## An exact quantum polynomial-time algorithm for Simon's problem (1997)

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Venue: | IN PROCEEDINGS OF THE 5TH ISRAELI SYMPOSIUM ON THEORY OF COMPUTING AND SYSTEMS (ISTCS'97 |

Citations: | 84 - 12 self |

### BibTeX

@INPROCEEDINGS{Brassard97anexact,

author = {Gilles Brassard and Peter Høyer},

title = {An exact quantum polynomial-time algorithm for Simon's problem},

booktitle = {IN PROCEEDINGS OF THE 5TH ISRAELI SYMPOSIUM ON THEORY OF COMPUTING AND SYSTEMS (ISTCS'97},

year = {1997},

pages = {12--23},

publisher = {Society Press}

}

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### Abstract

We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upper-bounded by a polynomial in the worst case. We show that a natural generalization of Simon’s problem can be solved in this way, whereas previous algorithms required quantum polynomial time in the expected sense only, without upper bounds on the worst-case running time. This is achieved by generalizing both Simon’s and Grover’s algorithms and combining them in a novel way. It follows that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box.

### Citations

841 | fast quantum mechanical algorithm for database search
- Grover, “A
- 1996
(Show Context)
Citation Context ... Of independent interest are the techniques developed to obtain our results. Two of the most fundamental techniques discovered so far in the field of quantum computation are Simon’s [18] and Grover’s =-=[15]-=-. Here, we generalize both techniques and we show for the first time that they can be made to work together toward a common goal: our algorithm crucially requires the availability of both these tools.... |

808 | Algorithms for quantum computation: Discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ...ity Campusvej 55 DK–5230 Odense M Denmark Email: u2pi@imada.ou.dk it may run on any given instance if it keeps being unlucky. (The same can be said about Shor’s celebrated quantum factoring algorithm =-=[17]-=-.) In this paper, we address the issue of exact quantum polynomial time, which concerns problems that quantum computers can solve in guaranteed worst-case polynomial time with zero error probability. ... |

477 | Quantum Complexity Theory
- Bernstein, Vazirani
- 1992
(Show Context)
Citation Context ...ated. Later, Bernstein and Vazirani provided a relativized problem that can be solved in exact quantum polynomial time, but not in time n o(log n) on any classical bounded-error probabilistic machine =-=[4]-=-. More recently, we constructed such a problem that would require exponential time on any classical bounded-error probabilistic machine [11]. None of these problems were decision problems. 1 Here we r... |

454 |
Logical reversibility of computation
- Bennett
- 1973
(Show Context)
Citation Context ...ubgroup is mapped to its orthogonal subgroup, and the phases translate to a coset and vice versa. A classical function f is evaluated reversibly by the operation Uf which maps |x〉|y〉 to |x〉|y ⊕ f(x)〉 =-=[3]-=-. Note that a second application of Uf will restore the second register to its original value since |x〉|y ⊕ f(x) ⊕ f(x)〉 = |x〉|y〉. Let T0 be a transversal of H0 in G, that is, a subset T0 ⊆ G that con... |

355 | On the power of quantum computation
- Simon
- 1997
(Show Context)
Citation Context ...can be computed in polynomial time on a probabilistic Turing machine with bounded error probability. This belief has been seriously challenged by the theory of quantum computing. In particular, Simon =-=[18]-=- provided the first example of a problem that can be solved in polynomial time on a quantum computer, yet any classical bounded-error probabilistic algorithm would require exponential time if the data... |

339 |
Rapid solutions of problems by quantum computation
- Deutsch, Jozsa
- 1992
(Show Context)
Citation Context ...ses”. The study of exact quantum polynomial time is not new. The very first algorithm ever designed to demonstrate an advantage of quantum computers over classical computers, due to Deutsch and Jozsa =-=[14]-=-, was of this exact nature. However, it solved a problem that could be handled just as efficiently with a classical probabilistic computer, provided an arbitrarily small (one-sided) error probability ... |

200 | Elementary gates for quantum computation
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ... the lemma follows. The phase-change operator S {0} can be applied in time linear in the number of qubits, while SA can be applied by computing Uχ twice and doing a constant amount of additional work =-=[2]-=-. Hence, Q runs in time linear in the number of qubits and in the times required to compute A and Uχ. □ 4.3 Composing our new QP–algorithm By Lemma 8, we can take a quantum algorithm A and construct a... |

146 | Quantum measurements and the Abelian stabilizer problem
- Kitaev
- 1995
(Show Context)
Citation Context ...subroutine can be used to sample random elements of H⊥ 0 . The time needed to apply operator UG is equal to twice the time to compute FG plus the time to compute the function ρ. By a result of Kitaev =-=[16]-=-, for all finite additive Abelian groups G, the Fourier transform FG can be applied in polynomial time in log |G|. However, his method applies the transform not with perfection, but only with arbitrar... |

