Fig. 6: Generation of one training data sample.
From: Quantum process tomography with unsupervised learning and tensor networks
a First, we sample a random input POVM state α = (α1, α2, …, αN) from a reference prior distribution Q(α). b The string α specifies an input product state \({{{{{{{{\boldsymbol{\rho }}}}}}}}}_{{{{{{{{\boldsymbol{\alpha }}}}}}}}}={t}_{{{{{{{{\boldsymbol{\alpha }}}}}}}}}^{-1}{{{{{{{{\boldsymbol{M}}}}}}}}}_{{{{{{{{\boldsymbol{\alpha }}}}}}}}}\) to the channel. c The output state of the channel is obtained by contracting the input state with the circuit MPO U, resulting into a new MPO \({{{{{{{\mathcal{E}}}}}}}}({{{{{{{{\boldsymbol{\rho }}}}}}}}}_{{{{{{{{\boldsymbol{\alpha }}}}}}}}})\). d The measurement POVM Mβ. e The process probability distribution \({P}_{{{{{{{{\mathcal{E}}}}}}}}}({{{{{{{\boldsymbol{\beta }}}}}}}}|{{{{{{{\boldsymbol{\alpha }}}}}}}})={{{{{{{{\rm{Tr}}}}}}}}}_{{{{{{{{\boldsymbol{\tau }}}}}}}}}[{{{{{{{{\boldsymbol{M}}}}}}}}}_{{{{{{{{\boldsymbol{\beta }}}}}}}}}{{{{{{{\mathcal{E}}}}}}}}({{{{{{{\boldsymbol{{\rho }}}}}}}_{{{{{{{{\boldsymbol{\alpha }}}}}}}}}}})]\). f Sampling scheme to obtain a single measurement outcome β from \({P}_{{{{{{{{\mathcal{E}}}}}}}}}({{{{{{{\boldsymbol{\beta }}}}}}}}|{{{{{{{\boldsymbol{\alpha }}}}}}}})\). By tracing the indices β2, …, βN (i.e., contracting with a vector [1, …, 1]), the resulting tensor network with one open index is the probability \({P}_{{{{{{{{\mathcal{E}}}}}}}}}({\beta }_{1}\,|\,{{{{{{{\boldsymbol{\alpha }}}}}}}})\), which can be sampled to generate a measurement outcome \({\bar{\beta }}_{1}\). By sweeping left to right, this procedure is repeated for each qubits, generating an outcome \(\bar{{{{{{{{\boldsymbol{\beta }}}}}}}}}\) from the correct probability distribution \({P}_{{{{{{{{\mathcal{E}}}}}}}}}({{{{{{{\boldsymbol{\beta }}}}}}}}|{{{{{{{\boldsymbol{\alpha }}}}}}}})\). The final result of this procedure is one single training sample (α, β). The data set are generated by repeating these steps consecutively.
