## Quantum complexity of integration (2001)

Venue: | J. COMPLEXITY |

Citations: | 35 - 3 self |

### BibTeX

@ARTICLE{Novak01quantumcomplexity,

author = {Erich Novak},

title = { Quantum complexity of integration},

journal = {J. COMPLEXITY},

year = {2001},

pages = {2--16}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on [0, 1] d and define γ by γ = (k + α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are and comp(F k,α

### Citations

844 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
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Citation Context ...omes from the trivial classical algorithm. This term is n (log n + log ε −1 ) in the classical bit number model. The output of a quantum algorithm is a random variable A(x, ε), we always request that =-=(6)-=- |A(x, ε) − Sn(x)| ≤ ε with probability at least 3/4. Of course we can run the algorithm several times and, taking the median from several measurements, we increase the probability of success.QUANTUM... |

223 | Information-Based Complexity
- Traub, Wo´zniakowski, et al.
- 1988
(Show Context)
Citation Context ... is a good measure for the smoothness and appears in all the estimates. First of all, the optimal orders for deterministic and (general) randomized methods are known, see, e.g., Novak (1988). We have =-=(17)-=- and (18) comp(F k,α d , ε) ≍ ε −1/γ comp random (F k,α d , ε) ≍ ε −2/(1+2γ) . Therefore we have to study only the quantities comp quant , comp quant query, and comp coin . For the upper bounds we use... |

114 |
Tight bounds on quantum searching. Fortschritte der Physik
- Boyer, Brassard, et al.
- 1998
(Show Context)
Citation Context ...e consider the worst case setting, with the worst case cost and the worst case error. With randomized methods we can do much better, at least if n is large compared to ε −2 . The cost is of the order =-=(2)-=- comp random (n, ε) ≍ min(n, ε −2 ). Now the error of a method is a random variable and the requirement is that its expectation is bounded by ε. The statements (1) and (2) follow easily from well know... |

101 | Quantum counting
- Brassard, Høyer, et al.
(Show Context)
Citation Context ...per and lower bounds. See, for example, Novak (1988). If we allow only random bits (restricted Monte Carlo methods, coin tossing) instead of arbitrary randomized methods then one gets the upper bound =-=(3)-=- comp coin (n, ε) ≤ C · min(n, ε −2 log n) which follows easily from (2). For the results (1–3), and for all classical algorithms, we use the real number model of computation, with unit cost for each ... |

84 | A framework for fast quantum mechanical algorithms. To appear
- Grover
- 1998
(Show Context)
Citation Context ... ℓ 2ℓ i=1 and for d > 1 the respective tensor product Q d n that uses n = ℓd function values. By well known estimates for d = 1 together with the technique of Haber (1970, p. 489) we get the estimate =-=(7)-=- e(Q d n, F α d ) ≤ C · d · n −α/d for the worst case error of the product rule. To obtain e(Qd n , F α d n(F α ( ) d/α C d (8) d , ε) ≈ ε ) ≈ ε, we have to take function values. We can now use the re... |

65 | The quantum query complexity of approximating the median and related statistics
- NAYAK, WU
- 1999
(Show Context)
Citation Context ...p random (F α d , ε) ≤ C · ε −2 and if we only allow random bits then we obtain, using (3) and (8), α ε−2 (log d + log ε −1 (11) ). In the same way we obtain for the quantum computer the upper bounds =-=(12)-=- and (13) comp coin (F α d , ε) ≤ C d comp quant query (F α d , ε) ≤ C ε−1 comp quant (F α d , ε) ≤ C d α ε−1 (log d + log ε −1 ).4 ERICH NOVAK Observe that all these bounds (9)–(13) are just upper b... |

65 |
Complexity and Information
- Traub, Werschulz
- 1998
(Show Context)
Citation Context ...d measure for the smoothness and appears in all the estimates. First of all, the optimal orders for deterministic and (general) randomized methods are known, see, e.g., Novak (1988). We have (17) and =-=(18)-=- comp(F k,α d , ε) ≍ ε −1/γ comp random (F k,α d , ε) ≍ ε −2/(1+2γ) . Therefore we have to study only the quantities comp quant , comp quant query, and comp coin . For the upper bounds we use a techni... |

