Feynman’s prescription for a quantum simulator was to find a Hamitonian for a system that could serve as a computer. The Pólya-Hilbert conjecture proposed the demonstration of Riemann’s hypothesis through the spectral decomposition of Hermitian operators. Here we study the problem of decomposing a number into its prime factors, $N=xy$, using such a simulator. First, we derive the Hamiltonian of the physical system that simulates a new arithmetic function formulated for the factorization problem that represents the energy of the computer. This function rests alone on the primes below $\sqrt{N}$. We exactly solve the spectrum of the quantum system without resorting to any external ad hoc conditions, also showing that it obtains, for $x\ll \sqrt{N}$, a prediction of the prime counting function that is almost identical to Riemann’s $R(x)$ function. It has no counterpart in analytic number theory, and its derivation is a consequence of the quantum theory of the simulator alone.

• Received 1 December 2015

DOI:https://doi.org/10.1103/PhysRevLett.117.200502