A Quantum Genetic Algorithm with application to Cosmological Parameters Estimation

TL;DR

The general behavior of AEQGA as a function of its hyperparameters is tested and it is compared with a second quantum genetic algorithm found in the literature as well as with classical algorithms, finding consistent results.

Abstract

An Amplitude-Encoded Quantum Genetic Algorithm (AEQGA) has been developed to minimize $χ^2$ functions of different cosmological probes (Supernovae Type Ia, Baryon Acoustic Oscillations, Cosmic Microwave Background Radiation), to find the best-fit value for two cosmological parameters, namely the Hubble Constant and the density matter content of the Universe today. Our main aim is to pave the way to testing the adoption of quantum optimization in the inference of the cosmological parameters that describe the universe evolution. AEQGA computes the merit function classically, and then uses a quantum circuit to entangle the population and perform crossover and mutation operations. The results show consistency with the isocontours of the objective functions. We then tested the general behavior of AEQGA as a function of its hyperparameters and compared it with a second quantum genetic algorithm found in the literature as well as with classical algorithms, finding consistent results.

Figures (17)

• Figure 1: The quantum encoding step for a population of 8 individuals and thus in a 3-qubit quantum circuit. The numbers inside the rotation gates depend on the values of the starting classical population. This is the expanded version of the state preparation block shown in green in Fig. \ref{['Fig_Quantum_Circuit']}. For the insights on the gates used in this circuit, see $\S$\ref{['Sec_our_QGA']}.
• Figure 2: The quantum circuit for our algorithm, built for a population of 8 real values encoded in 3 qubits. Apart from the state preparation block previously detailed, the other gates are represent the operations of quantum mutation (on qubit 0) and quantum crossover (on qubits 1 and 2).
• Figure 3: Analytical contour map of the objective functions we study in this work. Left panel: for the SNe Ia. Right panel: for CMB + BAO. The black lines represent the $1,2,3,4$ and $5 \sigma$ analytical contours. With red dots, two results (one for SNe Ia, the other for CMB) found in the literature to compare with our objective functions.
• Figure 4: Results of the quantum genetic algorithm. Left panel: for the SNe Ia, $n_i=300, n_g=50,$ and $n_p=32$, crossover and mutation probabilities= 0.5. Right panel: the same for the CMB + BAO sets. We recall that $n_i$ is the number of iterations, $n_g$ is the number of generations, and $n_p$ is the number of individuals inside the population.
• Figure 5: The standard deviations on $H_0$ as a function of the crossover and mutation probabilities. The standard deviation as well as the contour levels are expressed in a logarithmic scale. Left panel: for the SNe Ia. Right panel: for the CMB + BAO set. The contour levels are shown only for the SNe IA results because they show a clear trend, which is not the case for the CMB + BAO ones.