Fitness landscape for quantum state tomography from neutron scattering

Abstract

Recently, a direct connection between static structure factors and quantum ground states for two-spin interaction Hamiltonians was proven. This suggests the possibility of quantum state tomography from neutron scattering. Here, we investigate the associated fitness landscape numerically. We find a linear relationship between the mean square distances of the structure factors and the associated state overlaps, implying a well-behaved fitness landscape. Furthermore, we find evidence suggesting that the approach can be generalized to thermal equilibrium states. We also extend the arguments to the cases of applied magnetic fields and finite clusters.

Figures (3)

• Figure 1: Fitness landscape for quantum state tomography using the ground-state's diffuse magnetic neutron scattering function. The target state $\psi$ is the ground state of a spin-1/2 ring-tetramer (L=4) with nearest-neighbour, antiferromagnetic Heisenberg interactions. (a) Cost function $\Delta\mathcal{S}$ as a function of the distance ${\rm dist}(\psi,\psi')$ of the trial state vectors $\psi'$ to the target ground state supp-mat. We display three sets of 1000 random state vectors each, with amplitudes drawn from Gaussian distributions centered on $\psi$. The color legend indicates the maximum standard deviation $\sigma_\text{max}$ used for the set, while each trial state is marked by a filled circle and has components generated from distribution with a random width $\sigma$, where $0<\sigma<\sigma_\text{max}$supp-mat. (b) The target diffuse magnetic scattering function $\mathcal{S}\left(\mathbf{q}\right)$ for the exact ground state. (c-e) Three scattering functions obtained from the three random states highlighted with stars on panel (a), in order of increasing state-distance.
• Figure 2: Dependence of the fitness landscapes on the number of spins $L$ in the cluster. For each $L$, 1000 target states were sampled with normal-distributed perturbations around the target wave function (see main text). Target states were chosen as ground states of: (a) A spin-1/2, nearest-neighbour anti-ferromagnetic Heisenberg model in a ring geometry, i.e., the same model as in Fig. \ref{['fig:fitness_landscape_gs']}, but with varying L. (b) A random long-range spin-1/2 Hamiltonian in a linear open chain geometry, given by Eq. (\ref{['eq:HeisenbergHamField']}) with random exchange constants $J_{i,j}^{\alpha,\beta}$ and a random global magnetic field with components $h_i^{\alpha} = h^{\alpha}$. For each $L$, Hamiltonian parameters $J_{i,j}^{\alpha,\beta}$ and $h^{\alpha}$ were drawn from a normal distribution with zero average and standard deviation equal to $J$ [the nearest-neighbour coupling of the Heisenberg Hamiltonian used in panel (a)]. Dashed lines in (a,b) show the linear relation in Eq. (\ref{['eq:straight_line']}).
• Figure 3: Dependence on $L$ (see legend) of the fitness landscape for finite-temperature quantum tomography for a model with random long-range interactions. The target Hamiltonian and geometry are defined in the same way as in Fig. \ref{['fig:Detailed_fitness_gs-both']}b). The target density matrix $\rho$ is the thermal equilibrium density matrix of that Hamiltonian at temperature $T=J$. 1000 trial configurations (random density matrices) were tested against this target, based on adding noise terms followed by restoring positive definiteness and normalization supp-mat. The horizontal axis shows a distance between random density matrices that for $T=0$ would have coincided with the overlap-induced distance $dist(\psi,\psi')$ between ground states supp-mat. The dashed line represents Eq. (\ref{['eq:straight_line']}), for comparison. The inset shows similar output with different random targets at each size.