Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules

Studies on quantum algorithms for ground-state energy estimation often assume perfect ground-state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here, we address that problem in two ways: by faster preparation of matrix-product-state (MPS) approximations and by more efficient filtering of the prepared state to find the ground-state energy. We show how to achieve unitary synthesis with a Toffoli complexity about <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>7</a:mn><a:mo>×</a:mo></a:math> lower than that in prior work and use that to derive a more efficient MPS-preparation method. For filtering, we present two different approaches: sampling and binary search. For both, we use the theory of window functions to avoid large phase errors and minimize the complexity. We find that the binary-search approach provides better scaling with the overlap at the cost of a larger constant factor, such that it will be preferred for overlaps less than about <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>0.003</c:mn></c:math>. Finally, we estimate the total resources to perform ground-state energy estimation of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Fe</e:mi></e:math>-<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mrow><g:mi mathvariant="normal">S</g:mi></g:mrow></g:mrow></g:math> cluster systems, including the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mi>Fe</j:mi><j:mi>Mo</j:mi></j:mrow></j:math> cofactor by estimating the overlap of different MPS initial states with potential ground states of the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mi>Fe</l:mi><l:mi>Mo</l:mi></l:mrow></l:math> cofactor using an extrapolation procedure. With a modest MPS bond dimension of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mn>4000</n:mn></n:math>, our procedure produces an estimate of approximately <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mn>0.9</p:mn></p:math> overlap squared with a candidate ground state of the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mrow><r:mi>Fe</r:mi><r:mi>Mo</r:mi></r:mrow></r:math> cofactor, producing a total resource estimate of <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mn>7.3</t:mn><t:mo>×</t:mo><t:msup><t:mn>10</t:mn><t:mn>10</t:mn></t:msup></t:math> Toffoli gates; neglecting the search over candidates and assuming the accuracy of the extrapolation, this validates prior estimates that have used perfect ground-state overlap. This presents an example of a practical path to prepare states of high overlap in a challenging-to-compute chemical system.