Factoring 51 and 85 with 8 qubits

We construct simplified quantum circuits for Shor's order-finding algorithm for composites N given by products of the Fermat primes 3, 5, 17, 257, and 65537. Such composites, including the previously studied case of 15, as well as 51, 85, 771, 1285, 4369, … have the simplifying property that the order of a modulo N for every base a coprime to N is a power of 2, significantly reducing the usual phase estimation precision requirement. Prime factorization of 51 and 85 can be demonstrated with only 8 qubits and a modular exponentiation circuit consisting of no more than four CNOT gates.

[1]  Jiangfeng Du,et al.  Quantum factorization of 143 on a dipolar-coupling nuclear magnetic resonance system. , 2012, Physical review letters.

[2]  A. Politi,et al.  Shor’s Quantum Factoring Algorithm on a Photonic Chip , 2009, Science.

[3]  Shor’s factoring algorithm and modern cryptography. An illustration of the capabilities inherent in quantum computers , 2004, quant-ph/0411184.

[4]  Michele Mosca,et al.  The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer , 1998, QCQC.

[5]  E. Lucero,et al.  Computing prime factors with a Josephson phase qubit quantum processor , 2012, Nature Physics.

[6]  W. Marsden I and J , 2012 .

[7]  B. Lanyon,et al.  Experimental demonstration of a compiled version of Shor's algorithm with quantum entanglement. , 2007, Physical review letters.

[8]  Griffiths,et al.  Semiclassical Fourier transform for quantum computation. , 1995, Physical review letters.

[9]  Jian-Wei Pan,et al.  Demonstration of a compiled version of Shor's quantum factoring algorithm using photonic qubits. , 2007, Physical review letters.

[10]  Graeme Smith,et al.  Oversimplifying quantum factoring , 2013, Nature.

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

[12]  X-Q Zhou,et al.  Experimental realization of Shor's quantum factoring algorithm using qubit recycling , 2011, Nature Photonics.

[13]  Christof Zalka Fast versions of Shor's quantum factoring algorithm , 1998 .

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

[15]  Stéphane Beauregard Circuit for Shor's algorithm using 2n+3 qubits , 2003, Quantum Inf. Comput..

[16]  M B Plenio,et al.  Efficient factorization with a single pure qubit and logN mixed qubits. , 2000, Physical review letters.

[17]  Göran Wendin,et al.  Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark , 2006, quant-ph/0610214.

[18]  Zhongyuan Zhou,et al.  qu an tph ] 3 0 M ar 2 01 3 Factoring 51 and 85 with 8 qubits , 2013 .

[19]  Dieter Suter,et al.  Quantum adiabatic algorithm for factorization and its experimental implementation. , 2008, Physical review letters.