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

Abstract

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.

Topics & Concepts