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arXiv:1206.3830 · IEEE Congress on Evolutionary Computation (CEC)

Optimizing qubit Hamiltonian parameter estimation algorithm using PSO

Alexandr Sergeevich, Stephen D. Bartlett

Published Jun 1, 2012• 10 citations
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Abstract

In this paper we develop qubit Hamiltonian single parameter estimation techniques using Bayesian approach. The algorithms considered are restricted to projective measurements in a fixed basis, and are derived under the assumption that the qubit measurement is much slower than the characteristic qubit evolution. The non-adaptive algorithm is optimized using particle swarm optimization and compared to previously developed locally-optimal scheme.

DOI
10.1109/cec.2012.6252948

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In this paper we develop qubit Hamiltonian single parameter estimation techniques using Bayesian approach. The algorithms considered are restricted to projective measurements in a fixed basis, and are derived under the assumption that the qubit measurement is much slower than the characteristic qubit evolution. The non-adaptive algorithm is optimized using particle swarm optimization and compared to previously developed locally-optimal scheme.
Using Bayesian experimental design techniques, we have shown that for a single twolevel quantum mechanical system under strong (projective) measurement, the dynamical parameters of a model Hamiltonian can be estimated with exponentially improved accuracy over offline estimation strategies. To achieve this, we derive an adaptive protocol which finds the optimal experiments based on previous observations. We show that the risk associated with this algorithm is close to the global optimum, given a uniform prior. Additionally, we show that sampling at the Nyquist rate is not optimal.
We investigate schemes for Hamiltonian parameter estimation of a two-level system using repeated measurements in a fixed basis. The simplest (Fourier based) schemes yield an estimate with a mean-square error (MSE) that decreases at best as a power law $\ensuremath{\sim}{N}^{\ensuremath{-}2}$ in the number of measurements NNN. By contrast, we present numerical simulations indicating that an adaptive Bayesian algorithm, where the time between measurements can be adjusted based on prior measurement results, yields a MSE which appears to scale close to $\mathrm{exp}(\ensuremath{-}0.3N)$. That is, measurements in a single fixed basis are sufficient to achieve exponential scaling in NNN.
To improve the efficiency of particle swarm optimization, a quantum particle swarm optimization algorithm is proposed on the basis of analyzing the search process of particle swarm optimization algorithm. In the proposed algorithm, particles are endoded by qubits described on the Bloch sphere, each particle occupy three locations of the search space, and each location represents a optimization solution. By employing the search method of general PSO to adjust the two parameters of qubit, the qubits rotation are performed on the Bloch sphere, which can simultaneously update three locations occupied by a qubit and quickly approach the global optimal solution. The experimental results of standard test function extreme optimization and fuzzy controller parameters optimization show that the proposed algorithm is superior to other similar algorithm in optimization ability and optimization efficiency.
To improve the efficiency of Particle Swarm Optimization(PSO),a quantum particle swarm optimization algorithm combined with quantum theory on the basis of Bloch sphere was proposed.In Bloch spherical coordinates,the particle automatically updated rotation angle and particle position,without setting the rotation angle in the form of look-up table(or setting fixed value of the interval),making up for the deficiency of quantum evolutionary algorithm and quantum genetic algorithm on the basis of Bloch sphere,and the algorithm is more generalizable.Using quantum Hadamard gate to realize the variation of particle enhanced the diversity of population,and prompted particle jump out of local extreme value.The simulation results of the typical function optimization problem show that the algorithm is stable with high precision and fast convergence rate,and it is practical.