Compressed sensing for Hamiltonian reconstruction

In engineered quantum systems, the Hamiltonian is often not completely known and needs to be determined experimentally with accuracy and efficiency. We show that this may be done at temperatures that are higher than the characteristic interaction energies, but not too much higher. The condition for this is that there are not too many multiparticle interactions: the Hamiltonian is sparse in a well-defined sense. The protocol that accomplishes this is related to compressed sensing methods of classical signal processing, in this case applied to sparse rather than low-rank matrices.

Figure 1

Quality of Hamiltonian determination for random couplings as a function of temperature. (a) and (b) The inverse dimensionless temperature is given by $\eta \beta ={10}^{-1}$ and $\eta \beta ={10}^{-4}$, respectively. Red indicates a high success rate, green indicates failure, and a “negative” success rate (blue) means that reconstruction is impossible. [That is, heat map values between 0 (green) and 1 (red) are to be taken as probabilities on a linear scale, while heat map values of $-1$ (blue) indicate that not only would reconstruction fail for our proposed procedure, but reconstruction would fail given any procedure, as these blue points correspond to measuring fewer observables than there are nonzero entries to be reconstructed.] Each pixel is the average of 100 trials.

Figure 2

Quality of Hamiltonian determination as a function of number of qubits and measured observables. (a)–(c) For Hamiltonians with entirely random couplings; (d)–(f) for Hamiltonians with random one- and two-qubit couplings only. (a) and (d) $n=3$; (b) and (e) $n=4$; (c) and (f) $n=5.$ Each plot gives the error in the estimated Hamiltonian as a function of the number of observables measured $\left(M\right)$, with compressed sensing (CS) and without (no CS), for varying small values of $s$ (number of nonzero couplings). [While the minimum error obtained by the no-CS protocol appears to be lower than the minimum error obtained with the CS protocol ($\sim {10}^{-14}$ and $\sim {10}^{-9}$, respectively), these differences are simply artifacts of the different noise floors of different numerical methods. For each algorithm, achieving its respective noise floor indicates that the Hamiltonian has been successfully reconstructed.] In each case, the CS protocol substantially decreases the $M$ required to accurately reconstruct the Hamiltonian; this improvement is more significant when the couplings are totally random. Additionally, we see that for both coupling schemes, the improvement increases with the number of qubits. Each data point is the median value of 100 trials.