Heat-Bath Algorithmic Cooling with optimal ther- malization 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 September 19, 2019 Heat-Bath Algorithmic Cooling is a set of physically realistic setups. techniques for producing highly pure quantum Cooling is a central problem in quantum physics and systems by utilizing a surrounding heat-bath in realizing technologies for quantum information pro- and unitary interactions. These techniques orig- cessing. The ability to produce a set of highly pure, inally used the thermal environment only to ‘cold’, quantum states is vital for the construction of a fully thermalize ancillas at the environment tem- quantum computer [1]. More generally, the observation perature. Here we extend HBAC protocols by of quantum effects often requires cooling and, as such, optimizing over the thermalization strategy. We many techniques have been developed to cool systems find, for any d-dimensional system in an arbi- efficiently in platforms ranging from cavity optomechan- trary initial state, provably optimal cooling pro- ics [2] to NMR [3,4], ion traps [5] and superconducting tocols with surprisingly simple structure and qubits [6]. exponential convergence to the ground state. Here we use powerful techniques, developed within Compared to the standard ones, these schemes the resource theory approach to thermodynamics [7], can use fewer or no ancillas and exploit memory to greatly extend an important class of cooling al- effects to enhance cooling. We verify that the gorithms known as Heat-Bath Algorithmic Cooling optimal protocols are robusts to various devia- (HBAC) [3,4]. The goal of these is to maximize the tions from the ideal scenario. For a single target purity of a target system S in a given number of cool- qubit, the optimal protocol can be well approxi- ing rounds. Each round of the algorithm starts with a mated with a Jaynes-Cummings interaction be- unitary applied to the target together with several aux- tween the system and a single thermal bosonic iliary systems A initialized in a thermal state, with the mode for a wide range of environmental temper- aim of pumping entropy away from the target. Next, the atures. This admits an experimental implemen- auxiliary systems are re-thermalized through coupling tation close to the setup of a micromaser, with a with a heat-bath, before the entire process is repeated performance competitive with leading proposals in the next round. The asymptotically optimal protocol in the literature. The proposed protocol pro- of this form (in terms of the purity reached in infinitely vides an experimental setup that illustrates how many rounds) is the Partner Pairing Algorithm (PPA), non-Markovianity can be harnessed to improve introduced in [8], whose asymptotic performance has cooling. On the technical side we 1. introduce a arXiv:1807.07974v3 [quant-ph] 18 Sep 2019 been recently derived for a single target qubit starting new class of states called maximally active states in a maximally mixed state [9]. and discuss their thermodynamic significance in terms of optimal unitary control, 2. introduce a Here, we generalize HBAC in the following sense: in- new set of thermodynamic processes, called β- stead of using the thermal environment only to refresh permutations, whose access is sufficient to sim- the auxiliary systems through a complete thermaliza- ulate a generic thermalization process, 3. show tion, we allow strategies involving an incomplete ther- how to use abstract toolbox developed within malization of system and ancillas. In particular, we the resource theory approach to thermodynam- optimize the protocol over every ‘dephasing thermaliza- ics to perform challenging optimizations, while tion’, that is any quantum map on SA which 1. leaves combining it with open quantum system dynam- the thermal state on SA fixed and 2. dephases the in- ics tools to approximate optimal solutions within put state in the energy eigenbasis. This generalizes the ‘rethermalization’ used in previous protocols, and thus Accepted in Quantum 2019-09-16, click title to verify 1 extends HBAC to a larger set that will be called Ex- tended Heat-Bath Algorithmic Cooling (xHBAC). For example, the recent protocol introduced in Ref. [10], which proposes the use of the heat bath to implement a non-local thermalization ‘state-reset’ (SR) process re- lated to the Nuclear Overhauser Effect [11], can already Figure 1: A generic xHBAC protocol. be seen as a (non-optimal) protocol within our extended family of xHBAC. (k) For every finite dimensional target state in an arbi- 2. Dephasing thermalization. Any Λ applied (k) trary initial state, we optimize the cooling