Prime Numbers and Quantum Computers - CERN
Transcript of Prime Numbers and Quantum Computers - CERN
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Prime Numbersand
Quantum Computers
Workshop on Quantum Computing for High Energy PhysicsCERN, Geneva, 5-6 Nov 2018
Germán Sierra Instituto de Física Teórica CSIC-UAM, Madrid
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UCSB 2012
15 = 3 x 5
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RSA-129 = 114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541= 3490529510847650949147849619903898133417764638493387843990820577 ×32769132993266709549961988190834461413177642967992942539798288533
Factorized in 1994 using 1.600 computers connected in internet
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Quantum promise
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Prime numbers go quantum
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Prime Number Theorem (1896)
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Non trivial zeros of the zeta function have real part equal to 1/2
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Test
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Problem: given x determine if it is prime or not
Miller-Rabin primality test:
Choose in the range
Test then x is composite with certainty
Run a test that involves
(witness)
Test then
strong witness
2) x is composite strong lier1) x is prime with high probability
Solution: use several witnesses
For
With k witnesses the error is
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Quantum speed up
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Turing, computers and number theory
Turing did not believe in the Riemann hypothesis and wanted to disprove it.
In 1950 he used the electronic computer at the Manchester university
to find the first 1104 Riemann zeros who all lie on the critical line.
Then the machine broke down.
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Entanglement entropy of the Prime state(JI Latorre, GS, 2015)
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A random density matrix has
Volumen law entropy
The Prime state is not random
(Don Page)
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Prime correlations (Hardy-Littlewood)
Entanglement encode correlations between primes
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IBM quantum computer and the Prime state(Diego García-Martín, GS, 2018)
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Conclusions
- Use quantum computers to study fundamental quantities in number theory:
Counting by measuring
- Number theory provides interesting highly entangled states to test quantum computers:
Quantum Arithmetics
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”Quantum Computation of prime number functions”,J.I. Latorre and G.S., Quan. Info. Comm. 2014.
”There is entanglement in the primes”,J.I. Latorre and G.S., Quan. Info. Comm. 2015.
”Five experimental Tests on the 5-Qubit IBM Quantum Computer”,D. García-Martín and G.S., J. Applied Maths and Phys. 2018
Work done in collaboration with J.I. Latorre and D. García-Martín
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Thanks for your attention