Faster electronic structure quantum simulation by spectrum amplification

We discover that many interesting electronic structure Hamiltonians have a compact and close-to-frustration-free sum-of-squares representation with a small energy gap. We show that this gap enables spectrum amplification in estimating ground state energies, which improves the cost scaling of previous approaches from the block-encoding normalization factor $\lambda$ to just $\sqrt{\lambda E_{\text{gap}}}$. For any constant-degree polynomial basis of fermionic operators, a sum-of-squares representation with optimal gap can be efficiently computed using semi-definite programming. Although the gap can be made arbitrarily small with an exponential-size basis, we find that the degree-$2$ spin-free basis in combination with approximating two-body interactions by a new Double-Factorized (DF) generalization of Tensor-Hyper-Contraction (THC) gives an excellent balance of gap, $\lambda$, and block-encoding costs. For classically-hard FeMoco complexes -- candidate applications for first useful quantum advantage -- this combination improves the Toffoli gates cost of the first estimates with DF [Phys. Rev. Research 3, 033055] or THC [PRX Quantum 2, 030305] by over two orders of magnitude.

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