# Optimal Heat-Bath Algorithmic Cooling

This is a Perspective on "Heat-Bath Algorithmic Cooling with optimal thermalization strategies" by Álvaro M. Alhambra, Matteo Lostaglio, and Christopher Perry, published in Quantum 3, 188 (2019).

By Martí Perarnau-Llobet (Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany).

Cooling is arguably the most relevant thermodynamic task for quantum science and technologies, finding countless applications ranging from the preparation of pure states for quantum computation to the experimental observation of genuinely quantum effects. Algorithmic cooling (AC) is a cooling technique developed in the context of quantum information, which consists in reversibly transferring entropy from a system to an ancilla by means of a global unitary operation; this results in a cooling of the system while the ancilla heats up [1]. This single-shot protocol can be improved by using an external thermal bath to cool the ancilla (which also breaks the correlations between system and ancilla), so that it can be used again for another AC step, a technique known as heat-bath algorithmic cooling (HBAC) [2]. This two-step process (unitary plus thermalisation) can be iterated until an asymptotic temperature is reached [3,4,5]. These theoretical ideas have been implemented in NMR platforms (see e.g. [6]).

Recently, Rodríguez-Briones et al. pointed out in [7] that HBAC can be further optimised by having some control on the interaction with the thermal bath, in particular by considering relaxation processes where the bath only couples to certain energy transitions of the ancilla. Indeed, it was shown that these more refined protocols can lead to a lower asymptotic temperature than standard HBAC [7]. The new article by Alhambra, Lostaglio, and Perry pushes this idea to its limits by finding optimal strategies for HBAC under general engineered thermalisation processes, leading to ground state cooling with an exponential convergence with the number of steps [8].

The starting point of Alhambra et al. is to combine HBAC with recent ideas developed in the context of resource theories, where thermodynamic processes have been characterised in rather abstract terms (see e.g. the review [9]). The considered cooling protocols consist of two steps:

1. A unitary operation is applied on system and ancilla, which requires an external energy input.
2. A (dephasing) thermal map is applied on system and ancilla. Roughly speaking, this includes any (possibly engineered) process where system and ancilla are put in contact with the thermal bath without external energy sources.

Examples of the latter operation include finite-time dissipative processes (where system and ancilla interact with the bath without reaching the asymptotic equilibrium state), and engineered relaxations where only certain energy transitions are coupled to the bath.

Given this rather general framework, the authors consider protocols where such two operations are repeated for $k$ times; the goal is then to maximise the ground state population of the system. Remarkably, the optimal protocol for arbitrary $k$ and initial conditions (initial state of the $d$-dimensional system to be cooled, bath temperature, etc.) is derived, leading to an exponential convergence with the number of rounds $k$ to perfect cooling, i.e., ground state population equal to 1. This should be contrasted with the impossibility to obtain perfect cooling in standard heat bath algorithmic cooling, even after infinite rounds [3,4,5].

These results provide new fundamental limits on HBAC by engineered thermalisation processes. Two natural questions then arise: what is the underlying physical mechanism behind these enhancements? and, can they be realised in experimentally realistic set-ups? The authors give compelling answers to these questions when the system to be cooled is a qubit. First, they show that such enhancements can only be seen when the thermalisation dynamics are non-Markovian, making an appealing link between memory effects and enhanced cooling. Second, it is shown that perfect cooling can be obtained by coupling the qubit to a single bosonic mode (acting as a thermal bath, and with no need for an ancilla), which can be reutilised in each round of the HBAC cooling. The desired system-bath interaction can be realised by an intensity-dependent Jaynes-Cummings model [10], whereas the standard Jaynes-Cummings model enables one to obtain non-Markovian cooling enhancements despite not reaching perfect cooling.

Overall, the article of Alhambra, Lostaglio, and Perry derives new limits on Heat Bath Algorithmic Cooling contributing to our understanding of the fundamental limits of cooling [5], which is intimately connected with the third law of thermodynamics [11]. Furthermore, it develops an exciting experimental proposal for exploiting non-Markovian effects to enhance cooling protocols.

### ► References

[1] Leonard J. Schulman and Umesh V. Vazirani. Molecular scale heat engines and scalable quantum computation. In Proceedings of the thirty-first annual ACM symposium on Theory of computing - STOC \textquotesingle99. ACM Press, 1999.

[2] P. Oscar Boykin, Tal Mor, Vwani Roychowdhury, Farrokh Vatan, and Rutger Vrijen. Algorithmic cooling and scalable NMR quantum computers. Proceedings of the National Academy of Sciences, 99(6):3388–3393, March 2002.

[3] Leonard J. Schulman, Tal Mor, and Yossi Weinstein. Physical limits of heat-bath algorithmic cooling. Physical Review Letters, 94(12), April 2005.

[4] Nayeli Azucena Rodríguez-Briones and Raymond Laflamme. Achievable polarization for heat-bath algorithmic cooling. Phys. Rev. Lett., 116:170501, Apr 2016.

[5] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber. Unifying paradigms of quantum refrigeration: A universal and attainable bound on cooling. arXiv preprint arXiv:1903.04970, 2019.
arXiv:1903.04970

[6] J. Baugh, O. Moussa, C. A. Ryan, A. Nayak, and R. Laflamme. Experimental implementation of heat-bath algorithmic cooling using solid-state nuclear magnetic resonance. Nature, 438(7067):470–473, November 2005.

[7] Nayeli A Rodríguez-Briones, Jun Li, Xinhua Peng, Tal Mor, Yossi Weinstein, and Raymond Laflamme. Heat-bath algorithmic cooling with correlated qubit-environment interactions. New Journal of Physics, 19(11):113047, November 2017.

[8] Álvaro M. Alhambra, Matteo Lostaglio, and Christopher Perry. Heat-bath algorithmic cooling with optimal thermalization strategies. Quantum, 3:188, September 2019.

[9] Matteo Lostaglio. An introductory review of the resource theory approach to thermodynamics. Reports on Progress in Physics, September 2019.

[10] M. H. Naderi, M. Soltanolkotabi, and R. Roknizadeh. A theoretical scheme for generation of nonlinear coherent states in a micromaser under intensity-dependent jaynes-cummings model. The European Physical Journal D, 32(3):397–408, January 2005.

[11] Nahuel Freitas, Rodrigo Gallego, Lluís Masanes, and Juan Pablo Paz. Cooling to absolute zero: The unattainability principle. In Thermodynamics in the Quantum Regime, pages 597–622. Springer, 2018.

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