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The learning distributions
This protocol is extracted from research article:
Experimental learning of quantum states
Sci Adv, Mar 29, 2019;

Procedure

We used different learning distributions for the two experimental setups, $D(I)$ and $D(II)$. The distribution $D(I)$ is uniform over the set of stabilizer measurements (36) of the GHZ state minus the identity matrix. The distribution $D(II)$ is uniform over the set of stabilizer measurements in X and Z of the GHZ state minus the identity matrix. A GHZ state (20) is a type of stabilizer state. A stabilizer state |ψ〉 is the unique eigenstate with eigenvalue +1 of a set of N commuting multilocal Pauli operators Pis, that is, Pi|ψ〉 = |ψ〉, where Pi = ⊗ jwj and wj ∈ {I, σx, σy, σz} are the Pauli matrices. We define the Pi as the stabilizers of the state.

There are 2n different stabilizers for an n-qubit stabilizer state. Because one of the stabilizers is always the identity (whose eigenvalue is 1 for every state), we chose not to include this measurement in those sampled by D.

Each Pi is a two-outcome observable (with eigenvalues +1 or −1). We constructed the POVM elements $Ei(1)$ and $Ei(2)$ of the observable Pi by noting that $Ei(1)+Ei(2)=I$ and $Ei(1)−Ei(2)=Pi$. The POVM element $Ei(1)$ can be then written as $Ei(1)=(I+Pi)/2$.

The set of stabilizers of a state form a group under the operation of matrix multiplication. To represent a state, it is then sufficient to consider the n stabilizers that generate this group. For an n-qubit state, there are n elements in the set of generators.

The high variance around m = 4 in Fig. 3 can be explained in the following way: Each data point was obtained by averaging over a number of different configurations sampled from $D(I)$. It is then likely to sample a configuration that includes two generators and two other stabilizers that can be obtained by the product of the generators. It is easy to see how the information content of such a configuration is less than the one where four independent stabilizers are sampled. This will, in turn, limit the ability of σ to output good predictions and will generate the high variance in the data.

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