete Circuits
- Fijany, WilliamsQuantum
(Show Context)
Citation Context ...ator U2 as applied in the fourth and final step of our algorithm can thus be implemented by applying the inverse of the quantum Haar transform. Since the Haar transform can be efficiently implemented =-=[16, 20]-=-, so can U2. The quantum version of the Haar transform was first considered and defined in [20], motivated by the successes of the quantum Fourier transforms. Possible relationships between the quantu... |

11 | A limit on the speed of quantum computation for insertion into an ordered list
- Farhi, Goldstone, et al.
- 1998
(Show Context)
Citation Context ... length m = ⌈log 2(N)⌉. The first lower bound of Ω( � log(N)/ log log(N)) was proved by Buhrman and de Wolf [11] by an ingenious reduction from the or problem. Farhi, Goldstone, Gutmann, 6sand Sipser =-=[14]-=- improved this to log2(N)/2 log2 log2(N), and Ambainis [1] then proved the previously best known lower bound of 1 12 log2(N) − O(1). In [14, 1], they use, as we do here, an inner product argument alon... |

6 | A lower bounds for quantum search of an ordered list
- Buhrman, Wolf
- 1999
(Show Context)
Citation Context ...< N | xi = 1}, where we identify the result f(x) with its binary encoding as a bit-string of length m = ⌈log 2(N)⌉. The first lower bound of Ω( � log(N)/ log log(N)) was proved by Buhrman and de Wolf =-=[11]-=- by an ingenious reduction from the or problem. Farhi, Goldstone, Gutmann, 6sand Sipser [14] improved this to log2(N)/2 log2 log2(N), and Ambainis [1] then proved the previously best known lower bound... |

5 |
Quantum Algorithms
- Høyer
- 2000
(Show Context)
Citation Context ...ator U2 as applied in the fourth and final step of our algorithm can thus be implemented by applying the inverse of the quantum Haar transform. Since the Haar transform can be efficiently implemented =-=[16, 20]-=-, so can U2. The quantum version of the Haar transform was first considered and defined in [20], motivated by the successes of the quantum Fourier transforms. Possible relationships between the quantu... |

5 |
R.: A lower bound for quantum search of an ordered
- Buhrman, Wolf
- 1999
(Show Context)
Citation Context .... |A |2 = π. Hence, |BN |2 ≤ π. 3.2 Lower bound for ordered searching The first non-trivial quantum lower bound on ordered searching proven was Ω( √ log2(N)/ log2 log2(N)), due to Buhrman and de Wolf =-=[9]-=- by an ingenious reduction from the parity problem. Farhi, Goldstone, Gutmann, and Sipser [11] improved this to log2(N)/2 log2 log2(N), and Ambainis [1] then proved the previously best known lower bou... |

2 | Quantum algorithms for finding claws, collisions and triangles,” quant-ph/0007016
- Buhrman, Dürr, et al.
- 2000
(Show Context)
Citation Context ...of Element Distinctness has the same Ω(N log N) lower bound as sorting. Interestingly, their quantum complexities differ dramatically: the quantum algorithm for Element Distinctness by Buhrman et al. =-=[10]-=- uses only O(N 3/4 log N) comparisons. Space-time tradeoffs for sorting and related problems have been studied for the classical case. A Time · Space lower bound of Ω(N 2 ) is proved for comparison-ba... |

2 | Efficient quantum algorithms. quant-ph/9702028 - Hoyer - 1997 |

1 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer - Rohrig, communication |