Stationary states of a dissipative two-qubit quantum channel and their applications for quantum machine learning

Entangled-state preparation and preservation are the cornerstones of any quantum information platform. However, the strongest adversaries in quantum information science are unwanted environmental effects such as decoherence and dissipation. Here, we address how to control and harness these unwanted effects that arise from the coupling of a system with its environment, to provide stationary entangled states for quantum machine learning (QML). To do so, we design a dissipative quantum channel, i.e., a two-qubit system interacting with a squeezed vacuum field reservoir, and study the output state of the channel by solving the corresponding master equation, especially, in the small squeezing regime. We show that the time-dependent output state of the channel is the so-called two-qubit X-states that generalize many families of entangled two-qubit states. Also, by considering a general Bell diagonal state as the initial state of the system, we reveal that this dissipative channel generates two well-known classes of entangled mixed state and Werner-like states in the steady-state regime. Moreover, this channel provides an efficient way to determine whether a given initial state results in a stationary entangled state or not. Finally, we examine the potential application of the designed two-qubit channel for QML. In this line, we propose a general theoretical scheme for quantum neural networks (QNNs) implemented with the variational quantum circuits, which encode data in continuous variables (CVs) of the two-qubit states. The linear (also referred to as affine) and nonlinear (activation function) transformations are enacted in the QNN using the stationary states of the two-qubit channel and measurement process, respectively. Finally, we proceed to test our proposed model by solving some supervised binary classification tasks. Integrating the non-unitary transformation and parallel-processed neural computing on such a two-qubit channel establishes the requirements for a meaningful QNN. Such a CV-QNN model with sufficient layers may execute any algorithm implementable on a universal CV quantum computer.


Introduction
Quantum computers are devices that have the potential to revolutionize computing for certain classes of problems (Petit et al. 2020;Arute et al. 2019;Takeda and Furusawa 2019). They may solve NP-hard problems that classical computers are unable to deal with. Practically, it is not feasible to solve such problems using classical computer architectures (Rabanal et al. 2007). Hence, it is necessary to seek quantum algorithms for tackling NP-hard problems (Narayanan 1999). Deutsch showed that any physical process could be modeled perfectly on a quantum computer such that it would be able to perform tasks more quickly than traditional machines. Also, he stated that quantum machines exploit the phenomenon of quantum parallelism, where such machines compute each polynomial path in NP (non-deterministic polynomial) space in parallel. This implies that a quantum computer can be viewed as a massive, parallel processing machine which explores each path of an NP-complete or NP-hard problem in polynomial time (Deutsch 1985;Narayanan 1999). Entanglement detection in high-dimensional systems is an NP-hard problem because there is no efficient way to deal with such a problem (Hiesmayr 2021). However, several proposals have been proposed to address this issue thanks to the use of quantum neural networks (QNNs) (Cai et al. 2015;Lu et al. 2018;Yang and Zhang 2020;Levine et al. 2017). Very recently, a quantum algorithm has been developed to classify the entanglement of the input states, without any need to perform a measurement on the qubits encoding the state itself (Scala et al. 2022). Entanglement as a key quantum resource allows us to process and store exponentially more information than a classical computer (Nielsen and Chuang 2002;Chitambar and Gour 2019;Bub 2001). Indeed, the physical realization of a system in a general superposition of 2 n modes requires exponential resources classically and linear resources quantum mechanically because of the existence of entanglement (Ekert and Jozsa 1998;Jozsa 1997). Nevertheless, the noise sensitivity of quantum computers significantly affects their capabilities and advantages. For instance, background noise introduces errors, especially, on current-to-near-term quantum computers, and degrades their performance (Breuer and Petruccione 2002;Clerk and et al 2010).
Notably, unavoidable errors (effects) occur during the initialization phase of a qubit, which can be detrimental for quantum computing applications, only marginally affecting energy transfer and storage. Therefore, the inevitable noise may improve the performances of quantum batteries, i.e., the charging process (Gemme et al. 2022). Interestingly, local noise can enhance two-qubit teleportation. Since noise in general degrades quantum entanglement, hence, this enhancement may be due to an improvement in the classical correlations of the two-qubit states (Yeo 2008).
Generally, dissipation is always accompanied by fluctuations. The presence of noise along with dissipation is a manifestation of the so-called fluctuation-dissipation theorem of statistical mechanics (Scully and Zubairy 1999). Although unwanted environmental effects such as dissipation are the strongest adversaries of quantum information science, it has been shown that this statement always is not true. Dissipation can be exploited as a fully fledged quantum resource for universal quantum computation without any coherent dynamics needed to complement it. Notably, dissipation can drive a quantum system to its steady state which can be used for encoding the outcome of quantum computation (Verstraete et al. 2009).
The so-called dissipative quantum computing is of fundamental interest for QNN research, since it allows quantum computing algorithms based on dynamic attractors and steady states (Schuld et al. 2014;Rabinovich et al. 2006). Quantum mechanics is a linear theory. This linearity implies that mapping of one state onto another can be executed by linear operators (Jordan 2009). Therefore, the incompatibility between the nonlinear, dissipative dynamics of the NNs and the linear, unitary dynamics of coherent quantum computing may be addressed through dissipative quantum computing. Dissipative quantum computing is established based on the theory of open quantum systems. The idea of dissipative quantum computing is highly interesting for quantum machine learning (QML) (Schuld et al. 2014). While the total system, i.e., the principal system plus its environment still obeys the unitary evolution of quantum theory, the principal system alone undergoes non-unitary evolution (Verstraete et al. 2009;Schuld et al. 2014).
