## Quantum Algorithms for Element Distinctness (2001)

### Cached

### Download Links

- [www.lri.fr]
- [www.liafa.jussieu.fr]
- [www.liafa.univ-paris-diderot.fr]
- [www.liafa.jussieu.fr]
- [www.liafa.univ-paris-diderot.fr]
- [www.brics.dk]
- [www.cwi.nl]
- [homepages.cwi.nl]
- [www.lri.fr]
- [www.liafa.jussieu.fr]
- [www.liafa.jussieu.fr]
- [www.cpsc.ucalgary.ca]
- [www.lri.fr]
- [www.liafa.jussieu.fr]
- [www.liafa.univ-paris-diderot.fr]
- [www.liafa.jussieu.fr]
- [www.liafa.univ-paris-diderot.fr]
- DBLP

### Other Repositories/Bibliography

Venue: | SIAM Journal of Computing |

Citations: | 61 - 11 self |

### BibTeX

@INPROCEEDINGS{Buhrman01quantumalgorithms,

author = {Harry Buhrman and Christoph Dürr and Mark Heiligman and Peter Høyer and Frédéric Magniez and Miklos Santha and Ronald Wolf},

title = {Quantum Algorithms for Element Distinctness},

booktitle = {SIAM Journal of Computing},

year = {2001},

pages = {131--137}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with Θ(N log N) classical complexity. We also prove a lower bound of Ω ( √ N) comparisons for this problem and derive bounds for a number of related problems. 1

### Citations

1529 |
Chuang “Quantum Computation and Quantum Information
- Nielsen, Isaac
- 2000
(Show Context)
Citation Context ...hat isolated result, the second has been applied as a building block in quite a few other quantum algorithms [7, 9, 11, 12, 19, 10, 18, 8]. For a general introduction to quantum computing we refer to =-=[20]-=-. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [9] for finding a collision in a 2-to-1 function f. A collision is a pair of distinct elements x... |

952 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ... significantly faster than a classical computer. Here we are concerned with the second issue. Few good quantum algorithms are known to date, the two main examples being Shor’s algorithm for factoring =-=[22]-=- and Grover’s algorithm for searching [15]. Whereas the first so far has remained a seminal but somewhat isolated result, the second has been applied as a building block in quite a few other quantum a... |

913 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...puter. Here we are concerned with the second issue. Few good quantum algorithms are known to date, the two main examples being Shor’s algorithm for factoring [22] and Grover’s algorithm for searching =-=[15]-=-. Whereas the first so far has remained a seminal but somewhat isolated result, the second has been applied as a building block in quite a few other quantum algorithms [7, 9, 11, 12, 19, 10, 18, 8]. F... |

321 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...and�if and only if OR��. Thus if we can find a claw using comparisons, we can decide OR using queries to�(two�-queries suffice to implement a comparison). Using known lower bounds for the OR-function =-=[5, 3]-=-, this gives anªÅbound for exact quantum and anªÔÅbound for bounded-error quantum algorithms. ��ÇÆ�Å��ÐÓ�ÆifÆ�Å�Æ ÇÅ�ÐÓ�ÆifÅ�Æ The next theorem follows. Theorem 2 The comparison-complexity of the claw... |

280 | Quantum lower bounds by polynomials
- Beals, Buhrman, et al.
(Show Context)
Citation Context ...and�if and only if OR��. Thus if we can find a claw using comparisons, we can decide OR using queries to�(two�-queries suffice to implement a comparison). Using known lower bounds for the OR-function =-=[5, 3]-=-, this gives anªÅbound for exact quantum and anªÔÅbound for bounded-error quantum algorithms. ��ÇÆ�Å��ÐÓ�ÆifÆ�Å�Æ ÇÅ�ÐÓ�ÆifÅ�Æ The next theorem follows. Theorem 2 The comparison-complexity of the claw... |

