Optimal number-conserved linear encoding for practical fermionic simulation

Number-conserved subspace encoding reduces resources needed for quantum simulations, but scalable complexity trade-off bounds for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>M</a:mi></a:math> modes and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>N</b:mi></b:math> particles with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="script">O</c:mi><c:mo>(</c:mo><c:mi>N</c:mi><c:mo form="prefix">log</c:mo><c:mi>M</c:mi><c:mo>)</c:mo></c:mrow></c:math> qubits have remained unknown. We study qubit-gate-measurement trade-offs through the lens of classical/quantum error correction complexity and develop a framework of fermionic gate and measurement complexity based on classical encoder/decoder appearing in the error correction framework. We demonstrate optimal encoding with random classical parity check code and propose the Fermionic Expectation Decoder for scalable probability decoding in <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi mathvariant="script">O</f:mi><f:mo>(</f:mo><f:msup><f:mi>M</f:mi><f:mn>4</f:mn></f:msup><f:mo>)</f:mo></f:mrow></f:math> bases. The protocol is tested with variational quantum eigensolver on LiH in the STO-3G and 6-31G bases, and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mtext>H</h:mtext><h:mn>2</h:mn></h:msub></h:math> potential energy curve in the 6-311G* basis.