In this line, several investigations have focused on the dissipative dynamics of two-qubit systems. For instance, some authors analyzed the effects of squeezed parameters on quantum discord, quantum concurrence (Ji and Liu 2013), disentanglement, and decoherence (Ikram et al. 2007;Hernandez and Orszag 2008) in a two-qubit system subjected to dissipative environments, especially, whenever the two-qubit system is independently coupled to local reservoirs.
Yu and Eberly discussed the effects of quantum or classical noises on the dynamics of the quantum entanglement between two qubits interacting independently and demonstrated that entanglement displays a completely different behavior from the dynamics of the local coherence (Yu and Eberly 2004;. For both discrete-and CV-two-partite quantum systems, the initial entanglement in a two-mode squeezed state disappears in a finite time period in the thermal environment but may last for an infinite time in the vacuum environment (Duan et al. 2000;Fu-Li and Hong-Rong 2004).
So far a great deal of attention has been focused on the exploitation and mitigation of noise and dissipation (Kraus et al. 2008;Ticozzi and Viola 2012;Ghasemian and Tavassoly 2021). Recently, Smart et al. proposed a novel way for noise mitigation and also exploited noise for quantum simulations of open systems. In particular, they simulated stationary states on a quantum computer to obtain a unique spectroscopic fingerprint of the computer's noise (Smart et al. 2022). The simulation of open quantum systems encounters a challenge for universal quantum computing because the dynamical evolution of an open quantum system includes non-unitary operations. Nevertheless, the authors in Schlimgen and et al (2021) proposed a general quantum algorithm for simulating nonunitary time evolution on a quantum computer wherein any quantum operator can be decomposed into a linear combination of unitary operators.
On the other hand, the idea of open QNNs has been put forward based on dissipative quantum computing (Schuld et al. 2014). Some authors tried to deal with a QNN from the quantum computing perspective since complex quantum circuits can be constructed based on the structure of NNs. For example, S. Gupta and R. Zia proposed a quantum circuit in which each unitary quantum computation operation is followed by a dissipative operator (Gupta and Zia 2001). Also, it has been shown that a universal quantum processor can produce any desired unitary transformation on a two-qubit system when programmed with 15 classical inputs (Hanneke et al. 2010).
In this paper, we aim to generalize the latter situation to a more realistic case with non-unitary transformations. The idea is to design a dissipative quantum channel with free parameters to harness and exploit dissipation and decoherence for the generation of stationary entangled states. We demonstrate that the considered channel provides parameterized stationary entangled states such as the Werner-like and Bell states which are necessary for universal quantum computation. In analogy with the universal quantum processor that can produce any desired two-qubit unitary transformation, our dissipative quantum channel may be implemented for dissipative quantum computation based on the non-unitary transformations. We show that this dissipative quantum channel can be used to establish tractable QNNs that unify the advantages of superposition-based quantum computing and parallelprocessed neural computing, simultaneously.
The paper is organized as follows. Section 2 contains the dissipative dynamics of a two-qubit system under the influence of a squeezed vacuum field reservoir. Section 3 is allocated to the steady-state density matrix of the twoqubit system and the generation of stationary entangled states. Section 4 is allocated to the dynamics of quantum coherence between two qubits. In Section 5, we examine the application of the dissipative two-qubit channel for QML. Finally, the summary and conclusions are presented in Section 6.

Dissipative dynamics of a two-qubit system
In this section, at first, we briefly describe the time evolution of a quantum state through passing a quantum channel. Then, we study the dynamical evolution of a two-qubit system under the influence of a squeezed vacuum field reservoir. Indeed, we solve the corresponding quantum master equation and find its solution, i.e., the density matrix of the two-qubit system.

Quantum channel and quantum state evolution
Quantum computers can be implemented based on quantum gates similar to their classical counterparts that are constructed by combining the logical gates (Michielsen et al. 2017). Such an architecture is often called the circuit model of quantum computing (Mizel et al. 2007). It should be noticed that the evolution of a quantum state need not be unitary. In practice, this happens when one deals with an open quantum system where information about the quantum system is lost in the environment. In this case, the quantum state cannot generally be represented by a Hilbert space vector |ψ . In contrast, a density matrix ρ which is an operator on the Hilbert space well describes the state of the system. Any physical transformation that maps a density matrix ρ in into another one ρ out is referred to as a quantum channel (Nielsen and Chuang 2002;Jamiołkowski 1972;Choi 1975). Since physical systems are influenced by plausible experimental imperfections such as coupling to the environment, therefore, they undergo nonunitary evolutions. Both unitary and non-unitary operations occurring during quantum computation can be modeled by quantum channels.
In the continuation, we design a dissipative quantum channel using a two-qubit system and a squeezed vacuum field reservoir. We demonstrate when the two qubits interact globally with the reservoir, the system is driven to its stationary state.

