Quantum Genetic Algorithm 1 Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain 2 Quantum Mads, Uribitarte Kalea 6, 48001 Bilbao, Spain 3 IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain Genetic Algorithms (GAs) are extremely successful bioinspired optimisation algorithms, which emulate the natural selection process. Merging GAs with quantum computation is an old ambition which has been considered as a potential source of new heuristic optimisation methods. However, only restricted results have been achieved up to now due to the limitations imposed by quantum mechanics for cloning or erasing information. Here, we develop a fully quantum genetic algorithm (QGA) and study different subroutines for cloning or breeding by means of both a thorough numerical analysis and quantum- channel techniques. This approach paves the way for a new type of optimisation quantum algorithm which, additionally, can be straightforwardly parallelized among different quantum processors. R. Ibarrondo 1 , G. Gatti 1,2 , and M. Sanz 1,3 Roadmap: Classical to Quantum Genetic algorithms Optimization algorithms emulating darwinian evolution. Mappings to quantum The mapping from GA to QGA is not unique. Quantum population | ψ pop ⟩ = pmax ∑ p b p | u p k1 ⟩⊗ ... ⊗ | u p kn ⟩ H P | u k ⟩ = λ k | u k ⟩, k = 1,…,2 c Selection subroutine Quantum sorting network Initialization Selection Crossover Mutation Sort Reset Quantum Clone Swap Mutation State preparation Crossover Quantum cloning machine, T QCM Swap part of the genetic information Numerical-analytical results Mutation Single qubit rotations with probability in each qubit. p m p m 1 − p m X Y Z 1 /3 1 /3 1 /3 Aim: Find low energetic states Population | ψ pop ⟩ individuals n registers n genes c qubits/reg. c fitness criteria + constraints f (x) problem Hamiltonian (cost) H P where r1 r2 r3 r4 registers step 1 step 2 step 3 step 4 Comparison oracle: | u⟩ = { | uk⟩ | uk′⟩ | 0⟩ if λk ≥ λk′ | uk′⟩ | uk⟩ | 1⟩ if λk′ < λk | 0⟩ | u k ⟩ | u k′ ⟩ CMP | u⟩ Quantum Channels Numerical simulations • Selection without measuring the individuals. • Discarding individuals subject to the no-deleting theorem. |0⟩ |0⟩ |0⟩ r1 r2 r3 r4 |0⟩ |0⟩ |0⟩ |0⟩ |0⟩ • Perform replication subject to no-cloning theorem. • Combining genetic information of different individuals. QC techniques allow us to prove exponential convergence of the algorithm, with an exponent given by the spectral subradius of the channel [5]. Subroutines as QCs Generation QC ↓ T S , T C , T M → T = T M ∘ T C ∘ T S lim G→∞ T G (ρ)= Λ, T(ρ)= T M (T C (T S (ρ))) = ∑ k E k ρE † k [1] D. A. Sofge. (2008). “Prospective Algorithms for Quantum Evolutionary Computation.” arXiv:cs.NE/0804.1133 [2] R. Lahoz-Beltra, “Quantum Genetic Algorithms for Computer Scientists,” Computers, vol. 5, no. 4, p. 24, 2016. [3] U. Alvarez-Rodriguez, M. Sanz, L. Lamata, and E. Solano, “Biomimetic cloning of quantum observables,” Scientific Reports, vol. 4, pp. 4–7, 2014. [4] V. Bužek and M. Hillery, “Quantum copying: Beyond the no-cloning theorem,” Phys. Rev. A, vol. 54, no. 3, pp. 1844–1852, 1996. [5] M. Sanz, D. Pérez-García, M. M. Wolf, and J. I. Cirac, “A quantum version of Wielandt’s inequality,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4668–4673, 2010. ordered in parallel | u p k1 ⟩⊗ ... ⊗ | u p kn ⟩ r 1 r 2 r 3 r 4 T QCM T QCM + QCMs analyzed: • Biomimetic Cloning of Quantum Observables [3] • Universal Quantum Cloning Machine [4] BCQO (observable diagonal in ) σz UQCM F copy ( | ψ⟩)= 2 c ∑ j=0 | ⟨ j | ψ⟩ | 2 F copy ( | ψ⟩)= 1 2 + 1 1+2 c Probability of in the final population for different QGA variants, applied to randomly generated ’s. | u 0 ⟩ H P For BCQO Invariant Better for some states For UQCM • Previous attempts have only achieved partial success [1,2]. • Challenge: Non-linear behavior of genetic operators. Reset the lower registers |e0⟩ |e0⟩ r1 r2 r3 r4 where T(Λ)= Λ → F QGA = ⟨u 0 | Tr1⊥(ρ final ) | u 0 ⟩ Analysed cases with and . n =4 c =2 | u1⟩ | u2⟩ | u3⟩ | u4⟩ Probability of in each register, where λ 1 < λ 2 < λ 3 < λ 4 Only the best individuals survive NQUIRE