## Abstract

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 λ _{max} /O(log n), where λ _{max} 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 λ _{max} /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 λ _{max} /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 λ _{max} within a fraction 1 − O(n ^{−1} ) and O(n ^{−1} ), respectively.

Original language | English |
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Article number | 032203 |

Journal | Journal of Mathematical Physics |

Volume | 60 |

Issue number | 3 |

DOIs | |

State | Published - 1 Mar 2019 |