Analogue algorithm for parallel factorization of an exponential number of large integers: II—optical implementation

We report a detailed analysis of the optical realization of the analogue algorithm described in the first paper of this series (Tamma in Quantum Inf Process 11128:1190, 2015) for the simultaneous factorization of an exponential number of integers. Such an analogue procedure, which scales exponentially in the context of first-order interference, opens up the horizon to polynomial scaling by exploiting multi-particle quantum interference.

[1]  W. Schleich,et al.  Factorization of numbers with physical systems , 2006 .

[2]  W. Schleich,et al.  Gauss sum factorization with cold atoms. , 2007, Physical review letters.

[3]  Vincenzo Tamma Theoretical and experimental study of a new algorithm for factoring numbers , 2010 .

[4]  W. Schleich,et al.  Factoring numbers with a single interferogram , 2011, 1506.02907.

[5]  W. Schleich,et al.  Factorization of numbers with truncated Gauss sums at rational arguments , 2009, 1210.6471.

[6]  Vincenzo Tamma,et al.  Multi-boson correlation sampling , 2015, Quantum Inf. Process..

[7]  Vincenzo Tamma,et al.  From the Physics to the Computational Complexity of Multiboson Correlation Interference. , 2015, Physical review letters.

[9]  AMSShort Courses,et al.  Quantum Computation : The Grand Mathematical Challenge for the Twenty-First Century and the Millennium , 1999 .

[10]  J. Dowling,et al.  Factoring integers with Young's N slit interferometer: Classical-analog versus quantum-digital computers , 1996, Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference.

[11]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[12]  V. Tamma,et al.  Boson sampling with non-identical single photons , 2015, 1512.05579.

[13]  W. Schleich,et al.  Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers , 2012 .

[14]  Vincenzo Tamma,et al.  Toward real maximally path-entangled N -photon-state sources , 2008 .

[15]  New factorization algorithm based on a continuous representation of truncated Gauss sums , 2008, 0811.1595.

[16]  W. Schleich,et al.  Factorization with Gauss sums: scaling properties of ghost factors , 2007 .

[17]  W. Schleich,et al.  Factorization of numbers with the temporal Talbot effect: optical implementation by a sequence of shaped ultrashort pulses. , 2007, Physical review letters.

[18]  I. Chuang,et al.  Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance , 2001, Nature.

[19]  A. Rangelov Factorizing numbers with classical interference: several implementations in optics , 2008, 0811.2070.

[20]  Johann Summhammer Factoring and Fourier transformation with a Mach-Zehnder interferometer , 1997 .

[21]  W. Schleich,et al.  NMR experiment factors numbers with Gauss sums. , 2006, Physical review letters.

[22]  T. S. Mahesh,et al.  Factorizing numbers with the Gauss sum technique , 2007 .

[23]  Wave Packets Can Factorize Numbers , 2002, quant-ph/0208021.

[24]  Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses , 2011, 1210.6487.

[25]  Vincenzo Tamma,et al.  Analogue algorithm for parallel factorization of an exponential number of large integers: I. Theoretical description , 2015, Quantum Inf. Process..

[26]  M. S. Zubairy,et al.  Factorization with exponential sums , 2008 .

[27]  V. Tamma Sampling of bosonic qubits , 2014, 1506.04948.

[28]  Samuel J. Lomonaco,et al.  Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium: American Mathematical Challenge Society, Short Course, January 17-18, 2000, Washington, DC , 2002 .

[29]  B. M. Fulk MATH , 1992 .

[30]  B. Chatel,et al.  Factoring numbers with interfering random waves , 2008, 2008 Conference on Lasers and Electro-Optics and 2008 Conference on Quantum Electronics and Laser Science.

[31]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[32]  D. Suter,et al.  NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors , 2008, 0803.3396.

[33]  Vincenzo Tamma,et al.  Multiboson Correlation Interferometry with Arbitrary Single-Photon Pure States. , 2014, Physical review letters.

[34]  W. Schleich,et al.  Factorization of numbers with Gauss sums: I. Mathematical background , 2011, 1210.6474.

[35]  T. S. Mahesh,et al.  Factorizing numbers with the Gauss sum technique: NMR implementations , 2007, quant-ph/0701205.

[36]  W. Schleich,et al.  Factorization of numbers with Gauss sums: III. Algorithms with entanglement , 2012, 1210.6491.

[37]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .