Approximate diagonalization method for large-scale Hamiltonians

Physical Review A (Impact Factor: 2.81). 02/2012; 86(5). DOI: 10.1103/PhysRevA.86.052314
Source: arXiv


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 the results of our method with those obtained using
exact diagonalization and quantum Monte Carlo calculation for random problem
instances with up to 128 qubits.

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    • "Finally an important motivation of the current work is the presentation of precise numerical data for the energy gap at the quantum transition, which can be confronted to other approaches that deal with quantum search complexity. These include the semi-perturbative evaluation of the quantum energy gap [28], Quantum Monte Carlo annealing [11] [29] [15] [30] as well as annealing studies under Schrödinger wave function dynamics [31] [32]. We expect that algorithmic studies e.g. "
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    ABSTRACT: Using a recently constructed ensemble of hard 2SAT realizations, that has a unique ground-state we calculate for the quantized theory the median gap correlation length values $\xi_{GAP}$ along the direction of the quantum adiabatic control parameter $\lambda$. We use quantum annealing (QA) with transverse field and a linear time schedule in the adiabatic control parameter $\lambda$. The gap correlation length diverges exponentially $\xi_{\rm GAP} \propto {\rm exp} [+r_{\rm GAP}N]$ in the median with a rate constant $r_{\rm GAP}=0.553(6)$, while the run time diverges exponentially $\tau_{\rm QA} \propto {\rm exp} [+r_{\rm QA}N]$ with $r_{\rm QA}=1.184(16)$. Simulated classical annealing (SA) exhibits a run time rate constant $r_{\rm SA}=0.340(5)$ that is small and thus finds ground-states exponentially faster than QA. There are no quantum speedups in ground state searches on constant energy surfaces that have exponentially large volume. We also determine gap correlation length distribution functions $P(\xi_{\rm GAP})d\xi_{\rm GAP} \approx W_k$ over the ensemble that at $N=18$ are close to Weibull functions $W_k$ with $k \approx 1.2$ i.e., the problems show thin catastrophic tails in $\xi_{\rm GAP}$. The inferred success probability distribution functions of the quantum annealer turn out to be bimodal.


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