## Quantum hidden subgroup algorithms on free groups, (in preparation

Citations: | 7 - 2 self |

### BibTeX

@MISC{Lomonaco_quantumhidden,

author = {Samuel J. Lomonaco and Jr. and Louis H. Kauffman},

title = {Quantum hidden subgroup algorithms on free groups, (in preparation},

year = {}

}

### OpenURL

### Abstract

Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents

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Citation Context ...gorithm for this HSP cannot possibly find a solution. We begin with a question: Does Grover’s algorithm have symmetries that we can exploit? The problem solved by Grover’s algorithm [24], [11], [12], =-=[13]-=- is that of finding an unknown integer label j0 in an unstructured database with items labeled by the integers: given the oracle 0, 1, 2, . . ., j0, . . . , N − 1 = 2 n − 1 , f (j) = { 1 if j = j0 0 o... |

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Citation Context ...complexity of the non-abelian Fourier transform.4. THE NON-ABELIAN FOURIER TRANSFORM The Fourier transform on non-abelian groups is defined as follows: Let G be a finite non-abelian group, and let π =-=(1)-=- , π (2) , . . . , π (k) be a complete set of distinct irreducible representations of the group G. Each irreducible representation is a morphism π (i) : G −→ Aut (Wi) from G to the group of automophis... |

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Citation Context ...SN/Stabj0onto the set S. Find the hidden subgroup Stabj0with bounded probability of error. Now let us compare Shor’s algorithm with Grover’s. From section 6, we know that Shor’s algorithm [21], [25], =-=[35]-=-, [36] solves the hidden subgroup problem ϕ : Z −→ ZN with hidden subgroup structure Z −→ ZN ν ց ր ι Z/PZ Moreover, as stated in section 6, Shor has created his algorithm by pushing 15 the above hidde... |

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Citation Context ...This manuscript assumes that the reader is familiar with the class of quantum hidden subgroup algorithms. For an introductions to this subject, please refer, for example, to any one of the references =-=[22]-=-, [23], [26], [30], [35], [36]. 2. An example of Shor’s quantum factoring algorithm As an example of what we would like to make mathematically transparent, consider the following instance of Peter Sho... |

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Citation Context ...ion Uϕ defined by HA ⊗ HS Uϕ −→ HA ⊗ HS |a〉 |s0〉 ↦−→ |a〉 |ϕ(a)〉 We will use the above implementation to construct a quantum subroutine QRandϕ () which produces a probability distribution Probϕ : Â −→ =-=[0,1]-=- on the character group Â of the group A. Before doing so, we will, as explained in the previous section, make use of various identifications, such as respectively identifying the Fourier and inverse ... |

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Citation Context ... quantum computing. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev =-=[20]-=-. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum f... |

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Citation Context ... We will, on occasion, refer to the probability distribution Probϕ : Â −→ [0,1] on the character group Â as the stochastic source Sϕ (χ) which produces a symbol χ ∈ Â with probability Probϕ (χ). (See =-=[29]-=-.) Thus, QRandϕ (χ) is an algorithmic implementation of the stochastic source Sϕ (χ). Part 4. Vintage Simon Algorithms We now begin the development of the class of vintage Simon QHSAs. These are QHSAs... |

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20 | Introduction to quantum algorithms
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Citation Context ...bj0onto the set S. Find the hidden subgroup Stabj0with bounded probability of error. Now let us compare Shor’s algorithm with Grover’s. From section 6, we know that Shor’s algorithm [21], [25], [35], =-=[36]-=- solves the hidden subgroup problem ϕ : Z −→ ZN with hidden subgroup structure Z −→ ZN ν ց ր ι Z/PZ Moreover, as stated in section 6, Shor has created his algorithm by pushing 15 the above hidden subg... |

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Citation Context ...s, each successively more general than the previous. For the first algorithm, we assume that the unknown hidden period P is an integer. The algorithm is then constructed by using rigged Hilbert spaces=-=[4]-=-, [10], linear combinations of Dirac delta functions, and a subtle extension of the Fourier transform found in the generic QHS subroutine QRand(ϕ), which has been described previously in section 4 of ... |

12 |
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Citation Context ... produce a character sufficiently close to a primitive character of the maximal cyclic group ZP is Ω ( 1 lg lg Q ) . ( Step 5 This step is of algorithmic complexity O n (lg Q) 3) . (See, for example, =-=[27]-=-.) Step 6 For Step 5 to branch to this step, both Steps 2 and 4 must be successful. Thus the probability of branching to step 6 is ProbSuccess (Step 2) · ) ) n+1 ProbSuccess (Step 2) = Ω ( ( 1 lg lg Q... |

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7 |
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Citation Context ...ch successively more general than the previous. For the first algorithm, we assume that the unknown hidden period P is an integer. The algorithm is then constructed by using rigged Hilbert spaces[4], =-=[10]-=-, linear combinations of Dirac delta functions, and a subtle extension of the Fourier transform found in the generic QHS subroutine QRand(ϕ), which has been described previously in section 4 of this p... |

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Citation Context ...n QHS algorithm for SN cannot find the hidden subgroup Stabj0 for each of following two reasons: • Since the subgroups Stabj are not normal subgroups of SN, it follows from the work of Hallgren et al =-=[16]-=-, [17] that the standard non-abelian hidden subgroup algorithm will find the largest normal subgroup of SN lying in Stabj0. But unfortunately, the largest normal subgroup of SN lying in Stabj is the t... |

5 |
Shor’s quantum factoring algorithm
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(Show Context)
Citation Context ... with Probability Prob˜ϕ (y) = 〈 Υ (y) | Υ (y) 〉 (512) 2 the state will “collapse” to |y〉 with the value measured being the integer y, where 0 ≤ y < Q. A plot of Prob˜ϕ (y) is shown in Figure 1. (See =-=[21]-=- and [25] for details.) Figure 1. A plot of Prob˜ϕ(y).QHS ALGORITHMS 5 The peaks in the above plot of Prob˜ϕ (y) occur at the integers y = 0, 85, 171, 256, 341, 427. The probability that at least one... |

4 | Continuous quantum hidden subgroup algorithms
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(Show Context)
Citation Context ...sentation γ. Obviously, much more can be said about pushing. But unfortunately that would take us far afield from the objectives of this paper. For more information on pushing, we refer the reader to =-=[27]-=-. It would be amiss not to mention that the above algorithmic primitive of pushing suggests the definition of a second primitive which we will call lifting. Definition 3. Let ϕ : G −→ S be a map from ... |

4 |
Introduction to quantum algorithms, in “Quantum Computation: A Grand Mathematical Challenge for the Twenty-First
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Citation Context ...he reader is familiar with the class of quantum hidden subgroup algorithms. For an introductions to this subject, please refer, for example, to any one of the references [22], [23], [26], [30], [35], =-=[36]-=-. 2. An example of Shor’s quantum factoring algorithm As an example of what we would like to make mathematically transparent, consider the following instance of Peter Shor’s quantum factoring algorith... |

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3 |
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Citation Context ...e µ (k) = 0 for all integers k that are not squarefree. The last part of this theorem follows immediately from the fact that Q∑ ( ) n ∞∑ ⌊Q/k⌋ lim µ (k) = µ (k) Q−→∞ Q 1 kn = ζ (n)−1 . k=1 (See [33], =-=[34]-=-, or [19].) k=1 Q n . n ) n + ... , Corollary 7. Let n be an integer greater than 1, and let λ ′ 1 ,λ′ 2 ,... ,λ′ n be n integers randomly and independently selected with replacement from the set {1,2... |