performance to SA, where Λ is any quantum map such that (k) over every xHBAC for any given number of rounds. We i) Λ (τS ⊗ τA) = τS ⊗ τA (thermal fixed point); 0 give an analytical form for the optimal cooling opera- ii) If |Ei, |E i are states of distinct energy on SA, (k) 0 tions, uncovering their elegant structure, and show that hE|Λ (ρSA)|E i = 0 (dephasing). the ground state population goes to 1 exponentially fast At the end of the round, a refreshing stage returns the in the number of rounds. This opens up a new avenue auxiliary systems A to their original state ρA. We de- in the experimental realization of algorithmic cooling note the set of protocols whose rounds have this general schemes, one requiring control over fewer ancillas, but form by PρA . Typically ρA = τA, and this operation better control of the interaction with the environment. is simply a specific kind of dephasing thermalization. Our results suggest that xHBAC schemes provide new However, more generally, ρA may also be the output of cooling protocols in practically relevant settings. We some previous cooling algorithm, subsequently used as show that, for a single qubit target and no ancillas, the an auxiliary system. Note that A is distinguished by optimal protocol in xHBAC can be approximated by the rest of the environment in that it is under complete coupling the qubit by a Jaynes-Cummings (JC) interac- unitary control. tion to a single bosonic thermal mode, itself weakly cou- Some comments about xHBAC protocols are in or- pled to the external bath. The memory effects present der. First, if we skipped the dephasing thermaliza- in the JC interaction are crucial in approximating the tions and ρA = τA, we would get back to a standard theoretical optimal cooling. The performance of the HBAC scheme. xHBAC protocols include thermaliza- ideal cooling protocol is robust to certain kinds of noise tion of subsets of energy levels, as in the mentioned SR and imperfections, and it outperforms HBAC schemes protocol of Ref. [10]. But they are by no means lim- with a small number of auxiliary qubits. This makes it, ited to these strategies. Second, for every k we wish in our opinion, the most promising proposal for an ex- to find a protocol maximizing the ground state popu- perimental demonstration of cooling through xHBAC, (k) (k) lation p0 = h0|ρS |0i over all sequences of k-round as well as an experimental demonstration that non- operations. Note that often the literature on cooling Markovianity can be harnessed to improve cooling. is restricted to finding asymptotically optimal proto- cols (e.g., the PPA [8]), but here we find optimal k rounds protocols for every k. Finally, we note in passing 1 Results and discussion that within this work we will not make use of auxiliary ‘scratch qubits’ [9], i.e. S itself is the target system to 1.1 A general cooling theorem be cooled. To give the analytical form of the optimal protocols A general xHBAC will consist of a number of rounds we need to introduce two notions. First, that of maxi- and manipulate two types of systems, the target system mally active states. Given a state ρ with Hamiltonian S to be purified and the auxiliary systems A. Further- H, the maximally active state ρˆ is formed by diagonal- more, we will assume we can access a thermal environ- izing ρ in the energy eigenbasis and ordering the eigen- −1 ment at inverse temperature β = (kT ) , with k Boltz- values in increasing order with respect to energy. That Pd−1 mann’s constant. We denote by HS = i=0 Ei|iihi|, is, ρˆ is the most energetic state in the unitary orbit of (k) E0 ≤ · · · ≤ Ed−1, the Hamiltonian of S and by ρS ρ. the state after round k. Also, we denote by HA the Second, we define the thermal polytope in the follow- Hamiltonian of the auxiliary systems, initially in state ing way. Given an initial state ρ with populations in −βHX −βHX ρA, and by τX = e /Tr[e ] the thermal state the energy eigenbasis p, it is the set of populations on X = S, A. A protocol is made of k rounds, each p0 of Λ(ρ), with Λ an arbitrary dephasing thermaliza- allowing for (Fig.1): tion. Without loss of generality, we assume that ev- ery energy eigenspace has been diagonalized by energy 1. Unitary. Any unitary U (k) applied to SA. preserving unitaries. The thermal polytope is convex Accepted in Quantum 2019-09-16, click title to verify 2 and has a finite number of extremal points (Lemma 12, [12]). It is particularly useful in thermodynamics to classify out of equilibrium distributions according to their β-order [13]. Given a vector p and a correspondent Hamiltonian with energy levels Ei, the β−order of p is the permutation of π of 0, 1, 2,..