## Quantum algorithms for the triangle problem (2005)

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Venue: | PROCEEDINGS OF SODA’05 |

Citations: | 61 - 10 self |

### BibTeX

@INPROCEEDINGS{Magniez05quantumalgorithms,

author = {Frédéric Magniez and Miklos Santha and Mario Szegedy},

title = {Quantum algorithms for the triangle problem},

booktitle = {PROCEEDINGS OF SODA’05},

year = {2005},

pages = {1109--1117},

publisher = {}

}

### OpenURL

### Abstract

We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.

### Citations

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Citation Context ...ions. quantum algorithm, query complexity, triangle problem, quantum walk DOI. 10.1137/050643684 1. Introduction. Quantum computing is an extremely active research area (for introductions, see, e.g., =-=[22, 20]-=-), where a growing trend is the study of quantum query complexity. The quantum query model was implicitly introduced by Deutsch [15], Deutsch and Jozsa [16], Simon [25], and Grover [18], and explicitl... |

892 | A fast quantum-mechanical algorithm for database search
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Citation Context ...earch and makes Õ(n10/7 ) queries. The second algorithm uses Õ(n13/10 ) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search =-=[18]-=-. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm)... |

695 | Quantum Theory, the Church–Turing Principle and the Universal Quantum Computer
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Citation Context ...search area (for introductions see e.g. [22, 20]) where a growing trend is the study of quantum query complexity. The quantum query model was implicitly introduced by Deutsch, Jozsa, Simon and Grover =-=[15, 16, 25, 18]-=-, and explicitly by Beals, Buhrman, Cleve, Mosca and de Wolf [9]. In this model, like in its classical counterpart, we pay for accessing the oracle (the black box), but unlike in the classical case, t... |

362 | On the power of quantum computation
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Citation Context ...search area (for introductions see e.g. [22, 20]) where a growing trend is the study of quantum query complexity. The quantum query model was implicitly introduced by Deutsch, Jozsa, Simon and Grover =-=[15, 16, 25, 18]-=-, and explicitly by Beals, Buhrman, Cleve, Mosca and de Wolf [9]. In this model, like in its classical counterpart, we pay for accessing the oracle (the black box), but unlike in the classical case, t... |

351 |
Rapid solution of problems by quantum computation
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- 1992
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Citation Context ...search area (for introductions see e.g. [22, 20]) where a growing trend is the study of quantum query complexity. The quantum query model was implicitly introduced by Deutsch, Jozsa, Simon and Grover =-=[15, 16, 25, 18]-=-, and explicitly by Beals, Buhrman, Cleve, Mosca and de Wolf [9]. In this model, like in its classical counterpart, we pay for accessing the oracle (the black box), but unlike in the classical case, t... |

277 | Quantum lower bounds by polynomials
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Citation Context ...e study of quantum query complexity. The quantum query model was implicitly introduced by Deutsch, Jozsa, Simon and Grover [15, 16, 25, 18], and explicitly by Beals, Buhrman, Cleve, Mosca and de Wolf =-=[9]-=-. In this model, like in its classical counterpart, we pay for accessing the oracle (the black box), but unlike in the classical case, the machine can use the power of quantum parallelism to make quer... |

151 |
Regular partitions of graphs, Problemes combinatoires et theorie des graphes
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- 1976
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Citation Context ...r Triangle (Theorem 3.5) whose quantum query complexity is Õ(n10/7 ). Surprisingly, its quantum parts consist in only Grover Search subroutines. Indeed, Grover Search coupled with the Szemerédi lemma =-=[28]-=- already gives an o(n 3/2 ) bound. We exploit this fact using a simpler observation that leads to the Õ(n10/7 ) bound. Moreover, our algorithm uses only small quantum memory, namely O(log n) qubits (a... |

149 | Quantum lower bounds by quantum arguments
- Ambainis
- 2002
(Show Context)
Citation Context ...bounded error quantum query model, the Ω(n2 ) lower bound does not hold anymore in general. An Ω(n2/3 log 1/6 n) lower bound, first observed by Yao [31], can be obtained combining Ambainis’ technique =-=[4]-=- with the ∗ A preliminary version of this paper appeared in Proceedings of 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 1109–1117, 2005. † CNRS–LRI, UMR 8623 Université Paris–Sud, 91405 Orsay, ... |

