Hamiltonian Learning and Certification Using Quantum Resources

In recent years quantum simulation has made great strides, culminating in experiments that existing supercomputers cannot easily simulate. Although this raises the possibility that special purpose analog quantum simulators may be able to perform computational tasks that existing computers cannot, it also introduces a major challenge: certifying that the quantum simulator is in fact simulating the correct quantum dynamics. We provide an algorithm that, under relatively weak assumptions, can be used to efficiently infer the Hamiltonian of a large but untrusted quantum simulator using a trusted quantum simulator. We illustrate the power of this approach by showing numerically that it can inexpensively learn the Hamiltonians for large frustrated Ising models, demonstrating that quantum resources can make certifying analog quantum simulators tractable.

Figure 1

Flow chart for Hamiltonian learning algorithm.

Figure 2

IQLE. (Upper) Circuit diagram for drawing experimental data. (Lower) Circuit diagram for likelihood evaluation. QLE is similar but with $H_{-}=0$, no swap gate is used and the untrusted system is measured directly.

Figure 3

The quadratic loss plotted as a function of the number of inversion experiments for Ising models on the complete graph. The shaded areas show a 50% confidence interval for the quadratic loss.

Figure 4

The median decay exponent for the quadratic loss as a function of the number of parameters in the Ising model, $d$.

Figure 5

An approximate 1 parameter Ising model on the complete graph. The thin lines give the best fits to the exponential decays, which scale as $e^{-0.07N}$ and $e^{-0.23N}$ for $n=4$ and as $e^{-0.029N}$ and $e^{-0.23N}$ for $n=6$ qubits.