## algorithm proposed (2005)

### BibTeX

@MISC{Loo05algorithmproposed,

author = {Ken Loo},

title = {algorithm proposed},

year = {2005}

}

### OpenURL

### Abstract

The goals of this paper are to show the following. First, Grover’s algorithm can be viewed as a digital approximation to the analog quantum

### Citations

848 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...ion. In the semi-classical limit, does Shor’s algorithm have to run instantaneously? 2 The Grover Algorithm In this section, we will outline Grover’s algorithm. For more details on the algorithm, see =-=[4, 7, 11, 19]-=-. The material in this section is taken from [4]. We are given a function f : {0, . . .,N − 1} → {0, 1} such that there exist a w with f(w) = 1 and f(a) = 0 for a ̸= w. The goal is to use a quantum co... |

227 |
Techniques and applications of path integration
- Schulman
- 1981
(Show Context)
Citation Context ...e GFG evolution. Those who are familiar with Feynman path integral techniques should recognize that this is an opportunity for a sum-over-paths interpretation. For more on Feynman path integrals, see =-=[1, 2, 3, 8, 9, 13, 22]-=-. The traditional non-relativistic quantum mechanics interpretation of the propagator K (⃗x,⃗y, t) is that it is the probability amplitude of a particle that starts at position ⃗x at time zero and end... |

178 |
Protocol Analysis
- Ericcson, Simon
- 1984
(Show Context)
Citation Context ...nswering the question: what is a quantum algorithm? We start with the Lie-Trotter produce formula. For our purposes, we only need the finite dimensional version, which is the Lie product formula, see =-=[20]-=-. Theorem 4.1. Let A and B be self-adjoint, finite-dimensional matrices, then [ e −iηA/n e −iηB/n] n . (4.1) Furthermore, for any k, where [A, B] = AB − BA. e −iη(A+B) = lim n→∞ ||e −iη(A+B) − [e −iηA... |

147 |
Space-Time Approach to Non-Relativistic Quantum Mechanics
- Feynman
- 1948
(Show Context)
Citation Context ...aths techniques were applied to quantum algorithms, see [6]. The sum-over-paths interpretation of the work in [6] is somewhat unconventional compared to traditional Feynman path integral methods (see =-=[8, 9, 13, 22]-=-) in the sense that the number of paths in [6] are finite. It is the goal of this section to derive a sum-over-paths interpretation for the GFG algorithm using more traditional methods. In traditional... |

50 |
An Introduction to Nonstandard Real Analysis
- Hurd, Loeb
- 1985
(Show Context)
Citation Context ... give the GFG propagator a sum-over-paths interpretation. Using nonstandard analysis on Feynman path integrals is not a new concept, see [1, 14, 15, 16, 17, 18]. For more on nonstandard analysis, see =-=[1, 5, 12, 13, 23]-=-. The amount of nonstandard analysis that we will use is minimal. We will use the nonstandard reformulation of limits. Given a sequence rn such that lim n→∞ rn = r, (1.2) the nonstandard equivalent of... |

33 | Analog analogue of a digital quantum computation
- Farhi, Gutmann
- 1998
(Show Context)
Citation Context ...nics is an analog, continuous-time theory, and Grover’s algorithm is a digitization of analog quantum mechanics. In particular, it is a digitization of the analog quantum search algorithm proposed by =-=[7]-=-. The digitization is performed in such a way so that the success probability of the analog search algorithm is well preserved. We are not certain that this perspective of Grover’s algorithm is the co... |

28 |
Introduction to the Theory of Infinitesimals
- Stroyan, Luxemburg
- 1976
(Show Context)
Citation Context ... give the GFG propagator a sum-over-paths interpretation. Using nonstandard analysis on Feynman path integrals is not a new concept, see [1, 14, 15, 16, 17, 18]. For more on nonstandard analysis, see =-=[1, 5, 12, 13, 23]-=-. The amount of nonstandard analysis that we will use is minimal. We will use the nonstandard reformulation of limits. Given a sequence rn such that lim n→∞ rn = r, (1.2) the nonstandard equivalent of... |

