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Entropy

Spectral Signatures of Prime Factorization

Giuseppe Mussardo, Andrea Trombettoni

Published Mar 23, 2026• 0 citations
Publisher

Abstract

We present a protocol for integer factorization for all integers N below a certain cut-off Λ=2d, grounded in the theory of quantum measurement. In this framework, the factorization of an integer N≤Λ is achieved in a number of steps equal to the total number I of primes present in its factorization; explicitly, the procedure consists of a sequence of I quantum measurements. The method requires a single-purpose quantum device designed to perform measurements of an observable with a prescribed spectrum. Crucially, the construction of this device involves solving, once and for all, a set of approximately 2d differential equations, independently of the specific integer to be factorized. We argue that the initialization task of this device can be efficiently implemented on a quantum computer in d steps, thereby decoupling the computational cost of device preparation from the factorization process itself.

DOI
10.3390/e28030363

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We present a protocol for integer factorization for all integers N below a certain cut-off Λ=2d, grounded in the theory of quantum measurement. In this framework, the factorization of an integer N≤Λ is achieved in a number of steps equal to the total number I of primes present in its factorization; explicitly, the procedure consists of a sequence of I quantum measurements. The method requires a single-purpose quantum device designed to perform measurements of an observable with a prescribed spectrum. Crucially, the construction of this device involves solving, once and for all, a set of approximately 2d differential equations, independently of the specific integer to be factorized. We argue that the initialization task of this device can be efficiently implemented on a quantum computer in d steps, thereby decoupling the computational cost of device preparation from the factorization process itself.
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer NNN is achieved in a number of steps equal to the number kkk of its prime factors, -- e.g., if NNN is the product of two primes, two quantum measurements are enough, regardless of the number of digits nnn of the number NNN. Since kkk is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.
The arithmetic problem of factoring an integer NNN can be translated into the physics of a quantum device, a result that supports P\'olya's and Hilbert's conjecture to demonstrate Riemann's hypothesis. The energies of this system, being univocally related to the factors of NNN, are the eigenvalues of a bounded Hamiltonian. Here we solve the quantum conditions and show that the histogram of the discrete energies, provided by the spectrum of the system, should be interpreted in number theory as the relative probability for a prime to be a factor candidate of NNN. This is equivalent to a quantum sieve that is shown to require only o(lnN)3o{(ln\sqrt{N})}^{3}o(lnN​)3 energy measurements to solve the problem, recovering Shor's complexity result. Hence the outcome can be seen as a probability map that a pair of primes solve the given factorization problem. Furthermore, we show that a possible embodiment of this quantum simulator corresponds to two entangled particles in a Penning trap. The possibility to build the simulator experimentally is studied in detail. The results show that factoring numbers, many orders of magnitude larger than those computed with experimentally available quantum computers, is achievable using typical parameters in Penning traps.
Abstract We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probability one half. We conclude by discussing a protocol to obtain the desired prime factors employing a logarithmic energy spectrum which consists of prime numbers only.
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.
We implement a quantum protocol for prime number identification based on entanglement dynamics, using IBM quantum processors. The method links the primality of an integer to specific Fourier components extracted from the time evolution of entanglement in a bipartite quantum system. To mitigate experimental noise, we introduce a noise-mitigation method based on a global rescaling factor, which is calibrated on a subset of circuits and extrapolated across different configurations. Theoretical support is provided by a new analytical bound for the Fourier modes derived assuming an initial uniform superposition state. This new bound enhances the separation between prime and composite numbers under moderate experimental deviations. These results represent a step toward practical number-theoretic applications on noisy intermediate-scale quantum (NISQ) devices.