Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers

Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>A</a:mi><a:mover accent="true"><a:mi>x</a:mi><a:mo>⃗</a:mo></a:mover><a:mo>=</a:mo><a:mover accent="true"><a:mi>b</a:mi><a:mo>⃗</a:mo></a:mover></a:mrow></a:math> on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>κ</d:mi></d:math>) scaling of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>A</e:mi></e:math> matrix. In this work, we address the issue by introducing the postselection-improved HHL (Psi-HHL) framework that operates on a simple yet effective premise: subtracting mixed and wrong signals to extract correct signals while providing the benefit of optimal scaling in the condition number of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>A</f:mi></f:math> (denoted as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>κ</g:mi></g:math>) for large <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>κ</h:mi></h:math> scenarios. This approach, which leads to minimal increase in circuit depth, has the important practical implication of having to use substantially fewer shots relative to the traditional HHL algorithm. The term “signal” refers to a feature of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mo>|</i:mo><i:mi>x</i:mi><i:mo>〉</i:mo></i:mrow></i:math>. We design circuits for overlap and expectation value estimation in the Psi-HHL framework. We demonstrate performance of Psi-HHL via numerical simulations. We carry out two sets of computations, where we go up to 26-qubit calculations, to demonstrate the ability of Psi-HHL to handle situations involving large-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mi>κ</j:mi></j:math> matrices via (a) a set of toy matrices, for which we go up to size <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mn>64</k:mn><k:mo>×</k:mo><k:mn>64</k:mn></k:mrow></k:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>κ</l:mi></l:math> values of up to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>≈</m:mo><m:mn>1</m:mn><m:mo>×</m:mo><m:msup><m:mn>10</m:mn><m:mn>6</m:mn></m:msup></m:mrow></m:math>, and (b) application to quantum chemistry, where we consider matrices up to size <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mn>256</n:mn><n:mo>×</n:mo><n:mn>256</n:mn></n:mrow></n:math> that reach <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>κ</o:mi></o:math> of about 393. The molecular systems that we consider are <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:msub><p:mi>Li</p:mi><p:mn>2</p:mn></p:msub></p:math>, KH, RbH, and CsH.