Double Stochastic Gradient Descent

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In the Double Stochastic Gradient Descent method. The Hamiltonian was given as

H = 4 +2I \otimes X + 4 I ⊗ Z −X⊗X + 5 Y ⊗ Y + 2 Z ⊗ X

And so the idea of “Double” stochastic means when we iterate through our SGD, we only sample the Pauli terms from H and evaluate the expectation of those terms?

For instance, we can set to sample 3 out of 5 Pauli terms at each iteration. At the first iteration, we might sample, IX,XX, ZX then at the next step we might sample IX, XX, YY,… and so on. Am I thinking of this right? If so, how many terms were set to get the results presented below?

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Thanks!

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And so the idea of “Double” stochastic means when we iterate through our SGD, we only sample the Pauli terms from H and evaluate the expectation of those terms?

@KAJ226, exactly right! The stochasticity comes from two sources:

  1. The finite number of shots
  2. The sampling of a random subset of the Hamiltonian terms

Hence ‘doubly stochastic’.

In this particular example, the number of sampled Hamiltonian terms was set to 1, via n=1 in the below snippet:

def loss(params):
    return 4 + (5 / 1) * circuit(params, n=1)
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