Measuring the Loschmidt Amplitude for Finite-Energy Properties of the Fermi-Hubbard Model on an Ion-Trap Quantum Computer

Calculating the equilibrium properties of condensed-matter systems is one of the promising applications of near-term quantum computing. Recently, hybrid quantum-classical time-series algorithms have been proposed to efficiently extract these properties from a measurement of the Loschmidt amplitude <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mo fence="false" stretchy="false">⟨</a:mo><a:mi>ψ</a:mi><a:mrow><a:mo stretchy="false">|</a:mo></a:mrow><a:msup><a:mi>e</a:mi><a:mrow><a:mo>−</a:mo><a:mi>i</a:mi><a:mrow><a:mover><a:mi>H</a:mi><a:mo stretchy="false">^</a:mo></a:mover></a:mrow><a:mi>t</a:mi></a:mrow></a:msup><a:mrow><a:mo stretchy="false">|</a:mo></a:mrow><a:mi>ψ</a:mi><a:mo fence="false" stretchy="false">⟩</a:mo></a:math> from initial states <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:mrow><k:mo stretchy="false">|</k:mo></k:mrow><k:mi>ψ</k:mi><k:mo fence="false" stretchy="false">⟩</k:mo></k:math> and a time evolution under the Hamiltonian <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><q:mrow><q:mover><q:mi>H</q:mi><q:mo stretchy="false">^</q:mo></q:mover></q:mrow></q:math> up to short times <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><u:mi>t</u:mi></u:math>. In this work, we study the operation of this algorithm on a present-day quantum computer. Specifically, we measure the Loschmidt amplitude for the Fermi-Hubbard model on a <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><x:mn>16</x:mn></x:math>-site ladder geometry (32 orbitals) on the Quantinuum H2-1 trapped-ion device. We assess the effect of noise on the Loschmidt amplitude and implement algorithm-specific error-mitigation techniques. By using a thus-motivated error model, we numerically analyze the influence of noise on the full operation of the quantum-classical algorithm by measuring expectation values of local observables at finite energies. Finally, we estimate the resources needed for scaling up the algorithm. Published by the American Physical Society 2024