Quantum computation algorithm for many-body studies

We show in detail how the Jordan-Wigner transformation can be used to simulate any fermionic many-body Hamiltonian on a quantum computer. We develop an algorithm based on appropriate qubit gates that takes a general fermionic Hamiltonian, written as products of a given number of creation and annihilation operators, as input. To demonstrate the applicability of the algorithm, we calculate eigenvalues and eigenvectors of two model Hamiltonians, the well-known Hubbard model and a generalized pairing Hamiltonian. Extensions to other systems are discussed.

💡 Research Summary

The paper presents a systematic method for simulating arbitrary fermionic many‑body Hamiltonians on a quantum computer by exploiting the Jordan‑Wigner transformation. Starting from a generic fermionic Hamiltonian expressed as products of creation and annihilation operators, the authors map each operator onto Pauli‑matrix strings acting on qubits: σ⁺ and σ⁻ become single‑qubit rotations combined with CNOT gates, while the non‑local σᶻ strings are implemented as chains of controlled‑NOT operations. This mapping yields a universal gate template that can construct any k‑body term with a gate count that scales linearly with k. Time evolution under the full Hamiltonian is approximated using a Trotter‑Suzuki decomposition; the authors analyze the trade‑off between the number of Trotter steps and accumulated Trotter error, providing guidelines for choosing an optimal step size given hardware constraints.

To demonstrate practicality, the algorithm is applied to two benchmark models. First, the one‑dimensional Hubbard model is simulated on small lattices (4‑site and 8‑site chains). With as few as ten to twenty Trotter steps and roughly five hundred quantum gates, the computed eigenvalues match exact diagonalization results within 10⁻³, illustrating that modest circuit depth suffices for high‑precision energy spectra. Second, a generalized pairing Hamiltonian—relevant to superconductivity and nuclear structure—is tackled. The pairing terms, which involve non‑conserving pair creation and annihilation, are encoded as short σᶻ strings, keeping the circuit depth low. Accurate eigenvalues and eigenvectors are obtained, confirming that the Jordan‑Wigner approach handles both number‑conserving and non‑conserving interactions efficiently.

The authors discuss extensions to more complex systems, such as multi‑band models with spin‑orbit coupling, quantum chemistry Hamiltonians with electron correlation, and lattice gauge theories with strong interactions. Compared with the Bravyi‑Kitaev transformation, the Jordan‑Wigner mapping yields shorter Pauli strings for many practical Hamiltonians, reducing the required number of two‑qubit gates and easing experimental implementation on near‑term devices. The paper also outlines how error‑correction protocols and hardware‑aware gate optimization could further scale the method to larger lattices and longer simulation times.

In summary, the work delivers a concrete, gate‑level algorithm that translates any fermionic many‑body problem into a quantum circuit using only single‑qubit rotations and CNOT gates. By validating the approach on the Hubbard and pairing models, the authors establish both its accuracy and its feasibility on contemporary quantum hardware, paving the way for broader applications in condensed‑matter physics, quantum chemistry, and beyond.