Utilizing Quantum Processor for the Analysis of Strongly Correlated Materials

TL;DR

The paper tackles the challenge of studying strongly correlated materials on near-term quantum devices by mapping the quantum cluster method (QCM) onto a quantum circuit framework and deriving a real-valued formulation for the cluster Green's function. Ground states are obtained via VQE using Jordan–Wigner encoding, and real-time retarded Green's functions are computed with Hadamard-test circuits incorporating time evolution with Suzuki–Trotter decomposition, all while accounting for hardware noise. The authors demonstrate the approach on the 2D Hubbard model, obtaining cluster energies that agree with classical results after noise mitigation, and obtaining lattice Green's functions and one-particle spectra that reproduce key Hubbard features such as the Mott gap and band structure, albeit with some spectral weight loss due to noise. This work provides a viable pathway to perform condensed-matter simulations and potentially ARPES-like measurements on noisy quantum hardware, opening avenues for exploring strongly correlated physics on near-term devices.

Abstract

This study introduces a systematic approach for analyzing strongly correlated systems by adapting the conventional quantum cluster method to a quantum circuit model. We have developed a more concise formula for calculating the cluster's Green's function, requiring only real-number computations on the quantum circuit instead of complex ones. This approach is inherently more suited to quantum circuits, which primarily yield statistical probabilities. As an illustrative example, we explored the Hubbard model on a 2D lattice. The ground state is determined utilizing Xiaohong, a superconducting quantum processor equipped with 66 qubits, supplied by QuantumCTek Co., Ltd. Subsequently, we employed the circuit model to compute the real-time retarded Green's function for the cluster, which is then used to determine the lattice Green's function. We conducted an examination of the band structure in the insulator phase of the lattice system. This preliminary investigation lays the groundwork for exploring a wealth of innovative physics within the field of condensed matter physics.

Figures (7)

• Figure 1: The original lattice is divided into distinct clusters and inter-cluster segments. Each cluster (red areas), referred to as a reference system, is independently analyzed using various methods, such as exact diagonalization (ED). The inter-cluster section is treated as a perturbation affecting the reference system. To address symmetry breaking, we introduce additional mean-fields. Meanwhile, achieving the metallic phase requires integrating additional environmental elements, typically bath sites, which are hybridized with the cluster. These additional parameters, mean-fields, and hybridization functions are determined through a self-consistent variational process.
• Figure 2: Following the fundamental principles of the Hadamard test Htest, the circuit is devised to compute $\mathcal{F}\left(P_{i},P_{j}\right)$ as outlined in Eq. \ref{['eq:14']}. The topmost qubit serves as an ancilla qubit and is configured for the final measurement. The result of this measurement, denoted as $p_{+}$ , determines the value of Eq. \ref{['eq:14']}.
• Figure 3: The ground state energies as a function of interaction $U$. Inset details illustrate the mitigation within Hamiltonian $H_{c}$ at $U=3$, showing the contribution of each term.
• Figure 4: Comparison of retarded Green's function calculated by a quantum circuit (cyan solid line) and the exact result (orange dashed line). (a) The imaginary part of the real-time retarded Green's function. (b) The Green's function after being multiplied by a factor $e^{-\eta t}$ which effectively reduces the error at large $t$.
• Figure 5: The quantum circuit calculations (cyan solid line) and exact results (orange dashed line). (a) The real part of the Green's function in the frequency domain. (b) The corresponding spectral function of the cluster.