Preparing the Gutzwiller wave function for attractive SU(3) fermions on a quantum computer

We implement the Gutzwiller wave function for attractive SU(3) fermion systems on a quantum computer using a quantum-classical hybrid scheme based on the discrete Hubbard-Stratonovich transformation. In this approach, the nonunitary Gutzwiller operator is decomposed into a linear combination of unitaries constructed from two-qubit fermionic Givens-rotation gates, whose rotation angles are dictated by the auxiliary fields. We develop and reformulate two complementary methods to perform the sum over these auxiliary fields. In the first method, the Gutzwiller wave function is probabilistically prepared on the register qubits by projectively postselecting the desired state via measurements of ancilla qubits. We analyze the success rate both analytically and numerically as a function of the Gutzwiller variational parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>g</a:mi></a:math> for the Fermi-sea and BCS-like trial states at half filling. The success rate is found to decay exponentially for small <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>|</b:mo><b:mi>g</b:mi><b:mo>|</b:mo></b:mrow></b:math>, but remains finite in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mo>|</c:mo><c:mi>g</c:mi><c:mo>|</c:mo><c:mo>→</c:mo><c:mi>∞</c:mi></c:mrow></c:math> limit, with increasing <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>|</d:mo><d:mi>g</d:mi><d:mo>|</d:mo></d:mrow></d:math>. In the second method, we employ importance sampling to address the Gutzwiller variational problem, where the central objective is to estimate the expectation values of observables. We demonstrate the proposed scheme by calculating the energy and triple occupancy of the attractive SU(3) Hubbard model in the framework of digital quantum simulation. Moreover, we present experimental results obtained on a trapped-ion quantum computer for the two-site attractive SU(3) Hubbard model, showing good agreement with exact values within statistical errors.