# Heat-Bath Algorithmic Cooling with optimal thermalization strategies

Álvaro M. Alhambra1, Matteo Lostaglio2, and Christopher Perry3

1Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
3QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

### Abstract

Heat-Bath Algorithmic Cooling is a set of techniques for producing highly pure quantum systems by utilizing a surrounding heat-bath and unitary interactions. These techniques originally used the thermal environment only to fully thermalize ancillas at the environment temperature. Here we extend HBAC protocols by optimizing over the thermalization strategy. We find, for any $d$-dimensional system in an arbitrary initial state, provably optimal cooling protocols with surprisingly simple structure and exponential convergence to the ground state. Compared to the standard ones, these schemes can use fewer or no ancillas and exploit memory effects to enhance cooling. We verify that the optimal protocols are robusts to various deviations from the ideal scenario. For a single target qubit, the optimal protocol can be well approximated with a Jaynes-Cummings interaction between the system and a single thermal bosonic mode for a wide range of environmental temperatures. This admits an experimental implementation close to the setup of a micromaser, with a performance competitive with leading proposals in the literature. The proposed protocol provides an experimental setup that illustrates how non-Markovianity can be harnessed to improve cooling. On the technical side we 1. introduce a new class of states called $maximally$ $active$ $states$ and discuss their thermodynamic significance in terms of optimal unitary control, 2. introduce a new set of thermodynamic processes, called $\textit{$\beta$-permutations}$, whose access is sufficient to simulate a generic thermalization process, 3. show how to use abstract toolbox developed within the resource theory approach to thermodynamics to perform challenging optimizations, while combining it with open quantum system dynamics tools to approximate optimal solutions within physically realistic setups.

Access to pure `cold' quantum states is crucial in the realization of quantum technologies and the observation of quantum effects. Algorithmic cooling plays a prominent role in this task: it is a technique that uses a combination of unitary pulses and thermalizations to cool down a system. Standard techniques use the environment only to realize full thermalizations by thermal contact. Here, we overcome this limitation by providing the tools to optimize over generic thermalization strategies. This leads to optimal cooling protocols that, exploiting memory effects in the dynamics of the thermalization process, yield much more efficient cooling, beyond standard bounds. We illustrate the general results by providing an explicit cooling scheme involving simple light-matter interaction models widely realized in experiments.

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### Cited by

[1] Philip Taranto, Faraj Bakhshinezhad, Philipp Schüttelkopf, Fabien Clivaz, and Marcus Huber, "Exponential Improvement for Quantum Cooling through Finite-Memory Effects", Physical Review Applied 14 5, 054005 (2020).

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