Dissipative dynamics of a two-qubit system
Let us start with a chain link between contiguous qubits, shown in Fig. 1, where each qubit dissipates energy through two environments (one on the left and the other on the right). Indeed, two contiguous qubits dissipate energy into the same environment. Physically, a common environment mediates the interaction between the contiguous qubits. The qubit at the i-th site of the chain may be described by the ladder operators σ i and σ i † obeying the commutation relation [σ i , σ i † ] = σ z i . Besides, let us describe the radiation mode on the ith site of the chain by the ladder operatorŝ b i,j andb † i,j satisfying the relation corresponding to bosonic algebra [b i,j ,b † i,j ] = δ j,j . The interaction Hamiltonian of the chain can be obtained as (Memarzadeh and Mancini 2011) According to the general quantum reservoir theory (Scully and Zubairy 1999), the master equation of the chain in the standard Linbladian form reads as where L i = α(σ i + σ i+1 ) and α depend on the bath's parameter andρ denotes the reduced density operator corresponding the chain state. Here, we restrict our attention to a chain of two sites, i.e., a two-qubit system where its total Hamiltonian reads as (Bai and An 2021) where ω 0 is the transition frequency, andb is the annihilation operator of the kth field mode with frequency ω k . Also, g i,k denotes the coupling constant between the ith qubit with kth mode of the reservoir. Therefore, a twoqubit system plunged in a common thermal reservoir can be described by the following master equation where A 1 = √n + 1(σ 1 + σ 2 ) and A 2 = √n (σ † 1 + σ † 2 ) wheren is the number of thermal excitation in the common bath.
With the same procedure, a two-qubit system interacting with a squeezed vacuum field reservoir, schematically shown in Fig. 2, can be investigated by the following master equation (Ji and Liu 2013;Wu and Yu 2017) where is the spontaneous emission rate (global dissipation parameter) and the Lindblad operators read as Also, the quantities N + 1 = cosh 2 r, N = sinh 2 r, and M = e iθ cosh r sinh r refer to the quanta of the squeezed vacuum field reservoir where r being the squeezing parameter and θ is the squeezing angle (Ikram et al. 2007). It should be noted that the squeezed reservoir has been realized experimentally and widely applied in the relevant fields (Yanay and Clerk 2018;Zeytinoglu et al. 2017;Zippilli and Vitali 2021). Further reduction of the quantum fluctuations below the standard quantum limit can be achieved by squeezing (Drummond et al. 2004). Stable spin squeezing can be generated by squeezed-reservoir engineering (Bai and An 2021). Generally speaking, it can be shown that a squeezed vacuum reservoir considerably enhances the non-classical properties of a quantum system with respect to a thermal reservoir (Ji and Liu 2013; Kowalewska-Kudłaszyk and Leoński 2010; Ghasemian and Tavassoly 2017). In order to study the dynamics of the considered system, let us define the following computational basis {|1 := |e |e ; |2 := |e |g ; |3 := |g |e ; |4 := |g |g }, to recast the density matrix of the two-qubit as where ρ j,k (t) are the unknown time-dependent coefficients.
In what follows, we try to find the analytical solution of the master equation of the two-qubit system. Note that the analytical solution of the master equation (5) for a general initial state is too cumbersome. Here, we present the analytical expression of the density matrix of the two-qubit system by considering a Bell diagonal state (BDS) as the initial state of the system. The dynamical evolution of the density matrix of the system for a general initial state can be found in the Appendix. Note that all analytical expressions of the density matrix are obtained for the small squeezing regime (r 1), i.e., when the term L 1 contributes. The explicit expression of a BDS reads as (Horodecki 1996) where σ = (σ 1 , σ 2 , σ 3 ) is the vector of Pauli operators and T n are the diagonal elements of a 3 × 3 real matrix which denote the correlation between two qubits. Therefore, the time evolution of the density matrix of the two-qubit system in the small squeezing regime can be obtained as with where a = 2 cosh 2 r. Equation (10) introduces the socalled two-qubit X-states that generalize many families of entangled two-qubit states such as the Bell states (Nielson and Chuang 2000), Werner states (Werner 1989), isotropic states (Horodecki and Horodecki 1999), and maximally entangled mixed states (Ishizaka and Hiroshima 2000;Verstraete et al. 2001). From a pragmatic viewpoint, the universality of X-states relies upon their facility for both theoretical and experimental manipulation. Already, the two-qubit X-states have been realized with standard interactions arising in the context of nuclear magnetic resonance (Rau 2009; and with available technologies for generating Werner states (Zhang et al. 2002). Here, we propose a new practical approach for the generation of this well-known class of entangled state via the interaction of a two-qubit system with a squeezed vacuum field reservoir.

Steady-state density matrix
When a closed system reaches a stationary state, it will remain in that state for all time. However, if the closed quantum system is allowed to interact with an environment, it becomes an open quantum system. Now, we intend to find the stationary state of the considered dissipative twoqubit channel. Indeed, dissipation drives the system to its steady state. We show that for small squeezing, the steady state of the system depends on its initial state, so it will be of interest for encoding the outcome of quantum computing and stationary quantum state engineering as well as QML. Using Eq. (10), the steady-state density matrix of the twoqubit system in the small squeezing regime can readily be obtained by taking lim t→∞ ρ(t). So, the stationary density matrix of the system reads as where Let us rewrite this relation as Q = 1 8 (1 −T ) for the sake of simplicity. Now, we take a closer look at the steady state of the considered system. This state can be recast as which is an entangled mixed state. Since the outcome of the channel depends on its input state, therefore, Q ∈ [0, 1 2 ] is a free parameter. For instance, Q = 0 yields the ground state of the two-qubit system, i.e., |4 , while Q = 1 2 indicates that the outcome of the channel is the Bell state defined as |φ = 1 √ 2 (|2 − |3 ) which is a maximally entangled state. Otherwise, the outcome of the channel is a Werner-like state or an entangled mixed state for Q ∈ (0, 1 2 ). As a conclusion, the type of output states can be obtained by measuring the parameter Q. On the other hand, the input states can be classified by the outcome of the channel. Note that 0 ≤ Q ≤ 1 2 implies that −3 ≤T ≤ 1. Hence, input states can be classified into two general classes. The first class of initial states withT = 1 results in the pure state ρ(∞) = |4 4|. The second class of initial states is defined by −3 ≤T < 1, and provides the mixed states such as the Werner-like states. It is noteworthy to mention that the input and output of the channel are Bell states In other words, this so-called maximally entangled state remains unchanged through passing the channel. The results are itemized as below: • The initial states withT = 1 result in a stationary pure state ρ(∞) = |4 4|. • The initial states withT = −3 result in a stationary Bell state ρ(∞) = |φ φ|.
