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arXiv:1902.04439 · Physical Review Letters

Novel Technique for Robust Optimal Algorithmic Cooling

Sadegh Raeisi, Mária Kieferová, Michele Mosca

Published Jun 7, 2019• 15 citations
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Abstract

Heat-bath algorithmic cooling provides algorithmic ways to improve the purity of quantum states. These techniques are complex iterative processes that change from each iteration to the next and this poses a significant challenge to implementing these algorithms. Here, we introduce a new technique that on a fundamental level, shows that it is possible to do algorithmic cooling and even reach the cooling limit without any knowledge of the state and using only a single fixed operation, and on a practical level, presents a more feasible and robust alternative for implementing heat-bath algorithmic cooling. We also show that our new technique converges to the asymptotic state of heat-bath algorithmic cooling and that the cooling algorithm can be efficiently implemented; however, the saturation could require exponentially many iterations and remains impractical. This brings heat-bath algorithmic cooling to the realm of feasibility and makes it a viable option for realistic application in quantum technologies.

DOI
10.1103/physrevlett.122.220501

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Heat-bath algorithmic cooling provides algorithmic ways to improve the purity of quantum states. These techniques are complex iterative processes that change from each iteration to the next and this poses a significant challenge to implementing these algorithms. Here, we introduce a new technique that on a fundamental level, shows that it is possible to do algorithmic cooling and even reach the cooling limit without any knowledge of the state and using only a single fixed operation, and on a practical level, presents a more feasible and robust alternative for implementing heat-bath algorithmic cooling. We also show that our new technique converges to the asymptotic state of heat-bath algorithmic cooling and that the cooling algorithm can be efficiently implemented; however, the saturation could require exponentially many iterations and remains impractical. This brings heat-bath algorithmic cooling to the realm of feasibility and makes it a viable option for realistic application in quantum technologies.
The field of quantum information has inspired new methods for cooling physical systems at the quantum scale by manipulating entropy in an algorithmic way, such as heat-bath algorithmic cooling (HBAC). These methods not only provide fundamental insight into quantum thermodynamics, but they are also at the core of practical applications in quantum science and quantum technologies. Arguably, the most promising practical applications are in quantum computing, for the preparation of pure states. The ability to prepare highly pure states is required both for initializing qubits in most quantum algorithms and for supplying reliable low-noise ancilla qubits that satisfy the fault-tolerance threshold for quantum error correction (achieving the high levels of purity required represents one of the major challenges not only for ensemble implementations but also for technologies with strong but not perfect projective measurements). \n \n \nThe heat bath algorithmic cooling protocols have inspired the work within this thesis, which examines and proposes powerful new techniques that significantly enhance cooling by taking advantage of classical and quantum correlations. These new methods go beyond the limits of conventional cooling techniques, providing a novel way to cool that allows a generalized interaction of the system with the environment, which has not been taken into account in previous work. Concretely, I have contributed to elucidating our understanding of these algorithmic cooling mechanisms by using techniques from quantum information theory and quantum thermodynamics. First, I found the analytical solution of the maximum achievable cooling of these algorithmic cooling methods, which had been a longstanding problem that remained open for almost 15 years. Then, I showed how to circumvent the cooling limits of the conventional algorithmic cooling – which were widely believed to be optimal –, creating novel methods that show how correlations can be used to significantly improve cooling. On the one hand, we fundamentally changed the way previous methods considered the interactions between the system and environment and showed how correlated relaxation processes can be essential for enhancing cooling. On the other hand, we demonstrated that correlations present in the initial state due to internal interactions can be exploited to improve cooling. Finally, we showed how, by using ideas and concepts from resource theory, it is possible to find the optimal entropy compression required for HBAC by studying the n-to-1 distillation of athermality of two level systems.
Heat-Bath Algorithmic Cooling is a technique for producing pure quantum systems by utilizing a surrounding heat-bath. Here we connect the study of these cooling techniques to the resource theory of thermal operations, enabling us to derive provably optimal cooling protocols under a variety of experimental restrictions on the available control. For qubit systems, we find that a surprisingly simple, optimal protocol consisting of repeated application of a Pauli XXX unitary and a thermal operation can achieve purity converging exponentially quickly to one. What is more, this thermal operation can be well approximated using a Jaynes Cummings interaction between the system and a single thermal bosonic mode and we consider experimental implementations of this. In addition, we investigate the role of quantum coherence and non-Markovianity in cooling protocols and extend our results to higher dimensional systems. Finally, by considering the role of correlations with auxiliary systems in cooling, we show that purity arbitrary close to one can be achieved in a fixed number of operations. Our results serve to find practical applications for the resource theoretic approach to quantum thermodynamics and suggest relevant experimental implementations.
In this work we experimentally demonstrate the implementation of a recently proposed robust and state-independent heat-bath algorithmic cooling (HBAC) method [Sadegh Raeisi, M\'aria Kieferov\'a, and Michele Mosca, Phys. Rev. Lett. 122, 220501 (2019). doi:10.1103/PhysRevLett.122.220501] on an NMR quantum processor. While HBAC methods improve the purity of a quantum system via iterative unitary entropy compression, they are difficult to implement experimentally since they use sort operations that are different for each iteration. The new robust HBAC method proved that optimal HBAC is possible without prior state information and using a single fixed operation. We modified the protocol to experimentally perform efficient cooling of 13C{}^{13}\mathrm{C}13C and 15N{}^{15}\mathrm{N}15N spins and provide an optimal decomposition of this modified protocol in terms of quantum gates. We examined the relaxation dynamics of these algorithmically cooled spins, in order to ascertain the effect of decoherence on the cooled states.
The purity of quantum states is a key requirement for many quantum applications. Improving the purity is limited by fundamental laws of thermodynamics. Here, we are probing the fundamental limits for a natural approach to this problem, namely, heat-bath algorithmic cooling (HBAC). The existence of the cooling limit for HBAC techniques was proved by Schulman, Mor, and Weinstein. A bound for this value was found by Elias et al. and numerical testing supported the hypothesis that their bound may be the actual limit. A proof or disproof of whether their bound was the actual limit remained open for the past decade. Here, for the first time, we prove this limit. In the context of quantum thermodynamics, this corresponds to the maximum extractable work from the quantum system. We also establish, in the case of higher dimensional reset systems, how the performance of HBAC depends on the energy spectrum of the reset system.
Algorithmic cooling techniques provide tools to increase the purity of quantum states. It is known that the cooling with these techniques is limited. However, the physical root of the limit is still unclear. Here we show that the unitarity of the compression operation imposes the cooling limit of heat-bath algorithmic cooling. Specifically, we prove that the unitarity of the compression operation does not allow increasing the purity beyond the maximum of the individual purities. We formalize the limitations imposed by the unitarity of the compression operation in two theorems. We then introduce an optimal cooling technique and show that without the limitations of the unitary operations the cooling technique would exceed the limit of heat-bath algorithmic cooling. We also show that the limitations of unitarity lead to the exact cooling limit of algorithmic cooling. This work opens up new avenues to understanding the limits of dynamic cooling.
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 β-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.