Quantum Annealing for Prime Factorization

Shuxian Jiang,Alexander J Mccaskey,Keith A Britt,Travis S Humble,Sabre Kais

doi:10.1038/s41598-018-36058-z

Shuxian Jiang, Alexander J Mccaskey + Show 3 more

Open Access

https://doi.org/10.1038/s41598-018-36058-z

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Abstract

We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function then transforming the k-bit coupling (k ≥ 3) terms to quadratic terms using ancillary variables. Our resource-efficient method uses {mathscr{O}}({mathrm{log}}^{2}(N)) binary variables (qubits) for finding the factors of an integer N. We present how to factorize 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits, respectively. This method was tested using the D-Wave 2000Q for finding an embedding and determining the prime factors for a given composite number. The method is general and could be used to factor larger integers as the number of available qubits increases, or combined with other ad hoc methods to achieve better performances for specific numbers.

Highlights

Quantum Annealing was introduced[18] to solve optimization problems using quantum fluctuations to transit to the ground state, compared to simulated annealing which uses thermal fluctuations to get to the global minimum
We have presented two general methods for factoring integers using quantum annealing for optimizing a cost function that is reduced to an Ising Hamiltonian
The novelty of our demonstration of quantum annealing for prime factorization is based on the reduction in quantum resources required to execute factoring and the experimental verification of the algorithmic accuracy using currently available hardware

Summary

Background

Quantum Annealing was introduced[18] to solve optimization problems using quantum fluctuations to transit to the ground state, compared to simulated annealing which uses thermal fluctuations to get to the global minimum. Computation within the AQC model evolves the L-qubit quantum state under the time-dependent. According to the adiabatic theorem[22], the system state will remain in the instantaneous ground state of the time-dependent Hamiltonian provided the evolution is sufficiently slow to prevent excitations to higher-lying states. Under these idealized adiabatic conditions, the system will evolve into the energetic ground state of the problem Hamiltonian as |ψ(T)〉 = |φ0(T)〉. Interactions, and we develop an efficient transformation of the factoring problem Hamiltonian into pair-wise coupling

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