• The initial states with −3 ≤T < 1 result in a stationary mixture of the Bell state and the ground state of the two-qubit. Figure 3 shows some initial states of the two-qubit system that result in a stationary pure state. Although all considered initial states are BDSs, the first two plots are more similar to a standard X-type state because all their diagonal elements are non-zero. The initial state shown in Fig. 3(c) is a mixture of the ground and excited states of the two-qubit system. The last plot of Fig. 3 depicts a mixture of the state shown in Fig. 3(c) and the Bell state. Except for the third plot, the others possess non-zero off-diagonal elements which imply the initial coherence between the two qubits. In conclusion, we can state that the channel cannot protect the initial coherence and entanglement for these initial states, so they are classified into the first class of initial states. Now, let us consider some initial states that result in mixed states. According to Fig. 4, the initial states with T 1 = −1 and T 2 = −1 are X-type states where all their diagonal elements are non-zero in analogy with Fig. 3. The initial state with T 3 = −1 is a superposition of the semi-excited states of the two-qubit system without initial coherence, i.e., ρ(0) = 1 2 (|2 2| + |3 3|). The last case shown in Fig. 4(d) is an entangled mixed state constituted from a superposition of ground and excited states of the Indeed, there is no quantum coherence between two qubits in the steady-state regime when the system is initiated with these BDSs. The abreviations ee, eg, ge, and gg denote the two-qubit basis defined in Eq. (7). Also, T i with i = 1, 2, 3 are the diagonal elements of the two-qubit correlation matrix Fig. 4 The initial states with T = −1 which result in the entangled mixed state ρ(∞) = 1 2 (|φ φ| + |4 4|). In fact, the system may protect its initial coherence when the system is initiated with these BDSs. In other words, one can prepare a robust two-qubit system against the decoherence effect in this case. The abreviations ee, eg, ge, and gg denote the two-qubit basis defined in Eq. (7). Also, T i with i = 1, 2, 3 are the diagonal elements of the two-qubit correlation matrix two-qubit system and a Bell-like state. All these initial states lead to stationary mixed entangled states. Besides, it should be emphasized that the presence of off-diagonal elements in the density matrix is the signature of a quantum mechanical superposition, i.e., quantum coherence (Herbert 2011). Therefore, the quantum coherence between two qubits can be manipulated by controlling (i.e., choosing the proper) the initial state of the system. We analyze the dynamics of quantum coherence between two qubits in the next section. Note that, here, we have chosen the simplest free parameters of the dissipative two-qubit channel. Since the elements of the correlation matrix (T i ∈ R) are continuous parameters, various stationary entangled states can be generated by adjusting the initial states of the dissipative channel.
The problem of determining whether a given quantum state is entangled lies at the heart of quantum information processing. Since the realization of entanglement in highdimensional systems is an NP-hard problem, seeking new tractable ways for entanglement detection would be of fundamental interest. Here, we showed that the entanglement of the output state of the system can be obtained by evaluating the parameter Q which depends on the initial state of the two-qubit system. Therefore, this dissipative two-qubit channel provides an efficient way to determine whether a given initial state results in a stationary entangled state or not. This is an outstanding feature of a dissipative two-qubit channel since the stationary entangled states can be generated just by tuning the free parameters of a generic initial state. Also, the so-called Bell and Werner-like states can be generated and protected by this dissipative two-qubit channel. More importantly, the dissipative channel provides the stationary entangled states with the maximum degree of entanglement and facilitates the required quantum resources for realistic quantum information processing and dissipative quantum computation.

Quantum coherence
Generally speaking, all real systems are open to some extent. The coupling to the environment leads to very rapid decoherence, especially, in a macroscopic system. Roughly, decoherence implies the irreversible loss of quantum coherence, i.e., the conversion of a quantum superposition into a classical mixture (Wiseman and Milburn 2009). In other words, the state of an open system is subjected to decoherence due to interactions with the environment. In the open quantum context, the interaction between a system S and its environment E may be described by the following quantum master equation where H is the Hamiltonian of the system and L is a superoperator acting on a vector space of linear operators.
Note that the Hamiltonian H is usually used to describe the unitary dynamics of the closed systems. Nevertheless, this operator may generally include some environmental contributions. Since the coupling to the environment drives the system to a steady state, thus, the state ρ(t) asymptotically approaches some steady states ρ ss which can be obtained by the solution of the following stationary equation (Khrennikov 2020) i[H, ρ ss ] = Lρ ss .
Our analysis shows that the stationary solution of the system is not unique, especially, in the small squeezing regime, i.e., it depends on the initial conditions. Hence, it must be derived by first solving the corresponding master equation and then taking ρ ss = ρ(∞) = lim t→∞ ρ(t).
