## Circuit for Shor’s algorithm using 2n+3 qubits (2002)

Venue: | 54 |

Citations: | 11 - 0 self |

### BibTeX

@INPROCEEDINGS{Beauregard02circuitfor,

author = {Stéphane Beauregard},

title = {Circuit for Shor’s algorithm using 2n+3 qubits},

booktitle = {54},

year = {2002},

pages = {175}

}

### OpenURL

### Abstract

We try to minimize the number of qubits needed to factor an integer of n bits using Shor’s algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n 3 lg(n)) elementary quantum gates in a depth of O(n 3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. 1

### Citations

1506 |
Quantum Computation and Quantum Information
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- 2000
(Show Context)
Citation Context ... to study exactly how few qubits do we need to factor a number of n bits. Quantum factorization consists of classical preprocessing, a quantum algorithm for order-finding and classical postprocessing =-=[1, 2]-=- (fig. 1). We will concentrate on the quantum part of factorization and consider classical parts as being free as long as they are computable in polynomial time. The only use of quantum computation in... |

941 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
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Citation Context ...er and is completely general as it does not rely on any property of the number to be factored. 1 Introduction Since Shor discovered a polynomial time algorithm for factorization on a quantum computer =-=[1]-=-, a lot of effort has been directed towards building a working quantum computer. Despite all these efforts, it is still extremely difficult to control even a few qubits. It is thus of great interest t... |

148 | Quantum algorithms revisited
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(Show Context)
Citation Context ...est to study exactly how few qubits are needed to factor an n-bit number. Quantum factorization consists of classical preprocessing, a quantum algorithm for order-finding and classical postprocessing =-=[1, 2, 3]-=- (fig. 1). We will concentrate on the quantum part of factorization and consider classical parts as being free as long as they are computable in polynomial time. The only use of quantum computation in... |

143 |
Elementary gates for quantum computation // Phys
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(Show Context)
Citation Context ... qubit gates, up to doubly controlled conditionnal phase shifts and up to doubly controlled not gates. These gates can be implemented using a constant number of single qubit gates and controlled-nots =-=[12]-=-, so they can all be considered as elementary quantum gates. The φADD(a) circuit (fig. 3), where a is a classical value, requires n + 1 qubits and O(n) single qubit gates in constant depth. The number... |

95 | Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance
- Vandersypen, Ste®en, et al.
- 2002
(Show Context)
Citation Context ..., which amounts to many unused qubits. The number 15 was factored using NMR with seven qubits in an impressive display of quantum control by Vandersypen, Steffen, Breyta, Yannoni, Sherwood and Chuang =-=[14]-=-. The importance of reducing the number of qubits versus reducing the depth of a quantum computation is not clear as quantum computers of useful size are not yet available. We have to keep in mind tha... |

88 |
Quantum networks for elementary arithmetic operations
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(Show Context)
Citation Context ...) 0 Figure 2: The quantum addition as described by Draper [6]. 2 The Circuit The circuit for factorization that will be discussed here was inspired in part by a circuit from Vedral, Barenco and Ekert =-=[5]-=-. To reduce the number of qubits, we use a variant of a quantum addition algorithm described by Draper [6] (fig. 2). Other techniques used to reduce the number of qubits are the hardwiring of classica... |

59 | The hidden subgroup problem and eigenvalue estimation on a quantum computer
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(Show Context)
Citation Context ... advantage of using the C-Ua2j gates for Shor’s algorithm is the fact that we don’t really need the total 2n controlling qubits. In fact, it can be shown that only one controlling qubit is sufficient =-=[7, 8, 9]-=-. This is possible because the controlled-U gates all commute and the inverse QFT can be applied semi-classically. Indeed, we can get all the bits of the answer 80 m m 0 m H H 1 0 X H R0 m 1 X H R 2n... |

51 | Fast Parallel Circuits for the Quantum Fourier Transform - Cleve, Watrous - 2000 |

28 | Parallel quantum computation and quantum codes”, manuscript
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Citation Context ...imate QFT on n qubits requires O(n lg(n)) gates. There seems to be no obvious way to reduce the depth of either the exact QFT and the approximate QFT on n qubits below O(n) without using extra qubits =-=[11]-=-. The depth of the QFT on n + 1 qubits is thus O(n) with the little parallelization available without extra qubits. 2.6 The controlled-SWAP S W A P = Figure 10: The controlled-SWAP gate. The controlle... |