127 | Quantum amplitude amplification and estimation
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Citation Context ...is section we do not try to optimize log n factors and we will hide time in the Õ notation. The first observation is based on the Amplitude Amplification technique of Brassard, Høyer, Mosca, and Tapp =-=[10]-=- Lemma 3.1. For any known graph E ⊆ [n] 2 , a triangle with at least one edge in E can be detected with Õ(√E + � n|G ∩ E|) queries and probability 1 − 1 n . Perhaps the most crucial observation to the... |

113 | Topological quantum computation
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Citation Context ... an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G. 1. Introduction. Quantum computing is an extremely active research area (for introductions see e.g. =-=[22, 20]-=-) where a growing trend is the study of quantum query complexity. The quantum query model was implicitly introduced by Deutsch, Jozsa, Simon and Grover [15, 16, 25, 18], and explicitly by Beals, Buhrm... |

112 |
Quantum lower bounds for the collision and the element distinctness problems
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Citation Context ...de various collision type problems such as the 2-1 Collision Problem and the Element Distinctness Problem. The first polynomial lower bound for the 2-1 Collision Problem was shown by Aaronson and Shi =-=[1]-=- using the polynomial method of Beals, Buhrman, Cleve, Mosca and de Wolf [9]. For the Element Distinctness Problem, a randomized reduction from the 2-1 Collision Problem gives Ω(n 2/3 ). In this paper... |

97 | Quantum walk algorithm for element distinctness
- Ambainis
(Show Context)
Citation Context ... leads to the Õ(n10/7 ) bound. Moreover, our algorithm uses only small quantum memory, namely O(log n) qubits (and O(n 2 ) classical bits). Then, we generalize the new elegant method used by Ambainis =-=[6]-=- for solving the Element Distinctness Problem in O(n 2/3 ), to solve a general Collision Problem by a dynamic quantum query algorithm (Theorem 4.1). The solution of the general Collision Problem will ... |

89 | Quantum walks on graphs
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- 2001
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Citation Context ... operator acts only on the coin register and it is the identity on the node register. The shift operation only changes the node register, but it is controlled by the content of the coin register (see =-=[29, 2, 7]-=-). Often the coin flip is actually the Diffusion operator. Definition 2.2 (Diffusion over T ). Let T be a finite set. The diffusion operator over T is the unitary operator on the Hilbert space C T tha... |

87 | Finding and counting given length cycles
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Citation Context ...djacency matrix f of a graph G on n nodes. Output: a triangle if there is any, otherwise reject. Triangle has been studied in various contexts, partly because of its relation to matrix multiplication =-=[3]-=-. Its quantum query complexity was first raised in [12], where the authors show that in the case of sparse graphs the trivial (that is, using Grover Search) O(n 3/2 ) upper bound can be improved. Thei... |

85 | OneDimensional Quantum Walks - Ambainis, Bach, et al. - 2001 |

60 | Quantum algorithms for element distinctness
- Dürr, Høyer, et al.
(Show Context)
Citation Context ... of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in =-=[12]-=-, where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G. 1. Introduction. Quantum computing is an extremely active research area (for introductions se... |

60 | Polynomial degree vs quantum query complexity, in
- Ambainis
(Show Context)
Citation Context ...reaks down near the square root of the instance size [27, 21, 32, 26]: If the 1-certificate size of a boolean function on N boolean variables is K, then even the most general variants [8, Theorem 4], =-=[5]-=-, [21] of Ambainis’ quantum adversary technique [4] can prove only a lower bound of Ω( √ NK). Indeed only the Ω(n) lower bound is known for Triangle, which, because of the remark above, cannot be impr... |

52 |
On recognizing graph properties from adjacency matrices, Theor
- Rivest, Vuillemin
(Show Context)
Citation Context ...ministic query complexity exactly � � n 2 , where n is the number of nodes of the input graph. Though this conjecture is still open, an Ω(n2 ) lower bound has been established by Rivest and Vuillemin =-=[23]-=-. In randomized bounded error complexity the general lower bounds are far from the conjectured Ω(n2 ). The first non-linear lower bound was shown by Yao [30]. For a long time Peter Hajnal’s Ω(n4/3 ) b... |

