We used a cost function to quantify the difference between the target BAS distribution and the experimental measurements of the circuit. The cost functions used to implement the training are variants of the original DKL (26)DKL(p,q)=ip(i)log q(i)p(i)(3)

Here, p and q are two distributions. DKL(p, q) is an information theoretic measure of how two probability distribution differ. If base 2 for the logarithm is used, then it quantifies the expected number of extra bits required to store samples from p when an optimal code designed for q is used instead. It can be shown that DKL(p, q) is nonnegative and is zero if and only if p = q. However, it is asymmetric in the arguments and does not satisfy the triangle inequality. Therefore, DKL(p, q) is not a metric.

The DKL is a very general measure, but it is not always well defined, e.g., if an element of the domain is supported by p and not by q, then the measure will diverge. This problem may occur quite often if DKL(p, q) is estimated from samples and if the dimensionality of the domain is large. For PSO, we used the clipped negative log-likelihood cost function (16),Cnll=ip(i)log {max [ϵ,q(i)]}(4)

Here, we set p as the target distribution. Thus, Eq. 4 is equivalent to Eq. 3 up to a constant offset, so the optimization of these two functions is equivalent. ϵ is a small number (0.0001 here) used to avoid a numerical singularity when q(i) is measured to be zero. For BO, we used the clipped symmetrized DKL as the cost functionD˜KL(p,q)=DKL[max (ϵ,p),max (ϵ,q)]+DKL[max (ϵ,q),max (ϵ,p)](5)

This is found to be the most reliable variant of DKL for BO.

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