14 |
Nonstandard Analysis and its Applications
- Cutland
- 1988
(Show Context)
Citation Context ...ecomes meaningless. We proposed a nonstandard analysis formulation of proposition 5.3 which allows us to give proposition 5.3 a sum-over-paths interpretation. For details on nonstandard analysis, see =-=[1, 5, 12, 23]-=- and references within. Theorem 5.1. Let ω ∈ ∗N be an infinite nonstandard natural number, then the propagator for the GFG algorithm is given by { K(j, k, t) = st ∑ l1,l2,...,lω−1 exp [ −itE ∑ ω−1 m=0... |

13 |
A family of integrals serving to connect the Wiener and Feynman integrals
- Cameron
- 1960
(Show Context)
Citation Context ...e GFG evolution. Those who are familiar with Feynman path integral techniques should recognize that this is an opportunity for a sum-over-paths interpretation. For more on Feynman path integrals, see =-=[1, 2, 3, 8, 9, 13, 22]-=-. The traditional non-relativistic quantum mechanics interpretation of the propagator K (⃗x,⃗y, t) is that it is the probability amplitude of a particle that starts at position ⃗x at time zero and end... |

9 | Grover’s algorithm for multiobject search
- Chen, Fulling, et al.
(Show Context)
Citation Context ... the error between the two probabilities is bounded by inverse powers of the size of 1.1 In [7], Hs = E|s〉〈s|. Hence, our evolution differs from the one in [7] by a phase. 1.2 This term was coined by =-=[4]-=-, see equation 2.4. 2the database. Because of this, we take on the view that Grover’s algorithm is a digital approximation of the GFG algorithm. In this sense, we can view Grover’s algorithm as a spe... |

5 |
The Ilstow and Feynman
- Cameron
- 1962
(Show Context)
Citation Context ...e GFG evolution. Those who are familiar with Feynman path integral techniques should recognize that this is an opportunity for a sum-over-paths interpretation. For more on Feynman path integrals, see =-=[1, 2, 3, 8, 9, 13, 22]-=-. The traditional non-relativistic quantum mechanics interpretation of the propagator K (⃗x,⃗y, t) is that it is the probability amplitude of a particle that starts at position ⃗x at time zero and end... |

4 |
A Rigorous Real Time Feynman Path
- Loo
- 1999
(Show Context)
Citation Context ...imit. It is for this reason that we use nonstandard analysis to give the GFG propagator a sum-over-paths interpretation. Using nonstandard analysis on Feynman path integrals is not a new concept, see =-=[1, 14, 15, 16, 17, 18]-=-. For more on nonstandard analysis, see [1, 5, 12, 13, 23]. The amount of nonstandard analysis that we will use is minimal. We will use the nonstandard reformulation of limits. Given a sequence rn suc... |

4 |
Quantum-circuit model of Hamiltonian search algorithms,” Phys
- Roland, Cerf
(Show Context)
Citation Context ...the success probability of finding |w〉 in the GFG algorithm is well preserved (with error bounded by inverse powers of k) in the approximation. The analysis of the former error was previously done in =-=[21]-=-. We can attempt to derive a satisfying O (1) bound on the former error, see [21] (we will briefly outline this in section 4). Doing this leads to the results in [21], but this is not our goal. Our go... |

3 | A Rigorous Real Time Feynman Path Integral and Propagator
- Loo
(Show Context)
Citation Context ...imit. It is for this reason that we use nonstandard analysis to give the GFG propagator a sum-over-paths interpretation. Using nonstandard analysis on Feynman path integrals is not a new concept, see =-=[1, 14, 15, 16, 17, 18]-=-. For more on nonstandard analysis, see [1, 5, 12, 13, 23]. The amount of nonstandard analysis that we will use is minimal. We will use the nonstandard reformulation of limits. Given a sequence rn suc... |

2 |
Path Space measure for Dirac and Schrodinger Equations
- Nakmura
- 1997
(Show Context)
Citation Context ...imit. It is for this reason that we use nonstandard analysis to give the GFG propagator a sum-over-paths interpretation. Using nonstandard analysis on Feynman path integrals is not a new concept, see =-=[1, 14, 15, 16, 17, 18]-=-. For more on nonstandard analysis, see [1, 5, 12, 13, 23]. The amount of nonstandard analysis that we will use is minimal. We will use the nonstandard reformulation of limits. Given a sequence rn suc... |

1 | Rigorous Real-Time Feynman Path Integral for Vector Potentials
- Loo
- 1999
(Show Context)
Citation Context |