153 | Quantum lower bounds by quantum arguments
- Ambainis
- 2002
(Show Context)
Citation Context ...Note that the hardness of the parity-collision problem implies the hardness of exactly counting the number of colandÊ��¢� lisions. Our proofs use the powerful lower bound method developed by Ambainis =-=[2]-=-. Let us state here exactly the result that we require. Theorem 9 ([2]) Let������Æ℄��Æ℄�be the set of all possible input-functions, and¨����be a function (which we want to compute). Let���be two subse... |

134 | Quantum vs. classical communication and computation
- Buhrman, Cleve, et al.
- 1998
(Show Context)
Citation Context ...r’s algorithm for searching [15]. Whereas the first so far has remained a seminal but somewhat isolated result, the second has been applied as a building block in quite a few other quantum algorithms =-=[7, 9, 11, 12, 19, 10, 18, 8]-=-. For a general introduction to quantum computing we refer to [20]. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [9] for finding a collision in... |

133 | Quantum amplitude amplification and estimation
- Brassard, Høyer, et al.
(Show Context)
Citation Context ...r’s algorithm for searching [15]. Whereas the first so far has remained a seminal but somewhat isolated result, the second has been applied as a building block in quite a few other quantum algorithms =-=[7, 9, 11, 12, 19, 10, 18, 8]-=-. For a general introduction to quantum computing we refer to [20]. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [9] for finding a collision in... |

114 |
Quantum lower bounds for the collision and the element distinctness problems
- Aaronson, Shi
(Show Context)
Citation Context ... N) comparisons. We also prove a lower bound for this problem of Ω(N 1/2 ) comparisons for bounded-error quantum algorithms and Ω(N) for exact quantum algorithms. Very recently, in a series of papers =-=[1, 21]-=-, several nontrivial lower bounds have been proven for the collision-finding problem in a 2-to-1 function by Aaronson and Shi. The latest result of Shi actually gives a Ω(N 1/3 ) lower bound matching ... |

114 | Oracle quantum computing - Berthiaume, Brassard - 1994 |

106 | Quantum counting, in - Brassard, Høyer, et al. - 1998 |

101 | Quantum walk algorithm for element distinctness
- Ambainis
- 2004
(Show Context)
Citation Context ...the bounded-error quantum query complexity was shown to be Θ(N 2/3 ): The lower bound follows from Shi [23] by the observation above and an algorithm matching this bound was found in 2003 by Ambainis =-=[3]-=- using a quantum walk. The classical bounded-error query complexity is Θ(N) by a trivial reduction from the OR-problem: For an ORinstance x ∈ {0, 1} N we define the function f : [N +1] → [N +1] where ... |

86 | An exact quantum polynomial-time algorithm for Simon’s problem
- Brassard, Høyer
- 1997
(Show Context)
Citation Context ...nce. Email: santha@lri.fr. ÝÝCWI, P.O.Box 94079, Amsterdam, The Netherlands. Also affiliated with the University of Amsterdam. Email: rdewolf@cwi.nl. ing block in quite a few other quantum algorithms =-=[6, 8, 10, 11, 18, 7]-=-. For a general introduction to quantum computing we refer to [19]. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [8] for finding a collision in... |

68 | The Quantum Query Complexity of Approximating the Median and Related
- Nayak, Wu
- 1999
(Show Context)
Citation Context ...r’s algorithm for searching [15]. Whereas the first so far has remained a seminal but somewhat isolated result, the second has been applied as a building block in quite a few other quantum algorithms =-=[7, 9, 11, 12, 19, 10, 18, 8]-=-. For a general introduction to quantum computing we refer to [20]. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [9] for finding a collision in... |