47 |
Regular partitions of graphs, Colloque Inter
- Szemerédi
- 1978
(Show Context)
Citation Context ...r Triangle (Theorem 3.5) whose quantum query complexity is Õ(n10/7 ). Surprisingly, its quantum parts only consist in Grover Search subroutines. Indeed, Grover Search coupled with the Szemerédi Lemma =-=[28]-=- already gives a o(n 3/2 ) bound. We exploit this fact using a simpler observation that leads to the Õ(n10/7 ) bound. Moreover our algorithm uses only small quantum memory, namely O(log n) qubits (and... |

44 | MAGNIEZ: Lower bounds for randomized and quantum query complexity using Kolmogorov arguments
- LAPLANTE, F
- 2004
(Show Context)
Citation Context ...well as of many of its kins with small one-sided certificate size are notoriously hard to analyze, because one of the main lower bounding methods breaks down near the square root of the instance size =-=[27, 21, 32, 26]-=-: If the 1-certificate size of a boolean function on N boolean variables is K, then even the most general variants [8, Theorem 4][5][21] of the Ambainis’ quantum adversary technique [4] can prove only... |

34 | Hidden subgroup states are almost orthogonal
- Ettinger, Høyer, et al.
(Show Context)
Citation Context ... such bounds in the query model. For promise problems quantum query complexity indeed can be exponentially smaller than the randomized one, a prominent example for that is the Hidden Subgroup Problem =-=[25, 17]-=-. On the other hand, Beals, Buhrman, Cleve, Mosca and de Wolf [9] showed that for total functions the deterministic and the quantum query complexities are polynomially related. In this context, a larg... |

32 | All quantum adversary methods are equivalent
- Spalek, Szegedy
(Show Context)
Citation Context ...well as of many of its kins with small one-sided certificate size are notoriously hard to analyze, because one of the main lower bounding methods breaks down near the square root of the instance size =-=[27, 21, 32, 26]-=-: If the 1-certificate size of a boolean function on N boolean variables is K, then even the most general variants [8, Theorem 4][5][21] of the Ambainis’ quantum adversary technique [4] can prove only... |

29 | Quantum simulations of classical random walks and undirected graph connectivity
- Watrous
(Show Context)
Citation Context ... operator acts only on the coin register and it is the identity on the node register. The shift operation only changes the node register, but it is controlled by the content of the coin register (see =-=[29, 2, 7]-=-). Often the coin flip is actually the Diffusion operator. Definition 2.2 (Diffusion over T ). Let T be a finite set. The diffusion operator over T is the unitary operator on the Hilbert space C T tha... |

29 |
The quantum query complexity of the hidden subgroup problem is polynomial
- Ettinger, Høyer, et al.
- 2004
(Show Context)
Citation Context ...in the query model. For promise problems quantum query complexity indeed can be exponentially smaller than the randomized query complexity; a prominent example for that is the Hidden Subgroup Problem =-=[25, 17]-=-. On the other hand, Beals et al. [9] showed that for total functions the deterministic and the quantum query complexities are polynomially related. In this context, a large axis of research pioneered... |

27 | SZEGEDY: Quantum decision trees and semidefinite programming - BARNUM, SAKS, et al. - 2003 |

27 |
On the time required to recognize properties of graphs: a problem
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- 1973
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Citation Context ...ured, structured, or partially structured databases. The classical query complexity of graph properties has made its fame through the notoriously hard evasiveness conjecture of Aanderaa and Rosenberg =-=[24]-=- which states that every non-trivial and monotone boolean function on graphs whose value remains invariant under the permutation of the nodes has deterministic query complexity exactly � � n 2 , where... |

26 | Bounds for small-error and zero-error quantum algorithms
- Buhrman, Cleve, et al.
- 1999
(Show Context)
Citation Context ...time Peter Hajnal’s Ω(n4/3 ) bound [19] was the best, until it was slightly improved in [13] to Ω(n4/3 log 1/3 n). The question of the quantum query complexity of graph properties was first raised in =-=[11]-=- where it is shown that in the exact case an Ω(n2 ) lower bound still holds. In the bounded error quantum query model, the Ω(n2 ) lower bound does not hold anymore in general. An Ω(n2/3 log 1/6 n) low... |

