## Shor’s algorithm on a nearest-neighbor machine (2007)

Venue: | Asian conference on Quantum Information Science |

Citations: | 9 - 1 self |

### BibTeX

@INPROCEEDINGS{Kutin07shor’salgorithm,

author = {Samuel A. Kutin},

title = {Shor’s algorithm on a nearest-neighbor machine},

booktitle = {Asian conference on Quantum Information Science},

year = {2007},

pages = {12--13}

}

### OpenURL

### Abstract

We give a new “nested adds ” circuit for implementing Shor’s algorithm in linear width and quadratic depth on a nearest-neighbor machine. Our circuit combines Draper’s transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another version, with slightly larger depth, using only reversible classical gates. We do not know which version will ultimately be cheaper to implement. 1

### Citations

200 | Elementary gates for quantum computation - Barenco, Bennett, et al. - 1995 |

51 | Fast parallel circuits for the quantum Fourier transform
- Cleve, Watrous
- 2004
(Show Context)
Citation Context ...ponentiation circuit with depth 12n 2 + 60n log 2 2 n + O(n log n), width and size 3n + 6 log 2 n + O(1), 4n 3 + O(n 2 log n). We could further reduce the depth by using a parallel version of the QFT =-=[CW00]-=-, but each multiply would still have depth at least 5n+O(log 2 n). We could also consolidate the registers QY and QZ; we would get a slight increase in depth and a slight decrease in width. Acknowledg... |

29 |
Design of an efficient public-key cryptographic library for RISC-based smart cards
- Dhem
- 1998
(Show Context)
Citation Context ..., equivalently, one controlled integer multiplication. There are other schemes that give modular multiplication circuits at a cost of three times the cost of integer multiplication (see, for example, =-=[Dhe98]-=-). So it might seem that Zalka’s idea would save only a constant factor. However, Zalka’s idea is conceptually simpler; without it, we might not have found the linear-depth multiplier of Section 3. 2 ... |

26 | Fast version of Shor’s quantum factoring algorithm
- Zalka
- 1998
(Show Context)
Citation Context ...s do not affect the dominant terms in the expression for size or depth. Our contribution is a new approximate controlled modular multiplier with linear width and linear depth. We use an idea of Zalka =-=[Zal02]-=- for building approximate multipliers. While we still multiply by performing O(n) additions, we only perform a constant number of large QFTs for each multiply. When we insert our multiplier into the f... |

20 | Addition on a Quantum Computer
- Draper
- 2000
(Show Context)
Citation Context ...y of the techniques used to reduce circuit depth do not appear to apply to a nearest-neighbor architecture. Beauregard [Bea03] has given a simple exponentiation circuit using Draper’s transform adder =-=[Dra00]-=-. The adder requires two QFTs together with some controlled rotations. Beauregard’s circuit uses only 2n + O(1) qubits, but has cubic depth—the dominant cost is Θ(n 2 ) applications of the transform a... |

20 | Implementation of Shor’s algorithm on a linear nearest neighbour qubit array - Fowler, Devitt, et al. - 2004 |

19 |
A new quantum ripple-carry addition circuit
- Cuccaro, Draper, et al.
- 1998
(Show Context)
Citation Context ... occurred; if so, hj must have been 1. We then exchange each yr−j bit with Zj to move the control bits into position for the next round. Each of these steps can be performed with a ripple-carry adder =-=[CDKM04]-=-; the depth is Ct for a small constant C. We need 2k extra bits: the high bits hj and one scratch bit for each ripple. 4 To do modular multiplication, we use the same scheme as in our main constructio... |

12 | Circuit for Shor’s algorithm using 2n+3 qubits,” arXiv:quant-ph/0205095
- Beauregard
(Show Context)
Citation Context ...or restriction). See, for example, [VI05, VIL05, Van06] for recent summaries. Many of the techniques used to reduce circuit depth do not appear to apply to a nearest-neighbor architecture. Beauregard =-=[Bea03]-=- has given a simple exponentiation circuit using Draper’s transform adder [Dra00]. The adder requires two QFTs together with some controlled rotations. Beauregard’s circuit uses only 2n + O(1) qubits,... |

1 | Shor’s algorithm with fewer (pure) qubits
- Zalka
- 2006
(Show Context)
Citation Context ... This is the same asymptotic depth achieved by Van Meter [Van06], but we require only linear width. ∗ Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540. kutin@idaccr.org 1 Zalka =-=[Zal06]-=- has recently pointed out this same idea of performing mulitple additions framed by a single QFT, but he does not work out any details or discuss the application to nearest-neighbor circuits. 12 Prel... |