Measuring the Loschmidt Amplitude for Finite-Energy Properties of the Fermi-Hubbard Model on an Ion-Trap Quantum Computer
Kévin Hémery, Khaldoon Ghanem, Eleanor Crane, Sara Campbell, Joan Dreiling, Caroline Figgatt, C. B. Foltz, John Gaebler, Jacob Johansen, Michael Mills, Steven A. Moses, Juan Miguel Rey Pino, Anthony Ransford, M. W. Rowe, Peter Siegfried, Russell Stutz, Henrik Dreyer, Alexander Schuckert, Ramil Nigmatullin
Abstract
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