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 $g$ for the Fermi-sea and BCS-like trial states at half filling. The success rate is found to decay exponentially for small $|g|$, but remains finite in the $|g|\to\infty$ limit, with increasing $|g|$. 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.

💡 Research Summary

This paper presents a novel quantum-classical hybrid scheme for implementing the Gutzwiller wave function, a conceptually simple yet powerful variational ansatz, for attractive SU(3) fermion systems on a quantum computer. The central challenge addressed is the nonunitary nature of the Gutzwiller projection operator, which is not directly executable on quantum hardware. The authors overcome this by employing a discrete Hubbard-Stratonovich (HS) transformation, specifically tailored for the three-component interaction. This transformation decouples the density-density interaction term, expressing the nonunitary Gutzwiller operator e^{-gD̂} as a linear combination of unitary operators, e^{±λ(iσ^y_iαβ)}, which correspond to on-site color-flip processes between fermion species α and β. These unitaries can be efficiently realized on a quantum computer using two-qubit fermionic Givens rotation gates.

The core of the work lies in developing two complementary methods to handle the sum over the auxiliary fields introduced by the HS transformation. The first method is a probabilistic state preparation scheme using a Linear Combination of Unitaries (LCU) circuit. It employs ancillary qubits to coherently sum over all auxiliary field configurations. The desired Gutzwiller wave function is then prepared on the main register qubits by projectively postselecting on a specific measurement outcome of the ancillas. The authors analytically and numerically analyze the success probability of this postselection as a function of the variational parameter g for both Fermi-sea and BCS-like trial states, finding it decays exponentially for small |g| but remains finite in the strong correlation limit (|g|→∞).

The second method circumvents the need for ancilla qubits and the associated postselection overhead by employing a classical importance sampling technique. Instead of performing the sum coherently on the quantum computer, auxiliary field configurations are sampled classically according to a suitable probability distribution. For each sampled configuration, a corresponding unitary circuit (implementing the product of e^{±λ(iσ^y_iαβ)} operators) is executed on the quantum device to prepare a state, and observables are measured. The expectation values are then estimated by averaging over the samples, weighted by importance factors. This approach is more resource-efficient for near-term quantum devices where qubit count is limited.

The proposed framework is demonstrated through numerical simulations of the attractive SU(3) Hubbard model on small one-dimensional chains and two-dimensional square clusters, calculating energies and triple occupancies. Furthermore, as a proof-of-concept experimental validation, the authors design and implement a quantum circuit for the minimal two-site model. This circuit is executed not only on noisy simulators but also on a trapped-ion quantum computer (Quantinuum H1-1). The experimental results for energy and triple occupancy show good agreement with exact theoretical values within statistical errors, successfully demonstrating the practical feasibility of the scheme.

In summary, this work provides a concrete and scalable pathway to prepare Gutzwiller-type correlated variational states for multi-component fermionic systems on quantum processors. It bridges advanced condensed matter theory (Gutzwiller wave function, HS transformation) with quantum algorithm design (LCU, importance sampling), offering a versatile tool for studying strongly correlated phases like color superfluids and trionic states in SU(3) systems using quantum simulation.