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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In particular, Ref. 15 shows that if an efficient algorithm achieves this approximation ratio then there exists an algorithm which solves any decision problem in NP on input size n using runtime npoly(log(n)). This is believed to be very unlikely.
Given a function Eq. (1), we can add an auxiliary variable y ∈ {±1} and consider the function of n + 1 variables G(x, y) = xTBx + ybTx which only contains quadratic terms. It is then easily seen that the range of G is equal to the range of F.
We note that all our results apply also to the problem of minimizing the energy of H and approximating the minimum eigenvalue λmin(H) = −λmax(−H) which is more relevant in many-body physics. We opted to consider a maximization problem to avoid a proliferation of minus signs.
The statement of Theorem 2 with H1 = 0 follows from Eq. (1.1) of Ref. 20 by considering spin-1/2 particles and taking the zero temperature limit.
The fact that the product states in Theorems 1 and 2 can be taken to have this special form is a consequence of the fact that the six eigenstates of single-qubit Pauli operators {X, Y, Z} form a 2-design. Any other single-qubit 2-design could alternatively be used in its place, such as, for example, the 4-state one which is used for similar purposes in Ref. 19.
Recall that the sign of h has to be flipped if one is interested in the minimum rather than maximum eigenvalue.
Each encoded logical operator Xa, Ya, Za commutes with ba = ic3a − 2c3a − 1c3a for each 1 ≤ a ≤ N and {bj, bk} = 2δjk. The Hamiltonian acts on a Hilbert space , where is the N-qubit system with logical operators {Xa, Ya, Za: 1 ≤ a ≤ N}, and is a system of N Majorana fermions {ba: 1 ≤ a ≤ N}. To complete the proof, note that the Hamiltonian acts trivially on , which has dimension 2n/2.