25 | On the power of Ambainis’s lower bounds
- Zhang
- 2004
(Show Context)
Citation Context ...well as of many of its kins with small one-sided certificate size are notoriously hard to analyze, because one of the main lower bounding methods breaks down near the square root of the instance size =-=[27, 21, 32, 26]-=-: If the 1-certificate size of a boolean function on N boolean variables is K, then even the most general variants [8, Theorem 4][5][21] of the Ambainis’ quantum adversary technique [4] can prove only... |

23 | An Ω(n 4/3 ) lower bound on the randomized complexity of graph properties
- Hajnal
- 1991
(Show Context)
Citation Context ...randomized bounded error complexity the general lower bounds are far from the conjectured Ω(n2 ). The first non-linear lower bound was shown by Yao [30]. For a long time Peter Hajnal’s Ω(n4/3 ) bound =-=[19]-=- was the best, until it was slightly improved in [13] to Ω(n4/3 log 1/3 n). The question of the quantum query complexity of graph properties was first raised in [11] where it is shown that in the exac... |

18 |
Quantum algorithms for subset finding
- Childs, Eisenberg
(Show Context)
Citation Context ...complexity is Õ(n 2−2/(k+1) ). In fact we can improve this bound to Õ(n2−2/k ). Note that only the trivial Ω(n) lower bound is known. This problem was independently considered by Childs and Eisenberg =-=[14]-=- whenever H is a k-clique. Beside the direct Ambainis’ algorithm, they obtained an Õ(n2.5−6/(k+2) ) query algorithm. For k = 4, 5, this is faster than the direct Ambainis’ algorithm, but slower than o... |

17 |
Lower bounds to randomized algorithms for graph properties
- Yao
(Show Context)
Citation Context ...s been established by Rivest and Vuillemin [23]. In randomized bounded error complexity the general lower bounds are far from the conjectured Ω(n2 ). The first non-linear lower bound was shown by Yao =-=[30]-=-. For a long time Peter Hajnal’s Ω(n4/3 ) bound [19] was the best, until it was slightly improved in [13] to Ω(n4/3 log 1/3 n). The question of the quantum query complexity of graph properties was fir... |

14 | On the quantum query complexity of detecting triangles in graphs. arXiv.org e-print quant-ph/0310107
- Szegedy
- 2003
(Show Context)
Citation Context |

11 |
Improved lower bounds on the randomized complexity of graph properties
- Chakrabarti, Khot
- 2007
(Show Context)
Citation Context ... bounds are far from the conjectured Ω(n2 ). The first non-linear lower bound was shown by Yao [30]. For a long time Peter Hajnal’s Ω(n4/3 ) bound [19] was the best, until it was slightly improved in =-=[13]-=- to Ω(n4/3 log 1/3 n). The question of the quantum query complexity of graph properties was first raised in [11] where it is shown that in the exact case an Ω(n2 ) lower bound still holds. In the boun... |

5 |
Private communication
- Yao
(Show Context)
Citation Context ...t case an Ω(n 2 ) lower bound still holds. In the bounded error quantum query model, the Ω(n 2 ) lower bound does not hold anymore in general. An Ω(n 2/3 log 1/6 n) lower bound, first observed by Yao =-=[31]-=-, can be obtained combining Ambainis’ technique [4] with the above randomized lower bound. We address the Triangle Problem in this setting. In a graph G, a complete subgraph on three vertices is calle... |

4 |
degree vs. quantum query complexity
- Polynomial
- 2003
(Show Context)
Citation Context ... breaks down near the square root of the instance size [27, 21, 32, 26]: If the 1-certificate size of a boolean function on N boolean variables is K, then even the most general variants [8, Theorem 4]=-=[5]-=-[21] of the Ambainis’ quantum adversary technique [4] can prove only a lower bound of Ω( √ NK). Indeed only the Ω(n) lower bound is known for Triangle, which, because of the remark above, cannot be im... |

1 |
walk algorithm for element distinctness
- Quantum
(Show Context)
Citation Context ...iangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7 ) queries. The second algorithm uses Õ(n13/10 ) queries, and it is based on a design concept of Ambainis =-=[6]-=- that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangl... |

1 | BSS03] [CE03] [CK01 - Barnum, Saks, et al. - 2002 |