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Here we present the details of training our RBM. The neural network parametrization for the wavefunction is defined by Eqs. (4) and (5). We introduce additional notation. First, because there is one-to-one correspondence between visible layer configurations and Fock states , as discussed in the main text, we will use the symbol to denote both these objects. Second, we denote the unnormalized Boltzmann probability
with , for the amplitude RBM, and the analogous quantity
for the phase RBM. We remind the reader that the letters and denote the respective parameter sets of these RBMs.
’s harmonic oscillator eigenfunction (Hermite-Gaussian polynomial)with a phase factor,
we obtain the following expression:
where , the summation with index is over all quadrature measurements and is the number of measurements.
For the network training, we evaluate the gradients of the above log-likelihood over the neural net parameters and as follows:
where we defined
with and . Ascending by these gradients, we can maximize the log-likelihood (11). Both RBMs are trained simultaneously.
We note that the above gradients contain exhaustive summation over possible configutations of the visible and hidden layers of both RBMs. In the present work, we are able to compute this sum directly since the number of RBM units is relatively small. However, in the case of high Hilbert space dimension, Boltzmann sampling using an annealing device or algorithm will be required.
As discussed in the main text, see Eq. (7), we treat the mixed state to be reconstructed as a partial state of a pure state in a tensor product Hilbert space with the dimension . We decompose this state in the Fock basis and apply the same parametrization as in the previous subsection:
The partial trace of this state over the environment is as follows:
The log-likelihood (6) is then given by
where the summation indices run over the truncated Fock basis, over all quadrature measurements, and
We note that the expression (17) is very similar to the pure state case (11), but requires two additional summations over the truncated Fock basis. The log-likelihood (17) gradients over and read similarly to those for the pure state (12), but with the parameters (13) redefined as follows:
The remainder of the treatment replicates that for pure states.
To correct for an imperfect homodyne detector efficiency in our neural net approach, we model it as a perfect detector preceded by beam splitter of transmission Lvovsky2004 , which changes the quantum state by means of generalized Bernoulli transformation to a new state according to
where . Now we can repeat the above procedure for the mixed state (purification) Ansatz, with the only difference that we use instead of to calculate the log-likelihood (6).