Fast Quantum Simulation of Electronic Structure by Spectral Amplification

The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectral amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectral amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Λ</a:mi> </a:math> to just <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msqrt> <d:mrow> <d:mn>2</d:mn> <d:mi mathvariant="normal">Λ</d:mi> <d:msub> <d:mi>E</d:mi> <d:mi>gap</d:mi> </d:msub> </d:mrow> </d:msqrt> </d:math> , where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>E</g:mi> <g:mi>gap</g:mi> </g:msub> <g:mo>≪</g:mo> <g:mi mathvariant="normal">Λ</g:mi> </g:math> is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes—namely, double factorization and tensor hypercontraction—that when combined with spectral amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of iron-sulfur complexes and a <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mrow> <j:msub> <j:mrow> <j:mi>CO</j:mi> </j:mrow> <j:mrow> <j:mn>2</j:mn> </j:mrow> </j:msub> </j:mrow> </j:math> -fixation catalyst.