Applying the HVA method to find the Fermi-Hubbard Ground State

[camponogaraviera]

Team name:

Hackeinberg

Team members:

Lucas Camponogara Viera

Project Description:

Various approaches have been proposed to create effective hardware-efficient and chemistry-inspired ansatz. One such ansatz is the trotterized UCCSD ansatz which, although based on the accurate Coupled-cluster theory, suffers from inconsistency under low-order trotterization steps [1] when shallow circuits friendly to NISQ devices provide a poor approximation to the UCCSD wavefunction.

On the optimization side, gradient-based algorithms for the optimization of variational quantum circuits have also been explored, such as the Bayesian model gradient descent (BayesMGD) [2], the model gradient descent (MGD) [3], the SPSA, Adam, L-BFGS, NFT, and the ADAPT-VQE [4] which strongly relies on the parameter-shift rule to compute the gradient of the gates in order to optimize the Euler angles and change the circuit structure on-the-fly [5]. Gradient-free optimization algorithms include the Rotosolve, Rotoselect [6], and the NelderMead method, to name a few. The mainstream literature states that gradient-based methods are more efficient with faster convergence if gradients can be computed directly [7] [8].

The characteristic of the Hubbard model suggests the application of the Hamiltonian variational method [9] which uses terms of the Hamiltonian to propose the circuit ansatz with a smaller circuit depth than the unitary coupled cluster method (UCC).

In this solution, gradient-free and gradient-based optimizations were compared, and the final algorithm was chosen based on its computational time (speed) and convergence.

[1] Grimsley, H. R.; Claudino, D.; Economou, S. E.; Barnes, E.; Mayhall, N. J. Is the trotterized uccsd ansatz chemically well-defined? J. Chem. Theory Comput. 2020, 16, 1.

[2] Stanisic, S., Bosse, J.L., Gambetta, F.M. et al. Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer. Nat Commun 13, 5743 (2022).

[3] Sung, K. J. et al. Using models to improve optimizers for variational quantum algorithms. Quantum Sci. Technol. 5, 044008 (2020).

[4] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, Nicholas J. Mayhall, "An adaptive variational algorithm for exact molecular simulations on a quantum computer". Nat. Commun. 2019, 10, 3007.

[5] Lucas Camponogara Viera and José Paulo Marchezi. "Extending Adaptive Methods for Finding an Optimal Circuit Ansatze in VQE Optimization". QHack2022 Open Hackathon Project - Quantum Chemistry Challenge.

[6] Mateusz Ostaszewski, Edward Grant, Marcello Bendetti. "Structure optimization for parameterized quantum circuits." Quantum 5, 391 (2021).

[7] J. Li, X. Yang, X. Peng, and C.-P. Sun, Phys. Rev. Lett. 118, 150503 (2017).

[8] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Phys. Rev. A 98, 032309 (2018).

[9] Wecker, D., Hastings, M. B. & Troyer, M. Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (2015).

[10] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic. Strategies for solving the Fermi-Hubbard model on near-term quantum computers. Phys. Rev. B 102, 235122.

[11] Jiang, Z., Sung, K. J., Kechedzhi, K., Smelyanskiy, V. N. & Boixo, S. Quantum algorithms to simulate many-body physics of correlated fermions. Phys. Rev. Appl. 9, 044036 (2018).

Presentation:

https://github.com/QuCAI-Lab/qagc/blob/dev/Presentation.ipynb

Source code:

https://github.com/QuCAI-Lab/qagc/blob/dev/problem/answer.py