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Title:
Quantum Computation of Prime Number Functions
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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 encodes 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, 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.
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