Enhanced Quantum Synchronization via Quantum Machine Learning F. A. Cardenas-L´ opez´ ∗ and J. C. Retamal Departamento de F´ısica, Universidad de Santiago de Chile (USACH), Avenida Ecuador 3493, 9170124, Santiago, Chile and Center for the Development of Nanoscience and Nanotechnology 9170124, Estacion´ Central, Santiago, Chile M. Sanzy Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain E. Solano Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain IKERBASQUE, Basque Foundation for Science, Maria D´ıaz de Haro 3, 48013 Bilbao, Spain and Department of Physics, Shanghai University, 200444 Shanghai, People’s Republic of China We study the quantum synchronization between a pair of two-level systems inside two coupled cavities. By using a digital-analog decomposition of the master equation that rules the system dynamics, we show that this approach leads to quantum synchronization between both two-level systems. Moreover, we can identify in this digital-analog block decomposition the fundamental elements of a quantum machine learning protocol, in which the agent and the environment (learning units) interact through a mediating system, namely, the register. If we can additionally equip this algorithm with a classical feedback mechanism, which consists of projective mea- surements in the register, reinitialization of the register state and local conditional operations on the agent and environment subspace, a powerful and flexible quantum machine learning protocol emerges. Indeed, numerical simulations show that this protocol enhances the synchronization process, even when every subsystem expe- rience different loss/decoherence mechanisms, and give us the flexibility to choose the synchronization state. Finally, we propose an implementation based on current technologies in superconducting circuits. PACS numbers: Machine Learning, Quantum Information processing, Quantum Synchronization. I. INTRODUCTION observables of these systems [21–28]. Artificial intelligence (AI) and machine learning (ML) have Synchronization phenomenon refers to a set of two or more concentrated much attention in the last years. ML consists of self-sustained oscillators with different frequencies that are adaptive computational algorithms which can improve their forced to oscillate with a common effective frequency as re- performance, employing the history of data record [29]. In sult of the weak interaction between the system component physics, several fields have profited from the advantages of- [1], which is characterized by its non-reversible behaviour fered by ML, such as material science [30], high energy due to the time-reversal symmetry breaking [2]. This phe- physics [31] and condensed matter physics [32], finances [33], nomenon has been observed and used in biological systems state discrimination [34], design efficiently experiments [35], [3,4], engineering [5], and geolocalization, to name a few. just to name a few. Essentially, there are three types of learn- During the last decade, significant progress has been made in ing in ML, namely, supervised learning, unsupervised learn- the development of quantum platforms such as trapped ions ing and reinforcement learning [36]. In supervised learn- [6,7], nanomechanical resonator [8–10] as well as supercon- ing, the system learns from initial data to make future deci- ducting circuit and circuit quantum electrodynamics (cQED) sions. Regression (continuous output) and classification (dis- [11–13]. This important progress has made possible to study crete output) are considered as the archetypical supervised the synchronization phenomena at the quantum level [14–22]. learning algorithm. In unsupervised learning, the classes are Initially, arrays of quantum harmonic oscillators were stud- not defined from the beginning (classification), but they natu- arXiv:1709.08519v2 [quant-ph] 16 Jan 2019 ied. These systems have a classical limit since they can be ef- rally emerge from the initial data. In other words, the data is fectively treated as classical systems when the oscillators are organized in subsets based on correlations found by the algo- highly, allowing a natural comparison between classical and rithm. Data clustering is the most usual example of the unsu- quantum synchronization. However, the study of synchroniza- pervised learning algorithm. In reinforcement learning [37], tion in quantum systems without a classical counterpart such there is a scalar parameter, named rewarding, which evaluates as two-level systems becomes non-trivial and controversial. It the performance of the learning process. Depending on the re- has to be studied, among other techniques, through the natural warding, the system can decide whether the learning process