Fermion determinants on a quantum computer

We present a quantum algorithm to compute the logarithm of the determinant of the fermion matrix, assuming access to a classical lattice gauge field configuration. The algorithm uses the quantum eigenvalue transform, and quantum mean estimation, giving a query complexity that scales like <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>V</a:mi><a:mi>log</a:mi><a:mo stretchy="false">(</a:mo><a:mi>V</a:mi><a:mo stretchy="false">)</a:mo><a:mo stretchy="false">)</a:mo></a:math> in the matrix dimension <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>V</g:mi></g:math>.