Generally, the density operator corresponding to a quantum state can be expressed as the sum of two terms, diagonal and off-diagonal elements, where P j is a complex number satisfying | P j | 2 = P j , i.e., it contains the phase, as well. Physically, the first term (sum) corresponds to the probability distribution {|j , P j }, while the process of vanishing of the second term is known as "decoherence." In other words, the distribution of the off-diagonal elements of the density matrix of the system during the interaction with the environment can be used to measure the coherence quantity. Using Eq. (10), this quantum coherence Fig. 5 The effect of squeezing parameter (r) on the time evolution of quantum coherence between two qubits against the scaled time parameter τ = t. In the first two plots in the absence of squeezing r = 0, the system completely loses its coherence as time passes, while in the second two plots the system shows quantum coherence even at the steady state regime. In all cases, the coherence between the two qubits can be improved by increasing the squeezing parameter. In right-hand plots, the coherence between the two qubits at the steady state surpasses its initial values, especially, in the presence of the squeezed reservoir. Note that T i with i = 1, 2, 3 denote the diagonal elements of the two-qubit correlation matrix between the two qubits in the small squeezing regime may be examined via the following relation which is known as the l 1 -norm quantum coherence (Baumgratz et al. 2014). It is worth noting that in the steady-state regime, the quantum coherence reduces as which depends on the initial state of the two-qubit system. In Fig. 5, we numerically study the time evolution of quantum coherence between the two qubits for different values of squeezing parameter and initial states of the system. At first, it can be seen that the time evolution of quantum coherent strongly depends on both parameters, i.e., the squeezing parameter and initial state. For zero squeezing (r = 0), the system gradually loses its initial coherence in the first three plots, but in the last plot quantum coherence gradually increases and finally the system reaches its maximum coherence in the steady-state regime. Indeed, in Fig. 5(a), (b), there is no quantum coherence between the two qubits in the steady-state regime because forT 1 = 1 the system is driven to the pure state ρ(∞) = |4 4|. On the contrary, in Fig. 5(c), (d), the system maintains its quantum coherence even in the steady-state regime since forT 1 = −1, the steady state of the system is a mixed state with non-zero off-diagonal elements, i.e., ρ(∞) = 1 4 (2|φ φ| + 3|4 4|). On the other hand, for non-zero squeezing r = 0, it can be seen the quantum coherence between the two qubits may be improved by increasing the squeezing parameter. It is worth noting that for r = 0 the squeezed vacuum reservoir transforms into a thermal reservoir at zero temperature. Therefore, the squeezed vacuum reservoir can be used to suppress the decoherence effect in the two-qubit system.

Machine learning based on dissipative two-qubit channel
This section is allocated to the application of the considered dissipative quantum channel for QML. First, we describe how a quantum classifier can be built based on the stationary state of the two-qubit channel. Then, we proceed to solve some supervised binary classification problems. Indeed, the output states of the channel are classified into pure and mixed entangled states. Especially, the task of quantum entanglement detection can be viewed as a binary classification problem. So far, several classical neural network (CNN) models have been proposed to solve this problem via both supervised and unsupervised approaches (Lu et al. 2018;Yang et al. 2019;Ma and Yung 2018;Chen et al. 2021;Mengoni and Di Pierro 2019). Here, we examine the potential application of a dissipative quantum channel for supervised ML by detecting the stationary entangled states of the quantum channel. In order to implement a device for ML, one should provide a system including free parameters for encoding the input data. Also, the coupling to the environment can be controlled and engineered to drive the system to a steady state where the outcome of the computation is encoded (Verstraete et al. 2009).
In what follows, we explain how the initial state of the two-qubit system can be used to encode classical data. It has been shown, both theoretically (Plenio and Huelga 2002;Clark et al. 2003) and experimentally (Krauter et al. 2011;Lin et al. 2013), that a global dissipative environment can establish stationary entanglement. The authors in Lin et al. (2014) experimentally studied the harnessing of dissipation for quantum information processing by producing an entangled state that is inherently stabilized against decoherence. Therefore, controlling the environmental effects, i.e., the global dissipation, allows us to provide desired steady states for QML. In practical application, both squeeze parameters r and θ take effects. The competition between the squeeze parameters r and θ determines the behavior the of system. Besides, the spectral density of the environment affects the dynamics of the system. It was shown that the quantum Fisher information can be enhanced for appropriate squeeze parameters (Wu and Yu 2017). Also, the absorptive and dispersive properties of the medium can be controlled using squeezed bath parameters and coupling field strength (Joshi et al. 2005).

Quantum classifier
Various methods may be exploited for building a quantum classification model, i.e., the so-called kernel methods and variational quantum circuits ). The main approach taken by the community is established by formalizing problems of interest as variational optimization problems and using hybrid systems of quantum and classical hardware to find approximate solutions (Benedetti et al. 2019). In the current work, we follow the latter approach and design a variational quantum circuit for CV-QML by encoding data into the initial states of a two-qubit system. It is inspired from two sides. On one hand, it is inspired by the structure of CNNs, which have demonstrated impressive performance on many practical problems. Particularly, one can emulate the recent work on photonics-based CNNs (Shen et al. 2017). On the other hand, one can leverage ideas from variational quantum circuits, which have recently become the predominant way of thinking about algorithms on near-term quantum devices Schuld and Killoran 2019;Moll et al. 2018;Havlíček et al. 2019).
