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:2005.02422 · Quantum

The Prime state and its quantum relatives

Diego García-Martín, E. Ribas, Stefano Carrazza, José Ignacio Latorre, Germán Sierra

Published Dec 11, 2020• 8 citations
arXivHTMLPDF

Abstract

The Prime state of n qubits, |Pn⟩ , is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n . This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

DOI
10.22331/q-2020-12-11-371

Related papers

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

There is entanglement in the primes

José Ignacio Latorre, Germán Sierra

Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a Twin Prime state shows that this new state is even more entangled than the Prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to Arithmetics in novel ways.

arXiv:1403.4765•Quantum Information and Computation•May 1, 2015•18 citations

Tutorial

John P. Hayes

Over the last decade a new way to compute has been defined which, unlike conventional methods, is based on quantum mechanics rather than classical physics. This fundamental change in paradigm can, in principle, solve some important and hitherto intractable problems such as prime factorization of large numbers. This tutorial presentation will review the basics of quantum computation and the design of circuits to implement quantum algorithms. The starting point is the notion of a quantum bit or qubit. Because of the superposition property of quantum states, n qubits can store 2n numbers simultaneously, implying a type of massive parallelism. Furthermore, quantum states allow powerful forms of interaction such as entanglement that have no classical counterparts. Qubits are fragile, however, and are altered by measurement; hence quantum circuits must follow very different rules from classical ones. The differences between quantum and classical logic circuits will be discussed and illustrated. Finally, the physical implementation of quantum devices will be considered, along with the prospects for practical quantum computers.

ACM/IEEE Design Automation Conference (DAC)•Jan 1, 2003•0 citations

Tutorial: basic concepts in quantum circuits

J.P. Hayes

Over the last decade a new way to compute has been defined which, unlike conventional methods, is based on quantum mechanics rather than classical physics. This fundamental change in paradigm can, in principle, solve some important and hitherto intractable problems such as prime factorization of large numbers. This tutorial presentation will review the basics of quantum computation and the design of circuits to implement quantum algorithms. The starting point is the notion of a quantum bit or qubit. Because of the superposition property of quantum states, n qubits can store 2n numbers simultaneously, implying a type of massive parallelism. Furthermore, quantum states allow powerful forms of interaction such as entanglement that have no classical counterparts. Qubits are fragile, however, and are altered by measurement; hence quantum circuits must follow very different rules from classical ones. The differences between quantum and classical logic circuits will be discussed and illustrated. Finally, the physical implementation of quantum devices will be considered, along with the prospects for practical quantum computers.

ACM/IEEE Design Automation Conference (DAC)•Mar 2, 2004•0 citations

Geometry of Discrete Quantum Computing

Andrew J. Hanson, Gerardo Ortíz, Amr Sabry +1 more

Abstract. Conventional quantum computing entails a geometry based on the description of an n-qubit state using 2n infinite precision complex numbers denoting a vector in a Hilbert space. Such numbers are in general uncomputable using any real-world resources, and, if we have the idea of physical law as some kind of computational algorithm of the universe, we would be compelled to alter our descriptions of physics to be consistent with computable numbers. Our purpose here is to examine the geometric implications of using finite fields Fp and finite complexified fields Fp2 (based on primes p congruent to 3 (mod 4)) as the basis for computations in a theory of discrete quantum computing, which would therefore become a computable theory. Because the states of a discrete n-qubit system are in principle enumerable, we are able to determine the proportions of entangled and unentangled states. In particular, we extend the Hopf fibration that defines the irreducible state space of conventional continuous n-qubit theories (which is the complex projective space CP2 n−1) to an analogous discrete geometry in which the Hopf circle for any n is found to be a discrete set of p+1 points. The tally of unit-length n-qubit states is given, and reduced via the generalized Hopf fibration to DCP2 n−1, the discrete analog of the complex projective space, which has p2 n−1(p − 1) ∏n−1k=1(p2k + 1) irreducible states. Using a measure of entanglement, the purity, we explore the entanglement features of discrete quantum states and find that the n-qubit states based on the complexified field Fp2 have p n(p − 1)n unentangled states (the product of the tally for a single qubit) with purity 1, and they have pn+1(p − 1)(p+ 1)n−1 maximally entangled states with purity zero.

•Jan 1, 2013•13 citations
The Prime state of n qubits, |Pn⟩ , is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n . This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.
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.
Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a Twin Prime state shows that this new state is even more entangled than the Prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to Arithmetics in novel ways.
Over the last decade a new way to compute has been defined which, unlike conventional methods, is based on quantum mechanics rather than classical physics. This fundamental change in paradigm can, in principle, solve some important and hitherto intractable problems such as prime factorization of large numbers. This tutorial presentation will review the basics of quantum computation and the design of circuits to implement quantum algorithms. The starting point is the notion of a quantum bit or qubit. Because of the superposition property of quantum states, n qubits can store 2n numbers simultaneously, implying a type of massive parallelism. Furthermore, quantum states allow powerful forms of interaction such as entanglement that have no classical counterparts. Qubits are fragile, however, and are altered by measurement; hence quantum circuits must follow very different rules from classical ones. The differences between quantum and classical logic circuits will be discussed and illustrated. Finally, the physical implementation of quantum devices will be considered, along with the prospects for practical quantum computers.
Over the last decade a new way to compute has been defined which, unlike conventional methods, is based on quantum mechanics rather than classical physics. This fundamental change in paradigm can, in principle, solve some important and hitherto intractable problems such as prime factorization of large numbers. This tutorial presentation will review the basics of quantum computation and the design of circuits to implement quantum algorithms. The starting point is the notion of a quantum bit or qubit. Because of the superposition property of quantum states, n qubits can store 2n numbers simultaneously, implying a type of massive parallelism. Furthermore, quantum states allow powerful forms of interaction such as entanglement that have no classical counterparts. Qubits are fragile, however, and are altered by measurement; hence quantum circuits must follow very different rules from classical ones. The differences between quantum and classical logic circuits will be discussed and illustrated. Finally, the physical implementation of quantum devices will be considered, along with the prospects for practical quantum computers.
Abstract. Conventional quantum computing entails a geometry based on the description of an n-qubit state using 2n infinite precision complex numbers denoting a vector in a Hilbert space. Such numbers are in general uncomputable using any real-world resources, and, if we have the idea of physical law as some kind of computational algorithm of the universe, we would be compelled to alter our descriptions of physics to be consistent with computable numbers. Our purpose here is to examine the geometric implications of using finite fields Fp and finite complexified fields Fp2 (based on primes p congruent to 3 (mod 4)) as the basis for computations in a theory of discrete quantum computing, which would therefore become a computable theory. Because the states of a discrete n-qubit system are in principle enumerable, we are able to determine the proportions of entangled and unentangled states. In particular, we extend the Hopf fibration that defines the irreducible state space of conventional continuous n-qubit theories (which is the complex projective space CP2 n−1) to an analogous discrete geometry in which the Hopf circle for any n is found to be a discrete set of p+1 points. The tally of unit-length n-qubit states is given, and reduced via the generalized Hopf fibration to DCP2 n−1, the discrete analog of the complex projective space, which has p2 n−1(p − 1) ∏n−1k=1(p2k + 1) irreducible states. Using a measure of entanglement, the purity, we explore the entanglement features of discrete quantum states and find that the n-qubit states based on the complexified field Fp2 have p n(p − 1)n unentangled states (the product of the tally for a single qubit) with purity 1, and they have pn+1(p − 1)(p+ 1)n−1 maximally entangled states with purity zero.