Efficient Quantum Cooling Algorithm for Fermionic Systems

Lucas Marti; Refik Mansuroglu; Michael J. Hartmann

doi:10.22331/q-2025-02-18-1635

Efficient Quantum Cooling Algorithm for Fermionic Systems

Lucas Marti, Refik Mansuroglu, and Michael J. Hartmann

Department of Physics, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Staudtstraße 7, 91058 Erlangen

Published:	2025-02-18, volume 9, page 1635
Editor:	Simon Apers
Eprint:	arXiv:2403.14506v2
Doi:	https://doi.org/10.22331/q-2025-02-18-1635
Citation:	Quantum 9, 1635 (2025).

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Abstract

We present a cooling algorithm for ground state preparation of fermionic Hamiltonians. Our algorithm makes use of the Hamiltonian simulation of the considered system coupled to an ancillary fridge, which is regularly reset to its known ground state. We derive suitable interaction Hamiltonians that originate from ladder operators of the free theory and initiate resonant gaps between system and fridge. We further propose a spectroscopic scan to find the relevant eigenenergies of the system using energy measurements on the fridge. With these insights, we design a ground state cooling algorithm for fermionic systems that is efficient, i.e. its runtime is polynomial in the system size, as long as the initial state is prepared in a low-energy sector of polynomial size. We achieve the latter via a pseudo-adiabatic sweep from a parameter regime whose ground state can be easily prepared. We estimate that our algorithm has a polynomial runtime for systems where the spectral gap decreases at most polynomially in system size, and is faster than the adiabatic algorithm for a large range of settings. We generalize the algorithm to prepare thermal states and demonstrate our findings on the Fermi-Hubbard model.

Featured image: Diagram of the Subspace Cooling Algorithm.

Popular summary

This work introduces a method to cool a quantum system of fermions, the particle family that includes electrons, down to its lowest energy state, known as the ground state. Being able to do this is important because understanding the ground state helps reveal fundamental properties of the system, properties which are incredibly useful to the chemical or materials industries. To achieve this, we use an additional quantum system that acts like a refrigerator. By carefully designing interactions between the system and the fridge, energy is gradually drained from the main system, effectively cooling it. The fridge is regularly reset to its own lowest energy state to ensure it continues to absorb energy from the system. Both systems are encoded on a quantum computer, so this method is called quantum digital cooling.

Two issues arise when performing quantum digital cooling. First, we can only successfully design the mechanism that brings energy from the system to the fridge in a way that requires information from the ground state. Yet, the ground state is exactly what we are trying to understand by using the algorithm. We solve this by studying a simpler ground state from a system that behaves like the system of interest, and using that information instead. Secondly, one needs to understand the excitation levels that the system can reach from the ground state. This again requires information which we don't have in the beginning. To identify these levels, we introduce a technique called a spectroscopic scan. This involves measuring how the fridge absorbs or releases energy, helping us map out the energy structure of the system. Our work is the first of its kind to solve these issues for fermionic systems.

With this information, we design an efficient cooling algorithm that can run, under certain conditions, in polynomial time, meaning its complexity grows in a manageable way as the system size increases. For a classical computer, this would not be the case. The algorithm works as long as the system starts in a relatively low-energy state. We also show it can be faster than the adiabatic sweep algorithm — the current standard — in many cases.

Beyond just cooling the system to its ground state, the method can also be extended to prepare thermal states, which means creating states with controlled temperatures rather than just zero temperature.

Finally, we demonstrate our approach on the Fermi-Hubbard model, a key theoretical type of system used to describe electron interactions in materials.

► BibTeX data

@article{Marti2025efficientquantum,
  doi = {10.22331/q-2025-02-18-1635},
  url = {https://doi.org/10.22331/q-2025-02-18-1635},
  title = {Efficient {Q}uantum {C}ooling {A}lgorithm for {F}ermionic {S}ystems},
  author = {Marti, Lucas and Mansuroglu, Refik and Hartmann, Michael J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1635},
  month = feb,
  year = {2025}
}

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