43 |
Deterministic and Stochastic Error Bounds in Numerical Analysis. (Lecture Notes in Mathematics 1349
- Novak
- 1988
(Show Context)
Citation Context ...−1 (log d + log ε −1 ).4 ERICH NOVAK Observe that all these bounds (9)–(13) are just upper bounds which we get by a particular proof technique. Actually it is known that the order in (9) is optimal, =-=(14)-=- comp(F α d , ε) ≈ Cd,α ε −d/α , while the upper bounds for Monte Carlo methods are not optimal, we have (15) and (16) comp random (F α d , ε) ≈ Cd,α ε −2d/(2α+d) comp coin (F α d , ε) ≤ Cd,α ε −2d/(2... |

29 | Random approximation in numerical analysis, in
- Heinrich
- 1993
(Show Context)
Citation Context ...sults from above to obtain upper bounds for the complexity of numerical integration. Using the (trivial) result (1) we get a bound for the (worst case) complexity of integration, comp(F α ( ) d/α C d =-=(9)-=- d , ε) ≤ . ε With (2) we obtain (10) comp random (F α d , ε) ≤ C · ε −2 and if we only allow random bits then we obtain, using (3) and (8), α ε−2 (log d + log ε −1 (11) ). In the same way we obtain f... |

15 |
Numerical evaluation of multiple integrals
- Haber
- 1970
(Show Context)
Citation Context ... together with the technique of Haber (1970, p. 489) we get the estimate (7) e(Q d n, F α d ) ≤ C · d · n −α/d for the worst case error of the product rule. To obtain e(Qd n , F α d n(F α ( ) d/α C d =-=(8)-=- d , ε) ≈ ε ) ≈ ε, we have to take function values. We can now use the results from above to obtain upper bounds for the complexity of numerical integration. Using the (trivial) result (1) we get a bo... |

15 |
The real number model in numerical analysis
- Novak
- 1995
(Show Context)
Citation Context ...et by a particular proof technique. Actually it is known that the order in (9) is optimal, (14) comp(F α d , ε) ≈ Cd,α ε −d/α , while the upper bounds for Monte Carlo methods are not optimal, we have =-=(15)-=- and (16) comp random (F α d , ε) ≈ Cd,α ε −2d/(2α+d) comp coin (F α d , ε) ≤ Cd,α ε −2d/(2α+d) log ε −1 . For the proof of (15) see Heinrich (1993), Novak (1988), or Traub, Wasilkowski, Wo´zniakowski... |

11 |
Fast quantum algorithms for numerical integrals and stochastic processes
- Abrams, Williams
- 1999
(Show Context)
Citation Context ...iscrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F k,α d on =-=[0, 1]-=- d and define γ by γ = (k + α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are and comp(F k,α d , ε) ≍ ε −1/γ comp random (F k,α d , ε) ≍ ε −2/(1+2... |

10 |
Quantum computing, Documenta Mathematica Extra Vol
- Shor
- 1998
(Show Context)
Citation Context ...articular proof technique. Actually it is known that the order in (9) is optimal, (14) comp(F α d , ε) ≈ Cd,α ε −d/α , while the upper bounds for Monte Carlo methods are not optimal, we have (15) and =-=(16)-=- comp random (F α d , ε) ≈ Cd,α ε −2d/(2α+d) comp coin (F α d , ε) ≤ Cd,α ε −2d/(2α+d) log ε −1 . For the proof of (15) see Heinrich (1993), Novak (1988), or Traub, Wasilkowski, Wo´zniakowski (1988). ... |

