Factoring numbers with a single interferogram

Abstract

We construct an analogue computer based on light interference to encode the hyperbolic function f (ζ) ≡ 1/ζ into a sequence of skewed curlicue functions. The resulting interferogram when scaled appropriately allows us to find the prime number decompositions of integers. We implement this idea exploiting polychromatic optical interference in a multi-path interferometer and factor seven digit numbers. We give an estimate for the largest number that can be factored by this scheme.To find the factors of a large integer number N is a problem of exponential complexity. Indeed, the security of codes relies on this fact but is endangered by Shor's algorithm [1], which employs entanglement between quantum systems [2]. In the present paper we report the optical realization of a new algorithm for factoring numbers, which takes advantage of interference only.A naive way of approaching the problem of factorization consists of dividing N by integers ℓ, starting from ℓ = 3 until N/ℓ is an integer. In the worst case this procedure requires √ N divisions before one would find a factor. On a digital computer division of large numbers is a rather costly process. However, in many physical phenomena division occurs in a rather natural way. For example, a wave of wavelength λ, propagating over a distance L, acquires a phase φ = 2πL/λ and therefore probes the ratio L/λ. In the optical domain λ is measured in nanometers (nm), that is λ = ℓ nm. When we also express the path length L in units of nm, that is L = N nm, the phase φ = 2πN/ℓ is sensitive to the ratio N/ℓ. For factors of N , φ is an integer multiple of 2π. Otherwise φ is a rational multiple of 2π.In order to enhance the signal associated with a factor relative to the ones corresponding to non-factors, we use interference of waves, which differ in their optical path length by an integer multiple. In this way we take advantage of constructive interference when ℓ is a factor of N , but destructive interference when ℓ is not a factor of N . The cancellation of terms is most effective when the individual optical paths increase in a nonlinear way. In this case, the intensity of the interfering waves is determined by the absolute value squared of a truncated exponential sum [3]. A polychromatic source of light, which contains several wavelengths λ = ℓ nm, allows us to test several trial factors simultaneously, taking advantage of the properties of truncated exponential sums [4] with continuous arguments.Our method is motivated by recent work on factorization using truncated exponential sums [5], which has been realized in several experiments [6]. However, it differs from the past realizations in three important points:(i) the division of N by the trial factors ℓ is not precalculated [7], but it is performed by the experiment it- self, (ii) all the trial factors are tested simultaneously in a single experiment, and (iii) a scaling property inherent in the recorded interferogram of a single number allows us to obtain the factors of several numbers.The optical setup used to implement th...

Multipath correlation interference and controlled-NOT gate simulation with a thermal source

“…We also showed how this second-order interference effect can be extended to arbitrary orders N, leading to the interference of more general configurations of correlated paths. This could be used to simulate on-demand 100%-visibility correlations typical of entangled states of N qubits, with possible applications in high precision metrology and imaging [19,21,22,45], and in the development of novel optical algorithms [46][47][48][49][50].…”

Factorization algorithm based on the periodicity measurement of a continuous truncated exponential sum

“…In Reference, 21 we have demonstrated the physical computability of our method factorizing several numbers with up to seven digits with a single experiment. Here we depict the case M = 3.…”

“…[20][21][22] We will also show that such an algorithm can be implemented with an optical analog computer, based on a multi-path Michelson interferometer and a spectrometer.…”

Optical Eratosthenes’ sieve for large prime numbers

“…A new and relatively unexplored testbed for the connection between physics and prime numbers is classical optics. On the factorization front, multi-wavelength interferometry [5] and optoelectronically assisted algorithms [6] schemes have been proposed, and experimental demonstrations were carried out based on the Talbot effect [7] and polychromatic interference in a multipath interferometer [8]. On the fundamental side, Berry [9,10] proposed the reading of Riemann function's zeros from radiation patterns using the property of the far field propagation to represent the Fourier transform.…”