Fig. 2: OBP experiments with 75 and 127-qubit spin models.

Benchmarking the OBP framework in the simulation of the one-dimensional XY model of 75 spins (a) and (b) and the two-dimensional XY model of 127 spins (c) and (d). a/c Expectation of the polarization M at different time steps. The polarization is a conserved quantity (dashed line) under the dynamics of the XY model. Due to noise, the experimental signal decays with the depth of the circuit and can be partially recovered using error mitigation. The signals at different noise amplification (by a factor of 1, 1.5, 2.25 and 3, indicated by bolder to more transparent dashed orange lines, respectively) are extrapolated to obtain the PEA estimate (solid orange lines). Using OBP with 5 Trotter steps backpropagated, the polarization can be measured to a higher accuracy in deep circuits (blue lines). The insets highlight the qubits of ibm_kyiv used to represent the spins. The qubits are initialized in either \(| 0\rangle\) (green circles) or \(| 1\rangle\) (red circles). b/d Dynamics of several individual Z_i under the XY model. The vertical dashed lines indicate Trotter steps at which the expectation values are measured. The orange scatter points indicate results from measurement at 5, 15, and 20 Trotter steps and applying PEA without OBP. The OBP framework helps recover the dynamics of intermediate time values (blue scatter points) from these coarse measurement data. The results agree with the reference values (solid gray lines) obtained via an MPS simulation. All error bars shown were obtained through bootstrapping with 100 batches, and are shown at a 2-σ confidence. The shaded blue region represents the additional L₂ error bound due to the classical approximation of the backpropagated observable.