Computation is the process that transforms an input x into an output y. The desired output state may be obtained by encoding the information within a physical system and by evolving it via a finite sequence of operations. Hence, computation yields the solution to the problem we wish to solve. As an interesting application, we address how the considered dissipative two-qubit channel can be exploited as a quantum classifier (QNN). In fact, the steady-state density matrix of the system depends on the free parameters of its initial state, i.e., the elements of the correlation matrix T . One can encode the classical data into these free parameters and consider the channel as a building block of the quantum circuit shown in Fig. 6.
The key components of this circuit are a sequence of electro-optical modulators such as phaseless beamsplitters ) and a number of two-qubit channels. The simplest classifier (a single-neuron network) can be implemented by using three modulators and only one dissipative channel. However, a fully fledged NN may be established by the concatenation of an arbitrary number of dissipative two-qubit channels and the corresponding modulators. The output of each channel can be obtained via the positive operator-valued measurement (POVM). Note that we consider a typical circuit and present a simple recipe for encoding data. In practice, more complicated circuits can be designed by considering additional quantum operations and changing the arrangement of the circuit's components.
In what follows, we briefly describe how the data are encoded and processed by the circuit.
At first, the classical data are encoded into the input signals of the circuit via the action of the displacement operator on the vacuum state, i.e., D(x i )|0 = |x i for i = 1, · · · , N. It is worth noting that the Pauli operators of the logical qubit may be obtained by displacement operators. In particular, the authors in Campagne-Ibarcq et al. (2019) demonstrated the successful preparation, readout, and full error correction of a logical qubit encoded in Gottesman, Kitaev, and Preskill (GKP) grid states of a superconducting microwave resonator. They showed that the two-dimensional square GKP code is stabilized by two displacement operators. Also, Touzard et al. reported a new interaction between superconducting qubits and a readout cavity that results in the displacement of a coherent state in the cavity, conditioned on the state of the qubit . Now, the signals enter their corresponding modulators. The action of modulators on the input signals reads as M|x i = |w i x i where w i is the coefficient corresponding to the i-th tunable modulator, i.e., a weight parameter that can be adjusted during the training process . The adjusted signals are fed into the initial parameters (w i x i → T i ) of the dissipative channel to be processed. Indeed, the output signals of the modulators are encoded into the free parameters of the BDS such that T i := w i x i . Then, the data are mapped to the steady In the intermediate layer, the signals (data) are initially adjusted by means of a sequence of tunable electro-optical modulators such that |wx = M(w)|x and then are fed into the dissipative channels. The channels map their input signals ρ(0) into the steady states ρ(∞) of the two qubits. Finally, the output signal can be obtained via the measurement process. Also, a classical computer is used to train the circuit and find the new set of free parameters. The circuit's parameters are updated accordingly for the subsequent round. Note that the input and intermediate layers can be expanded for a large-scale NN, i.e., a multi-layer classifier state by the action of the dissipative channels. In other words, dissipation and decoherence provide the following transformation where Q = − 1 8 (w 1 x 1 + w 2 x 2 + w 1 x 3 − 1), which recalls the affine transformation corresponding to QNN, i.e., |x → |wx+b . Finally, the outcome of the circuit can be obtained via the POVMs that are specified by a set of positive operators E m which sum to the identity such that E k > 0 and k E k = I. Once the POVM is applied and the outcome m is obtained, the quantum state transforms to a new state ρ m with probability p m given by In analogy to the nonlinear activation process based on projective measurement (Bausch 2020), we can obtain the outcome of the dissipative channel by measuring its steady states with respect to the basis vectors of the two-qubit system {|1 , |2 , |3 , |4 } . Superposition encapsulates the idea that the system is simultaneously in all possible states. Only when a measurement is performed, the system collapses in one of the candidate states. Quantum measurements may be performed by a collection {E m = |m m|, m = 1, 2, 3, 4} of measurement operators (Nielsen and Chuang 2002). These operators act on the state space of the system being measured, where the index m refers to the measurement outcomes that may occur in the experiment. In our case, the state of the system, immediately before the measurement, is ρ(∞), then the probability that outcome m occurs can be obtained as which implies that P 1 = 0, P 2 = P 3 = Q, and P 4 = 1 − 2Q. Note that the output of the circuit can be obtained by performing other kinds of measurements, as well as computing expectation values. The measurement operators are flexible. Based on the desired physical situations, different measurement processes may be chosen, i.e., homodyne, heterodyne, photon-counting . Generally, the outcome of the circuit (quantum neuron) can be obtained via two distinct approaches. On one hand, the measurement process can be performed on quantum hardware. In this case, the so-called quantum collapse occurs as a result of quantum measurements (Zak and Williams 1998) which provides the outcome of the computation. On the other hand, the measurement process can be simulated on a classical computer by considering a nonlinear function such as the sigmoid function or step function. In fact, it is possible to simulate classical neurons with sigmoid or step function activation while processing inputs in quantum superposition. The authors in Cao et al. (2017) described how to design and simulate the performance of a standard feedforward network and associative memories in the quantum regime. In fact, they simulated the measurement process by a class of circuits called repeat-until-success (RUS) circuits which recall a sigmoid-like nonlinear function.
It is worth noting that the NNs have two universal features: dissipativity and nonlinearity. A neural net can converge to an attractor due to dissipativity; however, this convergence is accompanied by a loss of information. Indeed, a NN filters out insignificant features of a pattern vector. The nonlinearity increases the NN capacity and allows one to store simultaneously many different patterns. Both universal features, i.e., dissipativity and nonlinearity, may be implemented in NNs by the sigmoid function. It is important to emphasize that a sigmoid function may be generalized into quantum mechanics formalism via the socalled quantum collapse, which describes what happens when a projective measurement is performed (Nielsen and Chuang 2002;Zak and Williams 1998).

