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

TL;DR

This work extends a quantum-classical hybrid scheme for preparing the nonunitary Gutzwiller wave function to attractive SU(3) fermions by expressing the Gutzwiller operator as a linear combination of fermionic Givens rotations via a discrete HS transformation. It develops two complementary implementations: a probabilistic ancilla-postselection method and a measurement-based importance-sampling approach, both validated on small lattices and demonstrated experimentally on a two-site trapped-ion quantum computer with results that agree with exact solutions within uncertainties. The study provides explicit circuit decompositions, resource estimates, and practical guidance for scaling up to larger lattices, highlighting the role of correlated initial states to mitigate exponential decay of the postselection success probability. These methods enable variational investigations of SU(3) color-superfluid and trionic physics on quantum hardware, with potential applications to ultracold-atom simulations and beyond.

Abstract

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.

Figures (6)

• Figure 1: (a) Decomposition of the fermionic Givens-rotation gate $\widehat{f\mathcal{G}\mkern1mu}_{ij}(\lambda)$ into a product of CZ gates and a standard Givens-rotation gate. The qubit indicated by $\delta_i$ refers to a qubit located between qubits $i$ and $j$, adjacent to qubit $i$. A solid line connecting two crosses denotes a SWAP gate. (b) Decomposition of the Givens-rotation gate $\hat{\cal G}_{ij}(\lambda)$ into two CZ gates and eight single-qubit rotation gates. Here, the single-qubit rotation gate $R_z(\theta)$ is defined as $R_z(\theta) = e^{-\mathrm{i}\mkern1mu\theta\hat{Z}/2}$, and other rotation gates are defined analogously.
• Figure 2: (a) Quantum circuits for evaluating the linear combination $\widehat{f\mathcal{G}\mkern1mu}(\lambda)+\widehat{f\mathcal{G}\mkern1mu}(-\lambda)$ using a Hadamard-test-like setup. $H$ denotes the Hadamard gate. Green triangles represent ancillary qubits initialized in the $\ket{0}$ state and the yellow square indicates the measurement in the Pauli basis. (b) Quantum circuit for generating the Gutzwiller wave function $\ket{\psi_g}$ using $3N_{\text{site}}$ ancillary qubits and $3N_{\text{site}}$ register qubits, shown for the case of $N_{\text{site}}=2$. The label $i_{\alpha}$ refers to the qubit encoding the fermion with color $\alpha$ at site $i$, based on the Jordan-Wigner transformation. The ancillary qubits are initialized in the $\ket{0}$ state, while the register qubits are initialized in the trial state $\ket{\psi_0}$, which is prepared using a separate circuit composed of Givens rotation gates.
• Figure 3: Success probability $p_0$ as a function of the variational parameter $g$ for (a) one-dimensional chains with open-boundary conditions and (b) a $2\times 2$ square lattice. Open symbols correspond to results using the Fermi-sea trial state, while solid symbols represent results using the BCS-like trial state with $\Delta/J=0.4$.
• Figure 4: Quantum circuit for preparing the state $\hat{\cal N}(\boldsymbol{s}) |\psi_0\rangle$ for $N_{\text{site}}=2$, corresponding to the numerator in Eq.(\ref{['eq:mc_expval']}). The shaded region represents a sequence of fermionic SWAP gates used to convert between color-uniform and color-alternating labelings.
• Figure 5: Expectation values of (a,c) the total energy, kinetic energy, and interaction energy, and (b,d) the triple occupancy per site as functions of $g$. Results are shown for (a,b) a 4-site one-dimensional chain with open boundary conditions and (c,d) a $2\times 2$ square lattice with $U=-1$. The Fermi-sea trial state at half filling is used. Open symbols represent results obtained by fully summing over all auxiliary field configurations (approach I; see Sec. \ref{['sec:implementation-I']}), while solid symbols are from Monte Carlo sampling (approach II; see Sec. \ref{['sec:implementation-II']}), with error bars indicating statistical errors. Solid lines correspond to the exact solutions. Note that, due to the scale of the plots, results from approaches I and II are often indistinguishable.