Quantum Computation of Prime Number Functions

TL;DR

This paper introduces a quantum algorithm that constructs states representing prime numbers and related conjectures, enabling efficient estimation of prime distribution and testing number theory hypotheses using quantum computing.

Contribution

It presents a novel quantum circuit for generating prime states and related conjectures, integrating Grover's algorithm with primality testing for number theory applications.

Findings

Prime state encodes prime distribution and twin primes.

Quantum algorithms estimate prime counting function efficiently.

Proposes experimental tests for Goldbach and twin prime conjectures.

Abstract

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.