Fig. 4: Time-evolution circuits.

a Time-evolution circuit: The probability of measuring \({\left|0\right\rangle }^{\otimes N}\) at the output, after post-selecting on the top two qubits on the right-hand side, gives an approximation to λ²—the square of the principal eigenvalue of the transfer matrix (indicated by the dashed red lines). This circuit provides a cost function whose optimisation over \(U^{\prime}\) gives the state at time t + dt after starting at time t with the state parametrised by U. b Factorisation of the MPS and Time-evolution unitaries: The unitary U describing the iMPS quantum state of the system is parametrised on the circuit as shown. This is reduced from the full parametrisation of a two qubit unitary in order to enable shallower circuits. There is an additional redundancy of the first z-rotation on the reference qubit state \(\left|0\right\rangle\) and two further angles contained in the parametrisation do not change through the dynamical quantum phase transition that we study. c Factorisation of the time-evolution unitary: The two-site time-evolution unitary is factorised to the Rainbow gate set as shown. This is one of the more costly parts of the simulation in terms of circuit-depth.