is optimized or not. The use of quantum algorithms to accomplish machine ∗ Corresponding authors: learning tasks as well as the use of machine learning algo- [email protected] rithms to solve quantum information processing tasks has y [email protected] led to the emergence of Quantum machine learning [38– 43]. In this field, arise a new paradigm concerning the na- 2 ture of the machine learning components, namely, the agent mon frequency. In sectionIV, we discuss the implementation and the environment [44]. Four categories related to the of the ML protocol in superconducting circuits, considering nature of the learning components can be identified in this the near-term available technologies. In sectionV, we discuss two-party system: classical-classical (CC), classical-quantum the numerical aspects of the digital-analog decomposition of (CQ), quantum-classical (QC) and quantum-quantum (QQ). the master equation governing the system dynamics. Finally, The first of them is related to the classical machine learning. in sectionVI, we present conclusions, and perspectives. The second corresponds to the case where classical machine learning can address quantum tasks. The third corresponds to the quantum variance of classical machine learning algo- II. DIGITIZED QUANTUM SYNCHRONIZATION rithms. In this case, the quantum algorithms which overtake the performance of their classical counterpart have already Let us consider a system composed of two dissipative cav- been shown in supervised and unsupervised machine learning ities containing each one a two-level system. Both cavities [44–50]. The last category corresponds to the case in which interact via a hoping interaction, and a coherent driving field quantum systems comprise both agent and environment. In acts in one of the two-level systems to counterbalance the dis- such a case, the definition of learning has not been explicitly sipation present in both cavities. The dynamics of the system defined and has to be interpreted as the optimization of certain is described by the master equation (~ = 1). figure-of-merit [43, 52, 53]. Recently, a novel perspective of N=2 using ML algorithms to enhance quantum tasks has emerged, X y y y ρ_(t) = i[ ; ρ] + κ (2a`ρa a a`ρ ρa a`); (1) such as the use of genetic algorithms to reduce errors in quan- − H ` − ` − ` tum gates or quantum simulations [48], to learn an unknown `=1 transformation [54] or speed up quantum tomography [55], where the Hamiltonian is expressed in the rotating frame to generate a quantum adder [56], or to construct a quantum with respect to the laserH field as autoencoder [57] Furthermore, the use of neural network has N=2 proven to be a useful tool to address many-body physics prob- X y δ` z y − + = ∆`a a` + σ + g`(a σ + a`σ ) lems, to solve variationally the time-independent and time- H ` 2 ` ` ` ` dependent Schrdinger equation [32], to perform quantum state `=1 y y x tomography [55] and to classify phases of matter [58]. + J a1a2 + a1a2 + Ωσ1 : (2) In this article, we address the following question: can we y understand synchronization phenomena as a machine learn- Here, a` (a`) is the creation (annihilation) boson operator k ing protocol? To answer this question, we rely on the digital- of the lth field mode, while σ` stands for the k-component ization of the master equation governing the system dynam- Pauli matrix. ∆` = !p;` !d is the detuning between − ics. We show that the digitized dynamics leads to the same the lth field mode !p;` and the driving frequency !d, while result obtained in the analog case. We can identify in this δ` = !q;` !d stands for the detuning between the lth qubit − digitalization, fundamental elements of a quantum machine frequency !q;` and the driving frequency. Moreover, g is the learning (QML) protocol [54]. We realize the presence of QQ coupling strength between the field mode and the two-level paradigm of QML. We find that synchronization of two qubits, when implemented as a QML protocol, can be enhanced by adding a feedback mechanism, which lead to a reinforcement learning protocol. The enhancement on the synchronization q 1 Uq1 relies on the increase of the mutual information in the Agent- U Environment subsystem. Furthermore, the application of the q1,p1 p1 classical feedback protocol induces complete synchronization E of two-level observables in dynamics which do not synchro- Up1,p2 nize in absence of feedback mechanism. Finally, we propose p2 E an implementation with current technology in superconduct- Uq2,p2 q ing qubits. 2 Uq2 This article is organized as follows: In sectionII, we present the digital-analog decomposition of the master equation gov- erning the system dynamics and show that this decomposi- FIG. 1. Schematic