Probabilistic Simulation of Quantum Circuits with the Transformer*

In this work, we present an exact probabilistic formulation of quantum dynamics through positive value-operator measurements (POVM). In this formulation, unitary dynamics and quantum channels are represented by quasi-stochastic matrices acting on true probability distributions which specify the quantum state univocally. The probability distribution representation of the quantum state opens up the possibility of bridging the state-of-the-art techniques from machine learning into the simulation of quantum mechanics. Using the POVM formalism, we have developed a practical algorithm for the probabilistic simulation of quantum circuits with the Transformer, a powerful ansatz responsible for the most recent breakthroughs in the natural language processing research. The method is applied to state preparation of GHZ state and Linear Graph state up to 60 qubits, as well as variational quantum circuit preparation of the ground state of the Transverse Field Ising Model.