Systematic comparison of deep belief network training using quantum annealing vs. classical techniques

A Survey on Quantum Computational Finance for Derivatives Pricing and VaR

“…where k denotes a generic integer. If one chooses k in (51) such that (2k + 1) ∼ 2 , then sin (2k + 1) ∼ 1 and so one maximizes the probability of getting a result along � Ψ1 ⟩ upon measurement. We have therefore reached the initial purpose of amplifying the probability of measuring � Ψ1 ⟩ in a way which explicitly depends on .…”

“…We have therefore reached the initial purpose of amplifying the probability of measuring � Ψ1 ⟩ in a way which explicitly depends on . Said otherwise, by measuring (51) we have a more efficient access to the estimation of a.…”

“…Once we have the amplified state (51), we still need to actually estimate a in the best possible way. We stress once more that (51) is the result of rotations-implemented by the Grover operator Q-aimed to enhance in a controlled way the probability to obtain � Ψ1 ⟩ upon measurement. So, a sampling by repeated preparations and measurements of (51), instead of the original vector �Ψ⟩ , already provides an advantage, although not quadratic.…”

“…One collects data corresponding to measuring states amplified by means of Q k with different k (see Fig. 4 for the circuit needed for a specific k); then one compares this dataset against a suitable classical prior distribution which depends on the angle (see (51)). Maximizing the likelihood that the distribution fits the data satisfactorily, provides an estimation of , here acting as a variational parameter.…”

“…The advances on training the so-called Boltzmann machines (particularly the Restricted Boltzmann Machines (RBMs)) are especially relevant. Some representative works in this area of interest are [4,5,51,88].…”