We discuss classical algorithms for approximating the largest eigenvalue of quantum spin and fermionic Hamiltonians based on semidefinite programming relaxation methods. First, we consider traceless 2-local Hamiltonians H describing a system of n qubits. We give an efficient algorithm that outputs a separable state whose energy is at least λmax/O(log n), where λmax is the maximum eigenvalue of H. We also give a simplified proof of a theorem due to Lieb that establishes the existence of a separable state with energy at least λmax/9. Second, we consider a system of n fermionic modes and traceless Hamiltonians composed of quadratic and quartic fermionic operators. We give an efficient algorithm that outputs a fermionic Gaussian state whose energy is at least λmax/O(n log n). Finally, we show that Gaussian states can vastly outperform Slater determinant states commonly used in the Hartree-Fock method. We give a simple family of Hamiltonians for which Gaussian states and Slater determinants approximate λmax within a fraction 1 − O(n−1) and O(n−1), respectively.
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Research Article| March 29 2019
Approximation algorithms for quantum many-body problems
Sergey Bravyi, David Gosset, Robert König, Kristan Temme; Approximation algorithms for quantum many-body problems. J. Math. Phys. 1 March 2019; 60 (3): 032203. https://doi.org/10.1063/1.5085428
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