Training process
A variational quantum circuit is characterized by free parameters that should be trained (Zoufal et al. 2021;Mari et al. 2020). There are various classification algorithms for training a quantum circuit. The variational QNNs can be trained via classical simulation or directly on quantum hardware. Here, we use the former for training our classifier. The classical simulation includes evaluating the cost functions and the gradients with respect to the free parameters. However, as the size of the network grows, this task becomes less tractable. Therefore, the direct simulation on hardware is likely necessary for large-scale networks .
In the classical simulation, we define a square cost (loss) function to measure the distance between the actual class label and the probability of measuring the positive class (Hur et al. 2022) where y i is the actual class label. Then, minimizing the loss function with respect to free parameters of the circuit forces p(y = 1) to be as larger as possible than p(y = 0) if the ith training data belongs to the positive class, and vice versa if it belongs to the negative class. Note that, here, we use Eq. (22) for solving binary classification problems. Generally, some loss functions Fig. 7 Schematic representation of a variational circuit for QML. A variational quantum circuit leverages and unifies the diverse dynamics of the CV quantum computing formalism (the linearity of quantum theory) and the dissipativity and nonlinearity of the NNs can be exploited for multi-label classification (Cappelletti et al. 2020). Finally, the outcomes of the circuit are post-processed by the classical computer which updates the parameters in order to minimize the loss function. The adjusted parameters are then provided again to the quantum circuit in a closed loop. Typically, the variational model is trained using the stochastic gradient descent optimization algorithm, and weights are updated using the backpropagation of the error algorithm. The quantum circuit parameters are updated accordingly for the subsequent round. A typical variational circuit for hybrid classicalquantum ML is schematically shown in Fig. 7. It is worth mentioning that, in a hybrid quantum-classical system, the variables of parameterized quantum circuits are optimized by popular classical routines (e.g., gradientbased optimization). This approach has been successfully applied in various practical applications (Benedetti et al. 2019;Chen and Yoo 2021). A complete overview of parameterized quantum circuit models that have been demonstrated experimentally on superconducting, trapped ion, and photonic hardware can be found in Benedetti et al. (2019). In order to have a better insight into the training process, it is convenient to have a look at the gradient descent algorithm and the backpropagation algorithm (Kingma and Ba 2014;MacKay et al. 2003).

Solving binary classification problem
Now, we are ready to implement the channel as a quantum classifier. Here, we focus on the binary classifiers. However, any binary classifier can be turned into a multilabel classifier, for example by using the one-versus-all strategy (a heuristic method for using binary classification algorithms for multi-class classification) (Lloyd et al. 2020). Let us assume the simplest case and describe how a dissipative channel can be implemented as a classifier. Consider a single-neuron network (perceptron) with only one dissipative two-qubit channel, i.e., three input signals. The output of the channel can be obtained by measuring where w i x i denotes the i-th signal after passing through its corresponding modulator. This relation recalls the weighted sum (i.e., the affine transformation) of CNNs which can be embedded into our quantum formalism. The affine transformations and nonlinear activation functions are two key elements in NNs .
In order to complete the action of a perceptron, one should consider a nonlinear transformation, |x → ρ(Q) → M(Q) where M(Q) : R → R plays the role of the nonlinear transformation corresponding to the activation function of a classical perceptron. In practice, such a nonlinear activation function can be implemented by performing a threshold measurement on the outcome of the channel. For instance, Eq. (13) implies that Q = 0 results in the non-entangled states; otherwise, we arrive at the entangled states. Hence, the classifier can be used for entanglement detection.
Generally, we use the multi-layer perceptron classifier algorithm for training the classifier. This model optimizes the log-loss function using the limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) or stochastic gradient descent (SGD) algorithm (Malouf 2002;Bottou 2010). Now, we proceed to solve some binary classification problems. As the first case, we want to know whether a given initial state of the two-qubit results in a stationary entangled state or not. At first, we provide a small dataset based on the steady-state density matrix of the system and train the circuit to find the corresponding optimized free parameters. Finally, we predict the stationary entangled states for some unseen initial states to examine the accuracy of the classifier. Here, we try to consider different initial states of the two-qubit including those that result in the entangled states (positive class) and non-entangled states (negative class) as can be found in Table 1. The classifier is trained with 75% of the data and tested with the rest 25% of the data. The results show that the whole dataset is correctly classified. The actual class labels can be obtained via Eq. (13) such thatT = 1 refers to negative class (N), Table 1 The dataset for training and test processes of the classifier. The classifier is trained and tested by this dataset. All cases are correctly classified into positive and negative classes denoted by P and N, including the entangled and non-entangled output states, respectively  while −3 ≤T < 1 indicate the positive class (P). The predicted labels are shown in Table 1. Note that the actual and predicted labels are the same for all cases. Naturally, misclassified results are always unavoidable due to exceptional records that establish a minimum error rate achievable by any classifier. To show this fact, we should prepare bigger data. Figure 8 shows the confusion matrices for a dataset including 1601 samples. The results are obtained with two typical circuits. The first circuit consists of only one two-qubit channel, while the second circuit is composed of two layers such that the first one has three dissipative channels and the second one has one dissipative channel. Indeed, the second circuit is a twolayer classifier whereas its second layer is a single-channel circuit. As can be found, a few samples are misclassified with a single-channel classifier, while the multi-channel classifier perfectly classifies all samples. The accuracies of the single-channel circuit on the train and test data are 96.8% and 97.0%, respectively.
