Efficient Simulation of Quantum Chemistry Problems in an Enlarged Basis Set

We propose a quantum algorithm to simulate the dynamics in quantum chemistry problems. It is based on adding fresh qubits at each Trotter step, which enables a simpler implementation of the dynamics in the extended system. After each step, the extra qubits are recycled, so that the whole process accurately approximates the correct unitary evolution. A key ingredient of the approach is an isometry that maps a simple diagonal Hamiltonian in the extended system to the original one, and we give a procedure to compute this isometry. We estimate the error at each time step, as well as the number of gates, which scales as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:msup><a:mi>N</a:mi><a:mn>2</a:mn></a:msup><a:mo stretchy="false">)</a:mo></a:math>, where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><f:mi>N</f:mi></f:math> is the number of orbitals. We illustrate our results with three examples: the hydrogen chain, small molecules, and the FeMoco (<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><i:mi>Fe</i:mi></i:math><l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><l:msub><l:mi/><l:mn>7</l:mn></l:msub></l:math><o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><o:mrow><o:mi>Mo</o:mi><o:mi mathvariant="normal">S</o:mi></o:mrow></o:math><s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><s:msub><s:mi/><s:mn>9</s:mn></s:msub></s:math><v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><v:mrow><v:mrow><v:mi mathvariant="normal">C</v:mi></v:mrow></v:mrow></v:math>) molecule. In the hydrogen chain and the hydrogen molecule, we observe that the error scales in the same way as the Trotter error. For FeMoco, we estimate the number of gates in a fault-tolerant setup.