62 | Quantum algorithms for the triangle problem
- Magniez, Santha, et al.
- 2007
(Show Context)
Citation Context ...s no triangle complexity We present a bounded-error quantum algorithm which needs O(n + √ nm) queries. A better algorithm has been found in 2003, with O(n 1.3 ) bounded-error query quantum complexity =-=[19]-=-, while Yao [26] showed a lower bound of Ω(n 2/3 log 1/6 n). Classically a simple reduction from the OR-problem shows that the bounded-error query complexity is Θ(n 2 ), even if m = O(n). 2. Prelimina... |

59 | Quantum lower bound for the collision problem
- Aaronson
- 2002
(Show Context)
Citation Context ... N) comparisons. We also prove a lower bound for this problem of Ω(N 1/2 ) comparisons for bounded-error quantum algorithms and Ω(N) for exact quantum algorithms. Very recently, in a series of papers =-=[1, 21]-=-, several nontrivial lower bounds have been proven for the collision-finding problem in a 2-to-1 function by Aaronson and Shi. The latest result of Shi actually gives a Ω(N 1/3 ) lower bound matching ... |

46 | Quantum and classical strong direct product theorems and optimal time-space tradeoffs - Spalek, Wolf - 2004 |

45 | Quantum query complexity of some graph problems
- Durr, Heiligman, et al.
- 2004
(Show Context)
Citation Context ...over classical search algorithms. Whereas the first so far has remained a seminal but somewhat isolated result, the second has been applied as a building block in quite a few other quantum algorithms =-=[6, 8, 9, 10, 21, 20, 7, 12]-=-. The security of the widely used cryptosystem RSA is based on the assumption that it is hard to factor integers. Shor’s algorithm solves precisely this task. In the same flavor, the security of digit... |

37 | Quantum algorithm for the collision problem
- Brassard, Høyer, et al.
- 1997
(Show Context)
Citation Context ...nce. Email: santha@lri.fr. ÝÝCWI, P.O.Box 94079, Amsterdam, The Netherlands. Also affiliated with the University of Amsterdam. Email: rdewolf@cwi.nl. ing block in quite a few other quantum algorithms =-=[6, 8, 10, 11, 18, 7]-=-. For a general introduction to quantum computing we refer to [19]. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [8] for finding a collision in... |

34 | Determinism versus non-determinism for linear time RAMs with memory restrictions
- Ajtai
- 1999
(Show Context)
Citation Context ...ing it will require new ideas. An interesting direction could be to take into account simultaneously time complexity and space complexity, as has been done for classical algorithms by Yao [21], Ajtai =-=[1]-=-, Beame, Saks, satisfiesÌ¡Ë�ªÆ �Æ� Sun, and Vee [4], and others. In particular, Yao shows that the time-space product of any classical deterministic comparison-based branching program solving element ... |

32 | Super-linear time-space tradeoff lower bounds for randomized computation
- Beame, Saks, et al.
- 2000
(Show Context)
Citation Context ...ion occurs in�is equivalent to deciding whether� maps allÜto distinct elements. ofÆÐÓ�ÆrequireÆÐÓ�Æ¢Æ This is known as the element distinctness problem and has been well studied classically, see e.g. =-=[21, 16, 13, 4]-=-. Element distinctness is particularly interesting because its classical complexity is related to that of sorting, which is well known to comparisons. If we sort�, we can decide element distinctness b... |

32 |
Bounds for small-error and zero-error quantum algorithms
- Zalka
- 1999
(Show Context)
Citation Context |

32 | Quantum complexities of ordered searching, sorting, and element distinctness
- Høyer, Neerbek, et al.
(Show Context)
Citation Context ... N log N comparisons [13], and it was recently shown that such a linear speed-up is the best possible: quantum sorting requires Ω(N log N) comparisons, even if one allows a small probability of error =-=[16]-=-. Accordingly, our O(N 3/4 log N) quantum up2sper bound shows that element distinctness is significantly easier than sorting for a quantum computer, in contrast to the classical case. In Section 4, we... |