4 | The Monte Carlo algorithm with a pseudorandom generator
- Traub, Wo´zniakowski
- 1992
(Show Context)
Citation Context ...Bakhvalov, together with the lower bound of Nayak and Wu, see (4). We obtain the following optimal rates of convergence.QUANTUM COMPLEXITY OF INTEGRATION 5 Theorem 1. Define γ = (k + α)/d, as above. =-=(19)-=- comp quant k,α query (Fd , ε) ≍ ε −1/(1+γ) , (20) (21) comp quant (F k,α d , ε) ≤ Cd,k,α ε −1/(1+γ) (log ε −1 ) 1/(1+γ) , comp coin (F k,α d , ε) ≤ Cd,k,α ε −2/(1+2γ) (log ε −1 ) 1/(1+2γ) . To summar... |

3 |
Tapp (2000): Quantum amplitude amplification and estimation. LANL preprint quant-ph/0005055
- Brassard, Høyer, et al.
(Show Context)
Citation Context ... (33) and get (6). With (5) and (31) one obtains the bound (30) for the complexity of the problem. Acknowledgments. I thank Peter Høyer very much for his helpful comments. Peter gave me the reference =-=[4]-=- and we had a very interesting discussion about different models of [quantum] computation and about different upper and lower bounds for the computation of the mean of n numbers. Peter also was so kin... |

2 |
Eingeschränkte Monte Carlo-Verfahren zur numerischen Integration
- Novak
- 1983
(Show Context)
Citation Context ...(F α d , ε) ≤ C · ε −2 and if we only allow random bits then we obtain, using (3) and (8), α ε−2 (log d + log ε −1 (11) ). In the same way we obtain for the quantum computer the upper bounds (12) and =-=(13)-=- comp coin (F α d , ε) ≤ C d comp quant query (F α d , ε) ≤ C ε−1 comp quant (F α d , ε) ≤ C d α ε−1 (log d + log ε −1 ).4 ERICH NOVAK Observe that all these bounds (9)–(13) are just upper bounds whi... |

1 |
On quantum algorithms. LANL preprint quant-ph/9903061
- Cleve, Ekert, et al.
- 1999
(Show Context)
Citation Context ...−1 and says that it is optimal “up to polylogarithmic factors”. If we consider, for the quantum computer, the bit number model then we need a slightly larger cost. Høyer (2000) proves the upper bound =-=(5)-=- comp quant (n, ε) ≤ C · min(n , ε −1 (log n + log log ε −1 ), see Section 2 for details on the model of computation. The term n on the right side of (5) comes from the trivial classical algorithm. Th... |

1 |
2000): Quantum complexity of the mean problem
- Høyer
(Show Context)
Citation Context ...ut different models of [quantum] computation and about different upper and lower bounds for the computation of the mean of n numbers. Peter also was so kind to give me his results that will appear in =-=[10]-=-. I also thank several other referees and editors for valuable remarks. Peter Hertling found some minor mistakes that are corrected in this version. This work was done during my time as a fellow at th... |

1 |
Quantum Computer Algorithms. Thesis
- Mosca
- 1999
(Show Context)
Citation Context ...tegration, comp(F α ( ) d/α C d (9) d , ε) ≤ . ε With (2) we obtain (10) comp random (F α d , ε) ≤ C · ε −2 and if we only allow random bits then we obtain, using (3) and (8), α ε−2 (log d + log ε −1 =-=(11)-=- ). In the same way we obtain for the quantum computer the upper bounds (12) and (13) comp coin (F α d , ε) ≤ C d comp quant query (F α d , ε) ≤ C ε−1 comp quant (F α d , ε) ≤ C d α ε−1 (log d + log ε... |

1 |
On quantum algorithms. Preprint. See also LANL preprint quant-ph/9903061
- Cleve, Ekert, et al.
- 1999
(Show Context)
Citation Context ...the quantum computer, the bit number model then we need a slightly larger cost. Here we need ε −1 log ε −1 oracle calls. An additional factor log n comes in by the definition of the cost and one gets =-=(5)-=- comp quant (n, ε) ≤ C · min(n log ε −1 , ε −1 log n log ε −1 ), see Section 2 for details on the model of computation. The output of a quantum algorithm is a random variable A(x, ε), we always reques... |