Quantum Genetic Algorithm1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

2Quantum Mads, Uribitarte Kalea 6, 48001 Bilbao, Spain 3IKERBASQUE, 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. Ibarrondo1, G. Gatti1,2, and M. Sanz1,3

Roadmap: Classical to Quantum

Genetic algorithmsOptimization algorithms emulating darwinian evolution.

Mappings to quantum

The mapping from GA to QGA is not unique.

Quantum population

|ψpop⟩ =pmax

∑p

bp |upk1

⟩ ⊗ . . . ⊗ |upkn

⟩

HP |uk⟩ = λk |uk⟩, k = 1,…,2c

Selection subroutine

Quantum sorting network

Initialization

Selection

Crossover

Mutation

Sort

Reset

Quantum Clone

Swap

Mutation

State preparation

Crossover

Quantum cloning

machine, TQCM

Swap part of the genetic

information

Numerical-analytical results

MutationSingle qubit

rotations with probability in each qubit.

pm pm

1 − pmX

Y

Z

𝕀

1/31/3

1/3

Aim: Find low energetic states

Population |ψpop⟩

individuals n registers n

genes c qubits/reg. c

fitness criteria + constraints

f(x) problem Hamiltonian (cost)

HP

where

r1

r2

r3

r4

regi

ster

s

step 1 step 2 step 3 step 4

Comparison oracle:

|u⟩ = {|uk⟩ |uk′ ⟩ |0⟩ if λk ≥ λk′

|uk′ ⟩ |uk⟩ |1⟩ if λk′ < λk

|0⟩

|uk⟩|uk′ ⟩

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↓

TS, TC, TM → T = TM ∘ TC ∘ TS limG→∞

TG(ρ) = Λ,

T(ρ) = TM(TC(TS(ρ))) = ∑k

EkρE†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|up

k1⟩ ⊗ . . . ⊗ |up

kn⟩

r1

r2

r3

r4

TQCM

TQCM

+

QCMs analyzed: • Biomimetic Cloning of

Quantum Observables [3] • Universal Quantum Cloning

Machine [4]

BCQO (observable diagonal in )σz

UQCM

Fcopy( |ψ⟩) =2c

∑j=0

|⟨j |ψ⟩ |2

Fcopy( |ψ⟩) =12

+1

1 + 2c

Probability of in the final population for different QGA variants, applied to randomly generated ’s.

|u0⟩HP

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⟩

r1r2r3r4

where T(Λ) = Λ→

FQGA = ⟨u0 |Tr1⊥(ρfinal) |u0⟩

Analysed cases with and .n = 4 c = 2

|u1⟩ |u2⟩ |u3⟩ |u4⟩

Probability of

in each register, where λ1 < λ2 < λ3 < λ4

Only the best individuals survive

NQUIRE