26 |
Quantum amplitude amplification and estimation. Quantum Computation and Quantum Information: A Millennium Volume
- Brassard, Høyer, et al.
- 2002
(Show Context)
Citation Context ...nce. Email: santha@lri.fr. ÝÝCWI, P.O.Box 94079, Amsterdam, The Netherlands. Also affiliated with the University of Amsterdam. Email: rdewolf@cwi.nl. ing block in quite a few other quantum algorithms =-=[6, 8, 10, 11, 18, 7]-=-. For a general introduction to quantum computing we refer to [19]. One of the earliest applications of Grover’s algorithm was the algorithm of Brassard, Høyer, and Tapp [8] for finding a collision in... |

25 |
Near-optimal time–space tradeoff for element distinctness
- Yao
(Show Context)
Citation Context ...iding if a collision occurs in f is equivalent to deciding whether f maps all x to distinct elements. This is known as the element distinctness problem and has been well studied classically, see e.g. =-=[23, 17, 14, 5]-=-. Element distinctness is particularly interesting because its classical complexity is related to that of sorting, which is well known to require N log N + Θ(N) comparisons. If we sort f, we can decid... |

22 | Invariant quantum algorithms for insertion into an ordered list
- Farhi, Goldstone, et al.
- 1999
(Show Context)
Citation Context ...onger model, see [14]), so sorting and element distinctness are equally hard for classical computers. On a quantum computer, the best known upper bound for sorting is roughly 0.53 N log N comparisons =-=[13]-=-, and it was recently shown that such a linear speed-up is the best possible: quantum sorting requires Ω(N log N) comparisons, even if one allows a small probability of error [16]. Accordingly, our O(... |

22 | An introduction to quantum complexity theory - Cleve - 2000 |

20 | Quantum searching, counting and amplitude amplification by eigenvector analysis
- Mosca
- 1998
(Show Context)
Citation Context |

10 | Quantum vs. classical communication and computation (preliminary version - Buhrman, Cleve, et al. - 1998 |

7 | Near-optimal time-space tradeo# for element distinctness - Yao - 1988 |

6 | Randomized complexity lower bounds
- Grigoriev
- 1998
(Show Context)
Citation Context ...iding if a collision occurs in f is equivalent to deciding whether f maps all x to distinct elements. This is known as the element distinctness problem and has been well studied classically, see e.g. =-=[23, 17, 14, 5]-=-. Element distinctness is particularly interesting because its classical complexity is related to that of sorting, which is well known to require N log N + Θ(N) comparisons. If we sort f, we can decid... |

6 | Super-linear time-space tradeo# lower bounds for randomized computation - Beame, Saks, et al. - 2000 |

5 | An Introduction to Quantum Complexity Theory, quant-ph/9906111. 8 - Cleve |

3 |
Bound on the Number of Functions That Can Be Distinguished With k Quantum Queries, Phys
- Fahri, Goldstone, et al.
- 1999
(Show Context)
Citation Context ...tness are equally hard for classical computers. On a quantum computer, the best known upper bound for sorting is roughly 0.53 N log N comparisons [FGGS99a], whereas the best known lower bound is Ω(N) =-=[FGGS99b]-=-. Accordingly, our O(N 3/4 log N) quantum upper bound shows that element distinctness is significantly easier than sorting for a quantum computer, in contrast to the classical case. In Section 4, we c... |

3 | Quantum amplitude ampli and estimation - Brassard, Hyer, et al. - 2000 |

2 |
A lower bound for the integer element distinctiveness problem
- Lubiw, Rácz
- 1991
(Show Context)
Citation Context ...iding if a collision occurs in f is equivalent to deciding whether f maps all x to distinct elements. This is known as the element distinctness problem and has been well studied classically, see e.g. =-=[23, 17, 14, 5]-=-. Element distinctness is particularly interesting because its classical complexity is related to that of sorting, which is well known to require N log N + Θ(N) comparisons. If we sort f, we can decid... |

1 | lower bounds for collision and element distinctness with small range - Quantum - 2003 |