# Enhanced Quantum Synchronization via Quantum Machine Learning

## Abstract

We study the quantum synchronization between a pair of two-level systems inside two coupledcavities. 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 measurements in the register, reinitialization of the register state and local conditional operations on the agent and register 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 experience different loss/decoherence mechanisms, and give us flexibility to choose the synchronization state. Finally, we propose an implementation based on current technologies in superconducting circuits.

###### pacs:

Machine Learning, Quantum Information processing, Quantum Synchronization.## I Introduction

Artificial intelligence (AI) and machine learning (ML) have attracted much attention in the last decade. ML consists of computational algorithms which can improve their performance, even though this improvement has not been explicitly programmed michalski2013machine (). Several fields such as economy MLeconomy (), medicine MLMedicine (); LAVRAC19993 (), pattern recognition systems Plamodon2000 (); Lee1989 (), or social media BELLOORGAZ201645 () profited from the advantages offered by ML. Essentially, there are three types of learning in ML, namely, supervised learning, unsupervised learning and reinforcement learning Russell (). In supervised learning the system learns from initial data to make future decisions. Regression (continuous output) and classification (discrete output) are considered as archetypical supervised learning algorithm. In unsupervised learning, the classes are not defined from the beginning (classification), but they naturally emerge from the initial data. In other words, the data is organized in subsets based on correlations found by the algorithm. Data clustering is the most usual example of unsupervised learning algorithm. In reinforcement learning sutton1998reinforcement () there is a scalar parameter, named rewarding, which evaluates the performance of the learning process. Depending on the rewarding, the system can decide whether the learning process is optimized or not. Recently, a novel perspective of using ML algorithms to enhance quantum tasks has emerge, particularly by using genetic algorithms adder2017 (); genetic2016 ().

Synchronization phenomena refers to a set of two or more self-sustained oscillators with different frequencies that are forced to oscillate with a common effective frequency pikovsky2003synchronization (); luo2013dynamical (). The interaction between systems modifies the frequency at which each system oscillates. This phenomenon has been observed and used in biological systems Yamaguchi1408 (); Steriade1997 (), engineering PhysRevLett.109.064101 (), geolocalization, just to name a few. During the last decade, a significant progress has been made in the development of quantum platform such as trapped ions RevModPhys.75.281 (); HAFFNER2008155 (), nanomechanical resonator Chan2011 (); LaHaye74 (); RevModPhys.86.1391 () as well as superconducting circuit and circuit quantum electrodynamics (cQED) PhysRevA.75.032329 (); Clarke2008 (); wendin2016quantum (). This important progress has made possible to study the synchronization phenomena at the quantum level PhysRevLett.111.234101 (); PhysRevA.91.061401 (); Manzano2013 (); PhysRevLett.112.094102 (); PhysRevE.89.022913 (); PhysRevLett.112.014101 (); PhysRevLett.76.404 (). Initially, arrays of quantum harmonics oscillators were studied. These systems show the advantages that they have a classical limit, since they can be effectively treated as classical systems when the oscillators have many excitations. This allows a natural comparison between classical and quantum synchronization. However, the study of synchronization in quantum systems without a classical counterpart such as two-level systems becomes non-trivial and controversial. It has to be studied, among other techniques, through the natural observables of these systems PhysRevLett.100.014101 (); PhysRevB.80.014519 (); PhysRevA.88.042115 ().

In this article, we address how synchronization phenomena can be understood as a machine learning protocol. Our proposal relies on the digitization of the master equation that governs the system dynamics. We show that the digitized dynamics leads to the same result obtained in the analog case. Furthermore, we can identify all the fundamental elements of a quantum machine learning protocol. In this sense, we find that the synchronization of the two qubits can be enhanced when we add a feedback mechanism to machine learning protocol. Finally, we propose an implementation with current technology in superconducting qubits.

## Ii Digitized quantum synchronization

Let us consider a system composed by two dissipative cavities containing each one a two-level system. Both cavities interact via hoping interaction, and a coherent driving field acts in one of the two-level system to counterbalance the dissipation present in both cavities. The dynamics of the system is described by the master equation ().

(1) |

where the Hamiltonian is expressed in the rotating frame with respect to the laser field as

(2) | |||||

Here, () is the creation (annihilator) boson operator of the th field mode, while stands for the -component Pauli matrix. is the detuning between the th field mode and the driving frequency , also stands for the detuning between the th qubit frequency and the driving frequency, and is the coupling strength between the field mode and the two-level system. Finally corresponds to the strength of the driving field, is the coupling strength between cavities, and is the decay rate for the cavities. In this system configuration, it is already proven that quantum synchronization between observables of two qubits is achieved eneriz2017degree ().

