[Paper Review] Solving the Hubbard model using density matrix embedding theory and the variational quantum eigensolver

Lana Mineh, Ashley Montanaro|arXiv (Cornell University)|Aug 19, 2021

Quantum Computing Algorithms and Architecture50 references18 citations

TL;DR

This paper proposes a hybrid quantum-classical approach combining density matrix embedding theory (DMET) with the variational quantum eigensolver (VQE) to solve the Hubbard model on near-term quantum computers. By mapping the full lattice problem to a smaller embedded Hamiltonian using DMET, the method reduces qubit requirements while enabling accurate ground state energy and observable calculations via VQE, achieving high-fidelity results in numerical simulations up to 16 qubits.

ABSTRACT

Calculating the ground state properties of a Hamiltonian can be mapped to the problem of finding the ground state of a smaller Hamiltonian through the use of embedding methods. These embedding techniques have the ability to drastically reduce the problem size, and hence the number of qubits required when running on a quantum computer. However, the embedding process can produce a relatively complicated Hamiltonian, leading to a more complex quantum algorithm. In this paper we carry out a detailed study into how density matrix embedding theory (DMET) could be implemented on a quantum computer to solve the Hubbard model. We consider the variational quantum eigensolver (VQE) as the solver for the embedded Hamiltonian within the DMET algorithm. We derive the exact form of the embedded Hamiltonian and use it to construct efficient ansatz circuits and measurement schemes. We conduct detailed numerical simulations up to 16 qubits, the largest to date, for a range of Hubbard model parameters and find that the combination of DMET and VQE is effective for reproducing ground state properties of the model.

Motivation & Objective

• To develop a scalable quantum algorithm for solving the Hubbard model using embedding techniques.
• To implement DMET on a quantum computer via the VQE algorithm for ground state energy and observable calculations.
• To analyze the quantum circuit complexity of the embedded Hamiltonian and optimize ansatz design and measurement procedures.
• To evaluate the accuracy and scalability of the DMET-VQE approach across varying fragment sizes and Hubbard model parameters.
• To provide a detailed complexity analysis of circuit depth and measurement rounds for practical implementation.

Proposed method

• Uses single-shot DMET to map the full Hubbard model Hamiltonian to a smaller embedded Hamiltonian of size 4Nfrag qubits, where Nfrag is the fragment size.
• Derives the exact form of the embedded Hamiltonian, including interactions between the fragment and bath, enabling precise quantum circuit construction.
• Employs the Hamiltonian variational (HV) ansatz with fermionic swap networks to efficiently implement entangling gates and reduce circuit depth.
• Designs measurement schemes that minimize the number of circuit preparations by grouping Pauli terms and exploiting symmetries.
• Applies statistical noise-aware measurement protocols to simulate realistic NISQ device constraints.
• Conducts numerical simulations using exact diagonalization benchmarks to validate VQE performance.

Experimental results

Research questions

• RQ1 Can DMET combined with VQE accurately reproduce ground state properties of the 1D and 2D Hubbard model on quantum hardware?
• RQ2 How does the quantum circuit complexity—measured in two-qubit gate depth and number of circuit preparations—scale with fragment size and system dimensionality?
• RQ3 What is the impact of statistical noise on VQE convergence and observable accuracy in the embedded DMET framework?
• RQ4 How do fermionic swap networks improve the efficiency of VQE ansatz implementation in the embedded system?
• RQ5 To what extent does the DMET-VQE approach outperform direct truncation methods in terms of resource efficiency and accuracy?

Key findings

• The DMET-VQE approach successfully reproduces ground state energy and observables of the Hubbard model with high accuracy, matching results from exact diagonalization and Bethe ansatz solutions.
• For a 4×4 fragment (16 qubits), the ansatz requires a two-qubit gate depth of 30 per layer and 32 circuit preparations to measure all Hamiltonian terms.
• The method achieves accurate results even with statistical noise in measurements, demonstrating robustness under realistic NISQ conditions.
• The circuit depth and measurement cost grow more steeply with fragment size than direct truncation methods—e.g., a 4×8 Hubbard model requires only depth 9 and 5 preparations.
• Fermionic swap networks enable efficient ansatz implementation, reducing circuit depth and improving scalability for larger fragments.
• The study provides the first full quantum circuit complexity analysis of DMET with VQE, offering practical guidelines for future hardware implementations.