Abstract
Integer factorization has been one of the cornerstone applications of the field of quantum computing since the discovery of an efficient algorithm for factoring by Peter Shor. Unfortunately, factoring via Shor’s algorithm is well beyond the capabilities of today’s noisy intermediate-scale quantum (NISQ) devices. In this work, we revisit the problem of factoring, developing an alternative to Shor’s algorithm, which employs established techniques to map the factoring problem to the ground state of an Ising Hamiltonian. The proposed variational quantum factoring (VQF) algorithm starts by simplifying equations over Boolean variables in a preprocessing step to reduce the number of qubits needed for the Hamiltonian. Then, it seeks an approximate ground state of the resulting Ising Hamiltonian by training variational circuits using the quantum approximate optimization algorithm (QAOA). We benchmark the VQF algorithm on various instances of factoring and present numerical results on its performance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The term ansatz means a trial solution of a specific form with some parameter(s) that need to be specified for the particular problem at hand.
- 2.
To lower the needed qubits for our numerical simulations, we assumed prior knowledge of \(n_p\) and \(n_q\).
- 3.
We note that other simple relations exist that can be used for preprocessing—the simplified clauses for \(m=56153,291311\) as used in our numerical simulations were given by [3] who utilized a different preprocessing scheme.
- 4.
References
Beauregard, S.: Circuit for Shor’s algorithm using 2n+3 qubits. Quant. Inf. Comput. 3(2), 175–185 (2003)
Burges, C.J.C.: Factoring as Optimization. Microsoft Research, MSR-TR-200 (2002)
Dattani, N.S., Bryans, N.: Quantum factorization of 56153 with only 4 qubits, arXiv:1411.6758 [quant-ph] (2014)
Dridi, R., Alghassi, H.: Prime factorization using quantum annealing and computational algebraic geometry. Sci. Rep. 7(1), 43048 (2017)
Ekerå, M.: Modifying Shor’s algorithm to compute short discrete logarithms. IACR Cryptology ePrint Archive (2016)
Farhi, E., Goldstone, J., Gutmann, S.: A quantum approximate optimization algorithm, arXiv:1411.4028 [quant-ph] (2014)
Farhi, E., Harrow, A.W.: Quantum supremacy through the quantum approximate optimization algorithm, arXiv:1602.07674 [quant-ph] (2016)
Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (2000)
Fowler, A.G., Devitt, S.J., Hollenberg, L.C.L.: Implementation of Shor’s Algorithm on a Linear Nearest Neighbour Qubit Array. Quantum Information & Computation 4, 237–251 (2004)
Fried, E.S., Sawaya, N.P.D., Cao, Y., Kivlichan, I.D., Romero, J., Aspuru-Guzik, A.: qTorch: The quantum tensor contraction handler, arXiv:1709.03636 [quant-ph] (2017)
Geller, M.R., Zhou, Z.: Factoring 51 and 85 with 8 qubits. Sci. Rep. 3(3), 1–5 (2013)
Gidney, C.: Factoring with n+2 clean qubits and n\(-\)1 dirty qubits, arXiv:1706.07884 [quant-ph] (2017)
Johansson, J., Nation, P., Nori, F.: QuTiP: an open-source Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 183(8), 1760–1772 (2012)
Jones, N.C., et al.: Layered architecture for quantum computing. Phys. Rev. X 2(3), 1–27 (2012)
Kitaev, A.Y.: Quantum measurements and the Abelian stabilizer problem, arXiv:quant-ph/9511026 (1995)
Lin, C.Y.-Y., Zhu, Y.: Performance of QAOA on typical instances of constraint satisfaction problems with bounded degree, arXiv:1601.01744 [quant-ph] (2016)
Martín-López, E., Laing, A., Lawson, T., Alvarez, R., Zhou, X.Q., O’brien, J.L.: Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nat. Photonics 6(11), 773–776 (2012)
Nannicini, G.: Performance of hybrid quantum/classical variational heuristics for combinatorial optimization, arXiv:1805.12037 [quant-ph] (2018)
Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information, 10th Anniversary edn. Cambridge University Press, Cambridge (2010)
Peruzzo, A., et al.: A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5(May), 4213 (2014)
Politi, A., Matthews, J.C.F., O’Brien, J.L.: Shor’s quantum factoring algorithm on a photonic chip. Science 325(5945), 1221–1221 (2009)
Romero, J., Olson, J., Aspuru-Guzik, A.: Quantum autoencoders for efficient compression of quantum data. Quant. Sci. Technol. 2(4), 045001 (2017)
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)
Zalka, C.: Shor’s algorithm with fewer (pure) qubits, arXiv:quant-ph/0601097 (2006)
Acknowledgments
We would like to acknowledge the Zapata Computing scientific team, including Peter Johnson, Jhonathan Romero, Borja Peropadre, and Hannah Sim for their insightful and inspiring comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Anschuetz, E., Olson, J., Aspuru-Guzik, A., Cao, Y. (2019). Variational Quantum Factoring. In: Feld, S., Linnhoff-Popien, C. (eds) Quantum Technology and Optimization Problems. QTOP 2019. Lecture Notes in Computer Science(), vol 11413. Springer, Cham. https://doi.org/10.1007/978-3-030-14082-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-14082-3_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-14081-6
Online ISBN: 978-3-030-14082-3
eBook Packages: Computer ScienceComputer Science (R0)