Fig. 2: Two-dimensional GPs.
We plot the joint probability density function, as well as its scaled marginals, for the measurement outcomes at the output of a unitary Haar-random QNN acting on n = 18 qubits. The measured observable is Oj = Z1, where Z1 denotes the Pauli z operator on the first qubit. Moreover, the input states are for the left column, ρ1 = |0〉 〈0|⊗n and ρ2 = |GHZ〉 〈GHZ| with \(\left\vert {\rm{GHZ}}\right\rangle =\frac{1}{\sqrt{2}}({\left\vert 0\right\rangle }^{\otimes n}+{\left\vert 1\right\rangle }^{\otimes n})\), and for the right column, ρ1 and ρ3 = |Ψ〉 〈Ψ| with \(\left\vert \varPsi \right\rangle =\frac{1}{\sqrt{d}}{\left\vert 0\right\rangle }^{\otimes n}+\sqrt{1-\frac{1}{d}}{\left\vert 1\right\rangle }^{\otimes n}\). In both cases we took 104 samples.
