Quantum simulation of Fermi-Hubbard model based on transmon qudit interaction

The Fermi-Hubbard model, a fundamental framework for studying strongly correlated phenomena could significantly benefit from quantum simulations when exploring non-trivial settings. However, simulating this problem requires twice as many qubits as the physical sites, in addition to complicated on-chip connectivities and swap gates required to simulate the physical interactions. In this work, we introduce a novel quantum simulation approach utilizing qudits to overcome such complexities. Leveraging on the symmetries of the Fermi-Hubbard model and their intrinsic relation to Clifford algebras, we first demonstrate a Qudit Fermionic Mapping (QFM) that reduces the encoding cost associated with the qubit-based approach. We then describe the unitary evolution of the mapped Hamiltonian by interpreting the resulting Majorana operators in terms of physical single- and two-qudit gates. While the QFM can be used for any quantum hardware with four accessible energy levels, we demonstrate the specific reduction in overhead resulting from utilizing the native Controlled-SUM gate (equivalent to qubit CNOT) for a fixed-frequency ququart transmon. We further transpile the resulting two transmon-qudit gates by demonstrating a qudit operator Schmidt decomposition using the Controlled-SUM gate. Finally, we demonstrate the efficacy of our proposal by numerical simulation of local observables such as the filling factor and Green's function for various Trotter steps. The compatibility of our approach with different qudit platforms paves the path for achieving quantum advantage in simulating non-trivial quantum many-body systems.