Approximate diagonalization method for large-scale Hamiltonians

An approximate diagonalization method is proposed that combines exact diagonalization and perturbation expansion to calculate low-energy eigenvalues and eigenfunctions of a Hamiltonian. The method involves deriving an effective Hamiltonian for each eigenvalue to be calculated, using perturbation expansion, and extracting the eigenvalue from the diagonalization of the effective Hamiltonian. The size of the effective Hamiltonian can be significantly smaller than that of the original Hamiltonian, hence the diagonalization can be done much faster. We compare our approximate diagonalization results with those obtained using exact diagonalization and quantum Monte Carlo calculation for random problem instances with up to 128 qubits.

Figure 1

Hamiltonian parameters $\Delta \left(s\right)$ and $\mathcal{E}\left(s\right)$ as a function of normalized time $s$.

Figure 2

Spectra of (a) 8-qubit and (b) 16-qubit random Ising Hamiltonians with transverse fields, relative to the ground-state energy $E_0$. Solid thin lines represent the approximate diagonalization results and dashed lines represent exact diagonalization results. Circles show the results of the quantum Monte Carlo simulations.

Figure 3

Approximate energy gap (solid lines) of example random Ising instances, with 32 to 128 qubits, compared with results from Monte Carlo simulations (circles).