Our approach to this problem is to analyze the quantum synchronization of the two qubits by considering a digital-analog version of the master equation given in Eq. (1). We show that both digital-analog simulation yield the quantum synchronization. The decomposition of the master equation into digital steps is shown in Fig 1. We can discriminate two different types of interactions in this decomposition, namely, non-local gates or analog block and local gates or digital terms. Analog blocks are associated with the interaction terms in Hamiltonian in Eq. (2), which correspond to Jaynes-Cummings and hopping terms. The dynamics associated with these terms can be implemented by the unitary operations () and defined as

(3a) | |||||

(3b) |

On the other hand, local gates are associated with the unitary dynamics of the free Hamiltonian in Eq. (2), and the dissipative dynamics of the Lindbladian terms of the master equation in Eq. (1). For both qubits, the local gates only correspond to the evolution of their respective free Hamiltonians,

(4a) | |||||

(4b) |

For cavities, the local operations are represented by the dynamical map given by the master equation

(5) |

for a time . Now we want to compute the expectation values of the observables of both two-level system ( and ) using the digitized master equation and compare them with the expectation values obtained by directly solving the master equation Eq. (1). In Fig 2, shows the numerical simulations for the expectation values of Pauli matrices for the qubit obtained by both methods. The calculation is carried out for a sufficiently large number of steps ( divided into 100 parts). As we infer from these results both approaches are equivalent.

We will show that if we interpret the proposed digitized terms of the master equation as a machine learning protocol in which qubits learn from each other, we would improve the speed of this learning procedure. Indeed, in a recent proposal for reinforcement quantum learning Lucas2017 (); adder2017 (), the agent and the environment do not directly interact, and the learning process is mediated by an ancillary system, namely the register. In our setup, the synchronization between both qubits is similarly carried out through the field modes. In this case, we can identify the agent and the environment with the qubit and , respectively, while the register is identified with the field modes. With the novelty that the register is now connected to a decoherence channel. In our case, the interactions among the elements are given by the digital-analog blocks. Therefore, under this identifications, the digitized master equation can be considered as a machine learning protocol, but without a feedback mechanism. In the following section, we will introduce the feedback mechanism in order to improve the learning protocol.

## Iii Enhanced quantum synchronization

In this section, we will study a more general situation of three two-level systems, each of them identified with a learning units; agent, environment and register. Notice that the previous case is equivalent to this one, since we study a case in which we are far below one excitation in the system, so the cavity can be well-approximated by a qubit. In this scheme, both agent and environment qubits interact only with the register. In addition, concerning dissipative and depolarizing noise channels, we consider two cases: (i) only the register is affected by noise, and (ii) all the qubits are affected by noise. The system dynamics can be described in general by the master equation

(6) | |||||

where are the collapse operators for each qubit given by for relaxation, and for depolarizing noise, and stands for agent, register and environment qubits, respectively, while is the system Hamiltonian written in the rotating frame with respect to the driving field

(7) | |||||

Here, is the -component Pauli matrix for the th two-level system, stands for the ladder operator for the th qubit, is the detuning between the th qubit with respect to the driving field , and is the exchange coupling strength. Additionally, is the time step in which the Hamiltonian acts in the system dynamics and is the rectangle function defined as

(8) |

The quantum machine learning protocol for this situation is depicted in Fig 3. The first stage in our protocol is to perform the Action i.e. we transfer information from the environment qubit and encode it in the register. To transfer the information we apply an analog block in the environment-register subspace represented by

(9) |

Afterwards, we perform the Percept which corresponds to transferring the register information towards the agent. Similar to the Action case, the transfer of information is done by applying a similar analog operation in the register-agent subspace,

(10) |

Local operations on each subsystem are applied depending on the case under study (i) or (ii). The digital steps will correspond to the following operations.

Case (i): for the agent and the environment, the digital steps correspond to the dynamics of their respective free Hamiltonian represented by

(11a) | |||||

(11b) |

while, for the register the digital step is the evolution given by the master equation

(12) | |||||

Case (ii): for all the learning units (agent, register and environment) the digital step acting in them is given by the master equation

(13) | |||||

The next stage is to perform the feedback process involving measurement, reinitialization of the register state, and conditional local operations on the agent and environment subspace. The first step in the feedback process is to project the register in an eigenstate of . Depending on the measurement outcome, there are two conditional operations: if the register is projected in the state , we initialize the register in the state . Otherwise, if the register state is projected in , we initialize the register in and we apply a local rotation in the agent and the register subspaces, represented by , where .

