# Factorization using a logarithmic energy spectrum

Prime factorization is the decomposition of an integer into its prime factors. At present no algorithm is known that can find a factor of a very large integer in polynomial time on a classical computer. The security of codes e.g. the RSA cryptographic scheme relies on this fact, however, it could be broken by the Shor algorithm provided an elaborate quantum computer were available.

We investigate a hypothetical device that could be called a ''quantum analogue computer'' with the only purpose to factor integers. It consists of a non-harmonic trap filled with a fixed number of bosonic atoms that can be excited to a definite energy characterized by the number to be factored. The trap is designed such that after excitation the individual atoms have single particle energies depending exactly on one of the factors.

A simultaneous measurement of all single particle energies then gives the factors. A trap with a single particle spectrum where the energies depend logarithmically on the quantum numbers can be used for this protocol.

A controlled interaction between the bosons that can be switched on and off is needed as well. In recent work several different logarithmic spectra were studied and potentials associated with them were determined. The protocol was refined for  traps in one or two dimensions and for the cases of two or more factors. The main results are published in the articles [1-3].

Fig. 1: Scaled trap-potential which creates the energy spectrum $$E_n=\ln(n/2+1)$$ for $$n=0,1,2,\ldots$$ together with the six lowest eigenfunctions as a function of the dimensionless position.

### Contributors

F. Gleisberg, W. P. Schleich

The work was performed with the bachelor students C. Ufrecht, M. Stetter, M. Volpp, G. Wolff, M. Carmesin, F. Di Pumpo and N. Staudenmaier.

### Collaborations

M. Boshier (Los Alamos National Laboratory, Los Alamos, USA)

### Funding

Texas A&M University Insitute for Advanced Study (TIAS)

### References

[1] F. Gleisberg, R. Mack, K. Vogel and W. P. Schleich, Factorization with a logarithmic energy spectrum, New J. Phys. 15, 023037 (2013)
[2] F. Gleisberg, M. Volpp and W. P. Schleich, Factorization with a logarithmic energy spectrum of a two-dimensional potential, Phys. Lett. A 179, 2556 (2015)
[3] F. Gleisberg, F. Di Pumpo, G. Wolff, and W. P. Schleich, Prime factorization of arbitrary integers with a logarithmic energy spectrum, J. Phys. B 51(3), 035009 (2018)