Characterization of variational quantum algorithms using free fermions

Gabriel Matos1, Chris N. Self2,3, Zlatko Papić1, Konstantinos Meichanetzidis4,5, and Henrik Dreyer6

1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
2Quantinuum, Partnership House, Carlisle Place, London, SW1P 1BX, United Kingdom
3Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
4Quantinuum, 17 Beaumont St., Oxford OX1 2NA, United Kingdom
5Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
6Quantinuum, Leopoldstrasse 180, 80804 Munich, Germany

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We study variational quantum algorithms from the perspective of free fermions. By deriving the explicit structure of the associated Lie algebras, we show that the Quantum Approximate Optimization Algorithm (QAOA) on a one-dimensional lattice – with and without decoupled angles – is able to prepare all fermionic Gaussian states respecting the symmetries of the circuit. Leveraging these results, we numerically study the interplay between these symmetries and the locality of the target state, and find that an absence of symmetries makes nonlocal states easier to prepare. An efficient classical simulation of Gaussian states, with system sizes up to $80$ and deep circuits, is employed to study the behavior of the circuit when it is overparameterized. In this regime of optimization, we find that the number of iterations to converge to the solution scales linearly with system size. Moreover, we observe that the number of iterations to converge to the solution decreases exponentially with the depth of the circuit, until it saturates at a depth which is quadratic in system size. Finally, we conclude that the improvement in the optimization can be explained in terms of better local linear approximations provided by the gradients.

Variational quantum algorithms employ a classical computer to optimize a quantum one, combining the power of both. However, our understanding about these algorithms is still incomplete. Efforts to study them run not only into the limitation that quantum computers are difficult to simulate, but also that the associated classical optimization is itself very hard. In our work, we study the Quantum Approximate Optimization Algorithm (QAOA), a special instance of these algorithms. We show that when the problem we are trying to solve using QAOA has a linear geometry, the quantum states that can be prepared are precisely those that correspond to "free fermions". These can be efficiently simulated using only a classical computer, allowing us to circumvent the limitations of current quantum computers and more comprehensively study variational algorithms. Among other things, we found that symmetries, which tend to simplify a problem and are generally regarded as beneficial, can in certain cases overly constrain the optimization and make it more difficult. We were also able to characterize a phenomenon called "overparameterization", where the optimization becomes significantly easier by increasing the number of parameters in the algorithm. In this regime, we found that the number of iterations to find a solution decreases exponentially with the number of parameters. Additionally, this number of iterations goes from being polynomial to linear in the size of the system. Our work fully characterises what states can be prepared by QAOA in a linear geometry and furthers the understanding of the optimization associated with variational algorithms.

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[1] Diego García-Martín, Martín Larocca, and M. Cerezo, "Effects of noise on the overparametrization of quantum neural networks", Physical Review Research 6 1, 013295 (2024).

[2] Nikhil Khatri, Gabriel Matos, Luuk Coopmans, and Stephen Clark, "Quixer: A Quantum Transformer Model", arXiv:2406.04305, (2024).

[3] Matthew L. Goh, Martin Larocca, Lukasz Cincio, M. Cerezo, and Frédéric Sauvage, "Lie-algebraic classical simulations for variational quantum computing", arXiv:2308.01432, (2023).

[4] Xie-Hang Yu, Zongping Gong, and J. Ignacio Cirac, "Free-fermion Page curve: Canonical typicality and dynamical emergence", Physical Review Research 5 1, 013044 (2023).

[5] Leela Ganesh Chandra Lakkaraju, Srijon Ghosh, Debasis Sadhukhan, and Aditi SenDe, "Mimicking quantum correlation of a long-range Hamiltonian by finite-range interactions", Physical Review A 106 5, 052425 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 06:19:31) and SAO/NASA ADS (last updated successfully 2024-06-22 06:19:32). The list may be incomplete as not all publishers provide suitable and complete citation data.