To elucidate how the quantum machine learning process with feedback mechanism enhances the synchronization, let us consider initial orthogonal states between agent and environment. Based on the figure of merit proposed in Ref Lucas2017 () (the maximal fidelity between the agent state with the environment state is an indicator of a successful learning process), this is the worse situation for the learning process. Without loss of generality, the initial state of the system is . Let us calculate the evolution for the expectation values of Pauli matrices for the agent and environment qubit, respectively, by iteratively applying the quantum machine learning protocol depicted in Fig 3. We will compare the result obtained for the aforementioned cases (i) and (ii) with the case without feedback mechanism. These results are depicted in Fig 4. In the machine learning protocol without feedback in Fig 4(a) the average of and do not exhibit any oscillation, since due to the interaction in Hamiltonian given by Eq. (7) and the initial state , the system only evolves in the subspace composed of states with one excitation i.e. . Thus, the expectation value for operators and always vanishes. On the contrary, acting on the state introduces a local phase on the state depending if the state is in or , then as depicted in Fig 4(a). As we can see in Fig 4(b) and Fig 4(c), including feedback process, the initialization of the register state in an eigenstate of yields the system to evolves in a subspace of one and two excitation states, hence, , and are different from zero. As a result, the synchronization is improved when compared with the case without feedback mechanism. Fig 5(b) (case (i)) differs from Fig 5(c) (case (ii)) mainly in that the presence of noisy channels acting in all the learning units produces a harmful effect on the evolution of the expectation values, i.e. the expectation values decreases faster than in case (i). However, despite this additional detrimental effect, the feedback mechanism still provides an enhancement in the synchronization.

Since the feedback mechanism described here produces a change in the subspace of the system evolution, this effect can be interpreted as an effective environment engineering process, in the sense that, the dark state at which the system must converge is changed because of the measurement and the local conditional operations. Thus, by changing the feedback mechanism (learning strategy), we could be capable of modifying the state in which the agent and the environment synchronizes.

## Iv Implementation in superconducting circuits

Our proposal can be implemented in a circuit quantum electrodynamic architecture with current technology. Indeed, for the realization of the qubit-cavity setup, current technology allows us to connect charge qubits and flux qubits to a microwave transmission line resonators. Our setup, depicted in Fig 5(a) is composed of two transmission line resonator coupled by a superconducting interference device (SQUID) through the current. This coupling allows us to tune the cavity frequency and the coupling strength between each resonator PhysRevLett.113.093602 (); PhysRevB.93.094514 (); Wulschner2016 (); nakashima2017adjustable (). Moreover, two transmon qubits PhysRevA.76.042319 () are capacitively coupled to the resonator through the voltage at the ends of the transmission line resonator. We choose charge qubits instead of flux qubit because charge qubits have coherence times larger than the flux qubits PhysRevB.86.100506 (); geerlings2013improving (); lucero2013extended (); barends2016 (). For the machine learning protocol implemented with qubits, our proposal based on circuit quantum electrodynamics architecture can be implemented by considering arrays of Xmon qubits PhysRevLett.111.080502 (); PhysRevA.92.012320 (); PhysRevLett.113.220502 (), as shown in Fig 5(b). Xmon qubits offer high coherence times and fast tunability.

Current technology has made possible the implementation of quantum feedback in superconducting circuits PhysRevLett.109.240502 (); riste2015digital (); Murch2016 (). A system based on a closed-loop circuit together with binary measurement has allowed to implement a protocol to rapidly reinitialize the state of a qubit, this process is done in a time scale at least one order of magnitude faster than the relaxation time of the two-level system PhysRevLett.109.240502 (); riste2015digital (). In addition, current technologies based on transmon qubits coupled to a microwave resonator has made possible to implement weak measurements, this weak measurement mechanism has allowed to monitor the Rabi oscillation between qubit states as well as to reconstruct a quantum state Murch2016 ().

## V Conclusion

We have shown that quantum synchronization between a pair of two-level system is achieved by considering the digital-analog decomposition of the master equation which governs the system dynamics. We can identify in this block decomposition the fundamental elements of a quantum machine learning protocol, namely, agent, environment and register. Afterwards, we have also equipped the machine learning protocol with a feedback mechanism based on measurements and reinitialization of the register state together with conditional local operations on the agent and environment subspace, substantially increasing its power and flexibility. Indeed, numerical simulations show an enhancement in the synchronization manifested in the number of operators that synchronizes and the rate in which the synchronization is achieved. Furthermore, by modifying the protocol, we may choose the state in which the system synchronizes. Finally, based on current technologies on superconducting circuit and circuit quantum electrodynamics, we have proposed and implementation of the quantum machine learning protocol with feedback.

## Vi Acknowledgments

We would like to thank L. Lamata and Daniel Z. Rossatto for the useful discussions. F.A.C.-L. acknowledges support from CEDENNA basal grant No. FB0807 and Dirección de Postgrado USACH. J. C. R. acknowledges the support from FONDECYT under grant No. 1140194. M. S. and E. S. acknowledge support from Spanish MINECO/FEDER FIS2015-69983-P and Basque Government IT986-16.

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