114 | Oracle quantum computing
- Berthiaume, Brassard
- 1994
(Show Context)
Citation Context ...ve it. This provides the first evidence of an exponential gap between the power of exact quantum computation and that of classical 1 The Deutsch–Jozsa problem gives rise to an oracle decision problem =-=[7, 8]-=-. Also, in the soon-to-be-published journal version of their paper, Bernstein and Vazirani extend their result to a decision problem [5]. 1bounded-error probabilistic computation, even for decision p... |

114 | Tight bounds on quantum searching
- Boyer, Brassard, et al.
- 1996
(Show Context)
Citation Context ... [15], we encapsulate A in a larger quantum algorithm Q such that the probability that Q returns a good solution is significantly better compared to the probability that A returns a good solution. In =-=[9]-=-, it is shown that if A = Wn 2 is the Walsh-Hadamard transform and the probability of success of A is exactly one quarter (a = 1/4) then Q can be constructed such that it succeeds with certainty. For ... |

89 |
An approximate Fourier transform useful in quantum factoring
- Coppersmith
- 1994
(Show Context)
Citation Context ... generalization of our QP–algorithm would require the solutions to two problems. The first problem is that we must be capable of computing the Fourier transform FG exactly. Cleve [12] and Coppersmith =-=[13]-=-, building on the work of Shor [17], showed that it can be applied exactly in polynomial time whenever G is of smooth order. Here the order of a group G is smooth if all its prime factors are at most ... |

53 | The Quantum Challenge to Structural Complexity Theory
- Berthiaume, Brassard
(Show Context)
Citation Context ...ve it. This provides the first evidence of an exponential gap between the power of exact quantum computation and that of classical 1 The Deutsch–Jozsa problem gives rise to an oracle decision problem =-=[7, 8]-=-. Also, in the soon-to-be-published journal version of their paper, Bernstein and Vazirani extend their result to a decision problem [5]. 1bounded-error probabilistic computation, even for decision p... |

18 | Quantum physics and computers
- Barenco
- 1996
(Show Context)
Citation Context ...so runs in polynomial time, the claimed running time follows. □ 3 Simon’s quantum algorithm We assume in this extended abstract that the reader is familiar with the basic notions of quantum computing =-=[1, 6, 10]-=-. We denote a register holding m qubits, all in the zero state, by |0m〉. When its dimension is of no importance, we sometimes just write |0〉. For any nonempty subset X ⊆ G, let |X〉 denote the equally-... |

15 |
Quantum Computation”, Complexity Theory Retrospective
- Berthiaume
- 1997
(Show Context)
Citation Context ...so runs in polynomial time, the claimed running time follows. □ 3 Simon’s quantum algorithm We assume in this extended abstract that the reader is familiar with the basic notions of quantum computing =-=[1, 6, 10]-=-. We denote a register holding m qubits, all in the zero state, by |0m〉. When its dimension is of no importance, we sometimes just write |0〉. For any nonempty subset X ⊆ G, let |X〉 denote the equally-... |

13 |
A note on computing Fourier transforms by quantum programs
- Cleve
(Show Context)
Citation Context ...oup problem. A direct generalization of our QP–algorithm would require the solutions to two problems. The first problem is that we must be capable of computing the Fourier transform FG exactly. Cleve =-=[12]-=- and Coppersmith [13], building on the work of Shor [17], showed that it can be applied exactly in polynomial time whenever G is of smooth order. Here the order of a group G is smooth if all its prime... |

12 |
A quantum jump in computer science
- Brassard
- 1995
(Show Context)
Citation Context ...so runs in polynomial time, the claimed running time follows. □ 3 Simon’s quantum algorithm We assume in this extended abstract that the reader is familiar with the basic notions of quantum computing =-=[1, 6, 10]-=-. We denote a register holding m qubits, all in the zero state, by |0m〉. When its dimension is of no importance, we sometimes just write |0〉. For any nonempty subset X ⊆ G, let |X〉 denote the equally-... |

1 |
Quantum complexity theory”, final version of [4
- Bernstein, Vazirani
(Show Context)
Citation Context ...utsch–Jozsa problem gives rise to an oracle decision problem [7, 8]. Also, in the soon-to-be-published journal version of their paper, Bernstein and Vazirani extend their result to a decision problem =-=[5]-=-. 1bounded-error probabilistic computation, even for decision problems. Of independent interest are the techniques developed to obtain our results. Two of the most fundamental techniques discovered s... |

1 |
P.: On the power of exact quantum polynomial time. Unpublished
- Brassard, Hyer
- 1996
(Show Context)
Citation Context ...(log n) on any classical bounded-error probabilistic machine [4]. More recently, we constructed such a problem that would require exponential time on any classical bounded-error probabilistic machine =-=[11]-=-. None of these problems were decision problems. 1 Here we recast Simon’s problem in a natural grouptheoretic framework, we generalize it, and we give an exact quantum polynomial-time algorithm to sol... |