25 | Fast version of Shor’s quantum factoring algorithm
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(Show Context)
Citation Context ... advantage of using the C-Ua2j gates for Shor’s algorithm is the fact that we don’t really need the total 2n controlling qubits. In fact, it can be shown that only one controlling qubit is sufficient =-=[7, 8, 9]-=-. This is possible because the controlled-U gates all commute and the inverse QFT can be applied semi-classically. Indeed, we can get all the bits of the answer 80 m m 0 m H H 1 0 X H R0 m 1 X H R 2n... |

25 |
An Approximate Fourier Transform Useful
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(Show Context)
Citation Context ... will in practice ignore the ones with k greater than a certain threshold kmax. This approximate QFT is in fact very close to the exact QFT even with kmax logarithmic in n. In fact, it has been shown =-=[10]-=- that the error introduced by ignoring all gates with k > kmax is proportional to n2 −kmax .We can thus choose kmax ∈ O(lg( n ǫ )). 9The implementation of the approximate QFT on n qubits requires O(n... |

20 | Addition on a Quantum Computer
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(Show Context)
Citation Context ...2 k an−1 a n−2 a a n−1 n−2 a1 a0 φ φ n−1 n−2 (b) (b) 1 2 n−1 n 1 n−2 n−1 a 1 a0 φ φ n−1 n−2 (a+b) (a+b) φ (b) 1 φ (b) 0 1 2 1 φ (a+b) 1 φ (a+b) 0 Figure 2: The quantum addition as described by Draper =-=[6]-=-. 2 The Circuit The circuit for factorization that will be discussed here was inspired in part by a circuit from Vedral, Barenco and Ekert [5]. To reduce the number of qubits, we use a variant of a qu... |

16 |
J.Preskill, Efficient networks for quantum factoring, Phys
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(Show Context)
Citation Context ... 5n+2 qubits with basic optimization and further reduced to 4n + 3 if unbounded Toffoli gates (n-controlled nots) are available. Beckman, Chan, Devabhaktoni and Preskill provided an extended analysis =-=[13]-=- of modular exponentiation, with a circuit of 5n + 1 qubits using elementary gates and 4n + 1 if unbounded Toffoli gates are available. Zalka also described a method for factorization with 3n + O(lg n... |

10 |
Efficient factorization with a single pure qubit
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(Show Context)
Citation Context ... advantage of using the C-Ua2j gates for Shor’s algorithm is the fact that we don’t really need the total 2n controlling qubits. In fact, it can be shown that only one controlling qubit is sufficient =-=[5, 6, 7]-=-. This is possible because the controlled-U gates all commute and the inverse QFT can be applied semi-classically. Indeed, we can get all the bits of the answer sequentially as in figure 8. Each measu... |

4 |
Chuang (2001): Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 14
- Vandersypen, Steffen, et al.
(Show Context)
Citation Context ..., which amounts to many unused qubits. The number 15 was factored using NMR with seven qubits in an impressive display of quantum control by Vandersypen, Steffen, Breyta, Yannoni, Sherwood and Chuang =-=[14]-=-. The importance of reducing the number of qubits versus reducing the depth of a quantum computation is not clear as quantum computers of useful size are not yet available. We have to keep in mind tha... |

1 |
Physical Review A 54
- Vedral, Barenco, et al.
- 1996
(Show Context)
Citation Context ...The quantum addition as described by Draper [4]. 2 The circuit The circuit for factorization that will be discussed here is derived from a quite straightforward circuit from Vedral, Barenco and Ekert =-=[3]-=-. To reduce the number of qubits, we use a variant of a quantum addition algorithm described in Draper [4] (fig. 2). This quantum addition takes as input n qubits representing a number a, and n more q... |

1 |
IBM Research Report No. RC19642
- Coppersmith
- 1996
(Show Context)
Citation Context ...e will in practice ignore the ones with k higher than a certain threshold kmax. This approximate QFT is in fact very close to the exact QFT even with kmax logarithmic in n. In fact, it has been shown =-=[8]-=- that by keeping all gates Rk with k ≤ kmax, the approximate QFT differs from the QFT by a multiplicative factor e iǫ where |ǫ| ≤ 2πn2 −kmax . Thus, to get the precision up to a multiplicative factor ... |