The performance of the model for QML can be evaluated by considering some other classification problems. The results of classification corresponding to three well-known Sonar signals, Banknote authentication (Dua and Graff 2017), and Moons datasets are presented in Table 2. The Fig. 9 The results of classification for a typical Moons dataset with 1000 points. The results show that 883 out of 1000 points are correctly classified by a multi-layer classifier. The accuracies corresponding to the train and test datasets are 0.91% and 0.87%, respectively. The symbols 0s and 1s denote the data corresponding to the zero (upper moon) and one (lower moon) classes, respectively detailed numerical results corresponding to a typical Moons dataset with 1000 points are given in Fig. 9. As can be found, the classifier based on the dissipative two-qubit channel can solve these problems with relatively high accuracies as the entanglement classification. Generally, the accuracy of the classifier may be examined as well as improved by providing bigger and more reliable datasets. Hence, in order to find the actual accuracy of the classifier, the dissipative channel should be implemented in the lab and trained with big data. Here, the goal is to demonstrate the capability of the dissipative quantum channels for QML applications.
It should be noted that quantum computing consists of the preparation, evolution, and measurement of qubits. In reality, the qubits are not only manipulated by unitary quantum gates but also interact with an environment. Hence, it is necessary to explore the non-unitary dynamics of open quantum systems to obtain more realistic quantum computing models, i.e., dissipative quantum computing. On the other hand, the quantum perceptrons based on quantum measurement seem to give a more comprehensive solution for integrating the linear, unitary dynamics of coherent quantum computing and the nonlinear, dissipative dynamics of neural computing. However, they do not lead to the construction of a mature QNN model. So it is vital to seek a more advanced formulation of quantum theory to overcome this challenge. A dissipative quantum channel is a promising candidate for simulating a CNN because it provides stationary states via its highly dependent dynamics on the initial conditions (Schuld et al. 2014).

Summary and conclusion
Although unwanted environmental effects such as dissipation and decoherence destroy and wash out the crucial quantum resources of quantum information processing, this statement is not true for all physical situations. Here, we have shown that these unwanted effects that arise through the coupling of a system with its environment can be exploited for the generation of stationary entangled states and the realization of meaningful dissipative quantum computation. For this reason, we have considered a dissipative quantum channel composed of a two-qubit system plunged in a squeezed vacuum field reservoir and found the density matrix of the system by solving the corresponding master equation, especially, in the small squeezing regime. By considering a generic initial state for the system, we have shown that the channel generates the so-called X-type states that generalize a number of entangled states such as the Bell state and Werner-like state. Also, the results demonstrate that the channel can be used to generate a robust Bell state that remains unchanged through passing the dissipative channel. Measuring these entangled states, i.e., Bell states are critical for performing many of the quantum communication protocols and entanglement distribution across a quantum network (Duan and Kimble 2003;Zhao et al. 2007). Also, this dissipative channel may be useful in designing a Bell analyzer which was recently introduced for spectrally distinct photons (Lingaraju et al. 2022).
More interestingly, the channel can be exploited to generate stationary entangled states. Note that the output states of the channel can be classified into pure states and mixed states. Since the steady states of the channel depend on the initial state of the two-qubit, the system can be driven to a desired stationary state. As a consequence, entanglement detection can be established provided that the output state of the channel is measured. In this line, we have examined the potential application of the considered channel for universal quantum computing. To do so, we have designed a quantum circuit based on some dissipative channels and electro-optical components. Before information can be sent over a quantum network, it must be encoded into a quantum state. At first, we encode the classical data into the initial states of the two-qubit system. Then, they are processed through passing the circuit and finally measured via POVM. We have implemented the circuit as a QNN for dealing with feedforward classification problems.
In contrast to the traditional computers where logical gates operate on digital zeros and ones individually, in quantum computers, quantum gates operate on simultaneous superpositions of zeros and ones, keeping the quantum information protected as it passes through, a phenomenon required to realize true quantum networking. Besides, the peculiar quantum features such as quantum entanglement and coherence could be used to accomplish tasks impossible for classical systems. For instance, ensuring the security of communications and speeding up certain hard computational tasks may be achieved by the exploitation of these quantum resources. For instance, taking entanglement is generically an NP-hard problem, especially in highdimensional systems (Gurvits 2003). Nevertheless, ML is a powerful tool for extracting features or patterns hidden in large multi-partite datasets to tackle the quantum detection problem (Hiesmayr 2021). ML is also a natural candidate to extract correlated features of high-dimensional quantum systems. For instance, it can be used for quantum control (Bukov et al. 2018), state tomography (Chapman et al. 2016), measurement (Magesan et al. 2015;Hentschel and Sanders 2010), and many-body problems (Carleo and Troyer 2017). Particularly, the task of quantum entanglement detection can be formulated as a binary classification problem . Here, we have proposed a tractable recipe to understand how the dissipative two-qubit channel can be used for entanglement detection and also its prediction based on dissipative two-qubit systems. Consequently, the proposed circuit could provide a powerful tool to extract quantum features hidden in multi-partite quantum systems.
Author contribution E. Gh is responsible for the inception of the project and its ongoing design and development.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations
Ethics approval and consent to participate Not applicable

Conflict of interest
The author declares no competing interests.