Gutzwiller wave function on a digital quantum computer

The determination of the ground state of quantum many-body systems via digital quantum computers rests upon the initialization of a sufficiently educated guess. This requirement becomes more stringent the greater the system. Preparing physically motivated Ansätze on quantum hardware is therefore important to achieve a quantum advantage in the simulation of correlated electrons. In this spirit, we introduce the Gutzwiller wave function (GWF) within the context of the digital quantum simulation of the Fermi-Hubbard model. We present a quantum routine to initialize the GWF that comprises two parts. In the first, the noninteracting state associated with the $U=0$ limit of the model is prepared. In the second, the nonunitary Gutzwiller operator that selectively removes states with doubly occupied sites from the wave function is performed by adding to every lattice site an ancilla qubit, the measurement of which in the $|0\rangle$ state confirms the operator was applied. Due to its nondeterministic nature, we estimate the success rate of the algorithm in generating the GWF as a function of the lattice size and the interaction strength $U/t$. The scaling of the quantum circuit metrics and its integration in general quantum simulation algorithms are also discussed.

Figure 1

Comparison of Gutzwiller wave function (GWF) to noninteracting and self-consistent mean-field ground states for the Fermi-Hubbard model (FHM) in a chain with open boundary conditions at half filling. (a) Fidelity of three reference states with respect to the exact ground state for a chain of $N=12$ sites against a normalized Hubbard parameter $U/t$. (b) Same as (a), but now the size of chain $N$ is varied between 2 and 12 sites, while $U/t$ takes a fixed value of 10, as highlighted in (a) by the vertical dotted line. Numerical results were fitted to exponential decay $c_1{e}^{-c_{2}N}$.

Figure 2

(a) High-level scheme of routine to initialize GWF on a quantum computer, exemplified for a system of $N=4$ sites. After the noninteracting ground state $|{\psi }_0\rangle$ is prepared, a single-qubit rotation is applied to each auxiliary qubit if the two qubits that encode the occupations of the corresponding site are in state $|11\rangle$. Measuring the auxiliary qubits and retrieving only the trials that yield 0000 confirms the Gutzwiller operator was applied. (b) Detailed scheme of the routine to initialize GWF assuming linear qubit connectivity. The solid red box corresponds to the preparation of $|{\psi }_0\rangle$, which is first initialized in the diagonal basis and then transformed back to the original one via a Givens rotation decomposition, followed by the reordering of the qubits by site instead of spin, which involves fermionic swaps. The dashed blue box is the Gutzwiller operator: A network of swaps places auxiliary qubits next to the control qubits. A detailed description can be found in the Supplemental Material [34].