Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1601.04896 · Physical Review Letters

Quantum Simulation of the Factorization Problem

José Luis Rosales, Vicente Martín

Published Nov 10, 2016• 4 citations
arXivHTMLPDF

Abstract

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≪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.

DOI
10.1103/physrevlett.117.200502

Related papers

Quantum simulation of the integer factorization problem: Bell states in a Penning trap

José Luis Rosales, Vicente Martín

The arithmetic problem of factoring an integer $N$ 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 $N$, 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 $N$. This is equivalent to a quantum sieve that is shown to require only $o{(ln\sqrt{N})}^{3}$ 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.

arXiv:1704.03174•Physical Review A•Mar 19, 2018•4 citations

Simulating Factorization with a Quantum Computer

José Luis Rosales

Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum computer would eventually weaken these algorithms. In this paper, one follows Feynman's prescription for a computer to simulate the physics corresponding to the algorithm of factoring a large number $N$ into primes. Using Dirac-Jordan transformation theory one translates factorization into the language of quantum hermitical operators, acting on the vectors of the Hilbert space. This leads to obtaining the ensemble of factorization of $N$ in terms of the Euler function $φ(N)$, that is quantized. On the other hand, considering $N$ as a parameter of the computer, a Quantum Mechanical Prime Counting Function $π_{QM}(x)$, where $x$ factorizes $N$, is derived. This function converges to $π(x)$ when $N\gg x$. It has no counterpart in analytic number theory and its derivation relies on semiclassical quantization alone.

arXiv:1505.00642•May 4, 2015•0 citations

A Parallel Quantum Computer Simulator

Kevin Obenland, Alvin M. Despain

A Quantum Computer is a new type of computer which can efficiently solve complex problems such as prime factorization. A quantum computer threatens the security of public key encryption systems because these systems rely on the fact that prime factorization is computationally difficult. Errors limit the effectiveness of quantum computers. Because of the exponential nature of quantum com puters, simulating the effect of errors on them requires a vast amount of processing and memory resources. In this paper we describe a parallel simulator which accesses the feasibility of quantum computers. We also derive and validate an analytical model of execution time for the simulator, which shows that parallel quantum computer simulation is very scalable.

arXiv:quant-ph/9804039•Apr 16, 1998•52 citations

Quantum computation of prime number functions

José Ignacio Latorre, Germán Sierra

We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

Quantum Information and Computation•May 1, 2014•14 citations

Quantum Computation of Prime Number Functions

José Ignacio Latorre, Germán Sierra

We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2^n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. We also propose a Twin Prime state to measure the number of twin primes and another state to test the Goldbach conjecture. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

arXiv:1302.6245•Feb 25, 2013•0 citations

Identification of prime numbers with quantum computers

Victor Penchel Alvarenga Balthar dos Santos

The identification of prime numbers is an ancient area of study with applications in cryptography and fundamental mathematical problems, such as the Riemann Hypothesis (RH). Classical algorithms, like the Sieve of Eratosthenes and the AKS Primality Test, are efficient examples for this purpose. However, quantum computing (QC), with its superior theoretical capacity, has inspired the development of quantum algorithms, many of which are probabilistic or rely on the RH. In this dissertation, we propose a deterministic and efficient quantum algorithm that generalizes a recent method from quantum optics to identify prime numbers through entanglement dynamics. We adapted this method for QC, removing several constraints from the original work, and showed that discrete bipartite quantum systems prepared in certain initial states can be used to identify prime numbers through the Fourier modes of the reduced purity of one of the subsystems. The proposed algorithm is viable for implementation in future fault-tolerant QC systems. The initial steps involve encoding the system in qubits and preparing a maximally superposed state. In the state evolution step, we use Walsh matrices to efficiently prepare diagonal unitary gates. The reduced purity is measured using a modified controlled-SWAP test, and the Fourier modes are obtained via the numerical calculation of a one-dimensional integral. Subsequently, we present the simulations performed in Qiskit, which demonstrate the validity of the algorithm in all its stages. Finally, we discuss the computational cost, quantifying the number of elementary gates as a function of the number of qubits and speculating the total computational complexity of the algorithm.

•Sep 30, 2024•0 citations
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≪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.
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.
Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum computer would eventually weaken these algorithms. In this paper, one follows Feynman's prescription for a computer to simulate the physics corresponding to the algorithm of factoring a large number NNN into primes. Using Dirac-Jordan transformation theory one translates factorization into the language of quantum hermitical operators, acting on the vectors of the Hilbert space. This leads to obtaining the ensemble of factorization of NNN in terms of the Euler function φ(N)φ(N)φ(N), that is quantized. On the other hand, considering NNN as a parameter of the computer, a Quantum Mechanical Prime Counting Function πQM(x)π_{QM}(x)πQM​(x), where xxx factorizes NNN, is derived. This function converges to π(x)π(x)π(x) when N≫xN\gg xN≫x. It has no counterpart in analytic number theory and its derivation relies on semiclassical quantization alone.
A Quantum Computer is a new type of computer which can efficiently solve complex problems such as prime factorization. A quantum computer threatens the security of public key encryption systems because these systems rely on the fact that prime factorization is computationally difficult. Errors limit the effectiveness of quantum computers. Because of the exponential nature of quantum com puters, simulating the effect of errors on them requires a vast amount of processing and memory resources. In this paper we describe a parallel simulator which accesses the feasibility of quantum computers. We also derive and validate an analytical model of execution time for the simulator, which shows that parallel quantum computer simulation is very scalable.
We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2n2^n2n, where nnn is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.
We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2^n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. We also propose a Twin Prime state to measure the number of twin primes and another state to test the Goldbach conjecture. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.
The identification of prime numbers is an ancient area of study with applications in cryptography and fundamental mathematical problems, such as the Riemann Hypothesis (RH). Classical algorithms, like the Sieve of Eratosthenes and the AKS Primality Test, are efficient examples for this purpose. However, quantum computing (QC), with its superior theoretical capacity, has inspired the development of quantum algorithms, many of which are probabilistic or rely on the RH. In this dissertation, we propose a deterministic and efficient quantum algorithm that generalizes a recent method from quantum optics to identify prime numbers through entanglement dynamics. We adapted this method for QC, removing several constraints from the original work, and showed that discrete bipartite quantum systems prepared in certain initial states can be used to identify prime numbers through the Fourier modes of the reduced purity of one of the subsystems. The proposed algorithm is viable for implementation in future fault-tolerant QC systems. The initial steps involve encoding the system in qubits and preparing a maximally superposed state. In the state evolution step, we use Walsh matrices to efficiently prepare diagonal unitary gates. The reduced purity is measured using a modified controlled-SWAP test, and the Fourier modes are obtained via the numerical calculation of a one-dimensional integral. Subsequently, we present the simulations performed in Qiskit, which demonstrate the validity of the algorithm in all its stages. Finally, we discuss the computational cost, quantifying the number of elementary gates as a function of the number of qubits and speculating the total computational complexity of the algorithm.