The learning distributions

AR Andrea Rocchetto SA Scott Aaronson SS Simone Severini GC Gonzalo Carvacho DP Davide Poderini IA Iris Agresti MB Marco Bentivegna FS Fabio Sciarrino

This protocol is extracted from research article:

Experimental learning of quantum states

**
Sci Adv**,
Mar 29, 2019;
DOI:
10.1126/sciadv.aau1946

Experimental learning of quantum states

Procedure

We used different learning distributions for the two experimental setups, ${\mathcal{D}}_{(\mathrm{I})}$ and ${\mathcal{D}}_{(\text{II})}$. The distribution ${\mathcal{D}}_{(\mathrm{I})}$ is uniform over the set of stabilizer measurements (*36*) of the GHZ state minus the identity matrix. The distribution ${\mathcal{D}}_{(\text{II})}$ 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 *P*_{i}s, that is, *P*_{i}|ψ〉 = |ψ〉, where *P*_{i} = ⊗ _{j}*w*_{j} and *w*_{j} ∈ {*I*, σ^{x}, σ^{y}, σ^{z}} are the Pauli matrices. We define the *P*_{i} as the stabilizers of the state.

There are 2^{n} 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 *P*_{i} is a two-outcome observable (with eigenvalues +1 or −1). We constructed the POVM elements ${\mathit{E}}_{\mathit{i}}^{(1)}$ and ${\mathit{E}}_{\mathit{i}}^{(2)}$ of the observable *P*_{i} by noting that ${\mathit{E}}_{\mathit{i}}^{(1)}+{\mathit{E}}_{\mathit{i}}^{(2)}=\mathit{I}$ and ${\mathit{E}}_{\mathit{i}}^{(1)}-{\mathit{E}}_{\mathit{i}}^{(2)}={\mathit{P}}_{\mathit{i}}$. The POVM element ${\mathit{E}}_{\mathit{i}}^{(1)}$ can be then written as ${\mathit{E}}_{\mathit{i}}^{(1)}=(\mathit{I}+{\mathit{P}}_{\mathit{i}})/2$.

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 ${\mathcal{D}}_{(\mathit{I})}$. 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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