Fig. 4: Many-body spectroscopy of model Hamiltonians.

a, A schematic quantum circuit diagram for the algorithm. The first step is to apply a state-preparation circuit S to prepare a reference state \(\left\vert S\right\rangle =S\left\vert 0\right\rangle\), followed by an ancilla-controlled perturbation R preparaing a superposition of \(\left\vert S\right\rangle\) and the probe state \(\left\vert R\right\rangle =R\left\vert S\right\rangle\). This superposition is time-evolved by the target Hamiltonian, and then each qubit is projectively measured to produce a snapshot. By repeating the procedure for various evolution times t, different perturbations R and potentially different measurement bases, this setup provides access to detailed information about the spectrum of H. b, Consider a spin Hamiltonian H₂ with two anti-ferromagnetically coupled spin-3/2 particles. We simulate 20,000 snapshot measurements and classical processing to calculate the density of states \({D}^{{\mathbb{1}}}(\omega )\) (black line) and total-spin resolved versions \({D}^{{P}_{S}}(\omega )\) (coloured lines). The vertical dashed lines correspond to exact energies, and the coloured regions represent 95% confidence intervals. The peaks are broadened due to finite (coherent) simulation time \(J{T}_{{\rm{sim}}}=0.26\), which sets the spectral resolution. Hardware-efficient simulation schemes, which extend the simulation time (for example, Fig. 3), are favourable because they improve spectral resolution. Many-body spectroscopy further improves the effective spectral resolution, by leveraging multiple observables to distinguish overlapping peaks. Here, we see that spin resolution sparsifies the signal, enabling accurate peak detection and energy estimation, while the bare spectrum \({D}^{{\mathbb{1}}}(\omega )\) is too broad to resolve all states. c, The magnetic susceptibility χ, can also be computed from snapshot measurements using S^z-resolved density of states (here, \(J{T}_{{\rm{sim}}}=1.04\)). For these calculations, it is important to prevent exponential amplification of shot noise. We therefore use a simple empirical truncation procedure that introduces a small amount of bias (Methods) but enables rapid convergence with number of snapshots to the ideal value (black line).