Extended Lipkin model: Proposal for implementation in a quantum platform and machine learning analysis of its phase diagram

The extended Lipkin model: proposal for implementation in a quantum platform and machine learning analysis of its phase diagram S. Baid1, A. S´aiz1,2, L. Lamata1, P. P´erez-Fern´andez2, A.M. Romero3,4,a , A. Rios3,4, J.M. Arias1and J.E. Garc´ıa-Ramos5,6 1Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 2Departamento de F´ısica Aplicada III, Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, E-41092 Sevilla, Spain. 3Departament de F´ısica Qu`antica i Astrof´ısica (FQA), Universitat de Barcelona (UB), c. Mart´ı i Franqu´es, 1, 08028 Barcelona, Spain 4Institut de Ci`encies del Cosmos (ICCUB), Universitat de Barcelona (UB), c. Mart´ı i Franqu´es, 1, 08028 Barcelona, Spain 5Departamento de Ciencias Integradas y Centro de Estudios Avanzados en F´ısica, Matem´atica y Computaci´on, Universidad de Huelva, 21071 Huelva, Spain. 6Instituto Carlos I de F´ısica Te´orica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain aNew address: Fujitsu Research of Europe, antonio.marquezromer[email protected]om (Dated: April 25, 2024) Background: In recent years, the implementation of Nuclear Physics models in quantum computers has emerged as a promising and novel area of research. Simultaneously, the study of quantum shape phase transitions in nuclear models has gained significant attention. Specifically, the phase diagram of the Interacting Boson Approximation (IBA) has been extensively explored, particularly in connection with large-particle-number-limit considerations. Interestingly, the Extended Lipkin Model (ELM) serves as a valuable alternative for mimicking the IBA phase diagram and holds the advantage of being more straightforward to implement within a quantum computing platform. Purpose: We explore the ELM and provide: i) calculations of the ground state energy using a variational quantum eigensolver; ii) a comprehensive formulation for implementing the dynamics of the ELM within a quantum computing platform, enabling the experimental exploration of the IBA phase diagram for systems with a small number of particles; and iii) a determination of the phase diagram of the model using different Machine Learning (ML) methods. We note that in the ELM, unlike the usual Lipkin model, both first- and second-order quantum shape phase transitions take place depending on the model parameters. Method: Initially, we employ the Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE) to calculate the ground-state energy of the ELM. Next, we introduce the essential formulation and procedures required to implement this model effectively in a quantum computing environment. Finally, we use ML techniques to identify the different phases and critical points of the ELM. Results: We successfully reproduce the ground-state energy of the ELM across the complete phase space of the model using the ADAPT-VQE algorithm. We provide the necessary framework for implementing the ELM in a quantum computing platform, ensuring that the model can be executed with controlled errors. Finally, we obtain meaningful ML predictions that allow us to determine the phase diagram of the model. Conclusions: Our findings offer compelling evidence that the implementation of a nuclear model like the ELM in a quantum computing environment is not only feasible but can also be achieved with manageable error rates. This realization opens the door to detailed experimental investigations of the phase diagram of the ELM (and indirectly of the IBA) in a quantum computer, further advancing our understanding of quantum shape phase transitions and nuclear structure. Keywords: Quantum Platforms Nuclear Models ADAPT-VQE Quantum Shape Phase Transitions Interacting Boson Approximation Extended Lipkin Model Machine Learning arXiv:2404.15558v1 [quant-ph] 23 Apr 2024 2 I. INTRODUCTION A phase is a specific state of matter characterized by a well-defined structure and physical properties. Phase transitions correspond to abrupt changes in some physical properties of a system as it passes from one phase to the other [1–3]. Phase transitions are associated with sudden shifts in the internal energy of the system that occur at specific values of a control parameter, that is, a physical magnitude that can be changed externally to the system. The point of change is called critical point and, in addition to changes in energy, qualitative changes in other physical quantities, called order parameters, also appear. Order parameters behave differently depending on the phase, so they can potentially be used to distinguish among different phases. Phase transitions are well defined and characterized in macroscopic systems [1–3]. When one looks into mesoscopic quantum systems, a natural question is whether one can find similar phase transitions to the macroscopic ones or not. One may also wonder whether the same concepts introduced to study classical phase transitions apply at the quantum level. The study of Quantum Phase Transitions (QPTs) is a popular and widely discussed topic that encompasses various branches of quantum many-body physics [3–6]. In recent years, there has been a growing interest in investigating low-dimensional lattice models due to advances in quantum computing [7–9]. In addition, concepts from quantum information theory have been used to characterize quantum critical phenomena [10–12]. On a different note, there has been a resurgence in the examination of structural changes in finite-size systems, where early signs of transitions can be observed [13]. In the realm of atomic nuclei, the Interacting Boson Approximation (IBA) [14] provides a straightforward, yet detailed, framework to study first- and second-order phase transitions. In the IBA, there are three limiting situations, called dynamical symmetries, in which exact analytical solutions to the complex nuclear many-body problem exist. These are called U(5), SU(3) and O(6) dynamical symmetries. Remarkably, each of these symmetries corresponds to a specific shape in terms of the usual nuclear deformation parameters: β, that measures the deviation from sphericity, and γ, that measures the deviation from axial symmetry. U(5) corresponds to spherical shapes, SU(3) to axial prolate deformed shapes and O(6) to γ-unstable deformed shapes [14]. Thus, the phase space of the IBA is usually represented as a triangle such that each dynamical symmetry is located at one vertex [15]. We show this so-called Casten triangle in Fig. 1. In this figure, the critical line delineating the phase transition from spherical to deformed shapes is depicted by a solid arc connecting the e(5) critical point (representing a second-order phase transition on the U(5)-O(6) line) and the x(5) critical point (representing a first-order phase transition on the U(5)-SU(3) line). The dashed line (anti-spinodal) and dotted line (spinodal) demarcate the regions where shape coexistence occurs within the model. At the mean-field limit, there are two phases in the model: a spherical phase and a deformed one. The transitions between these phases are of first or second order depending on the path followed in the model parameter space. For instance, the transition from the spherical U(5) dynamical symmetry to the deformed γ-unstable O(6) dynamical symmetry is exactly solvable. This corresponds to the boson pairing model initially solved by Richardson [16], and later applied to the IBA [17,18]. These methods provide access to solutions for a very large number of bosons along this line. Graphically, this line corresponds to one side of the Casten triangle U(5) →O(6). This is the only integrable line in the IBA phase diagram and there, a second order QPT exists. The interior region of the Casten triangle, as well as the other two sides, remain beyond the scope of large-scale numerical studies that are limited by the capabilities of standard IBA codes, which can handle systems with approximately 100 bosons [19]. In all this area a critical line of first order QPTs appears. Other phase diagrams corresponding to algebraic models have been studied, e.g., the Agassi model [20]. The model has been implemented in a quantum platform and the phases of the system was determined resorting to the analysis of the time evolution of a properly chosen operator [21]. In a further analysis, the phase diagram has been determined using Machine Learning (ML) methods [22]. Strictly speaking, QPTs only occur in macroscopic systems. One may consider different avenues to connect QPTs to the physics of few-body systems. First, we may want to gain access to numerical simulations that can address low as well as large particle numbers (large-Nlimit). A second complementary approach would be to encode the model in a quantum platform, where large-Nscalings may be tamed down in the long term. These two research avenues are particularly challenging for the IBA, but other models may not suffer from the same limitations. The purpose of this paper is precisely to overcome the limitations of the two analyses suggested above with a model that is similar, but not as challenging, as the IBA. We use here the extended Lipkin model (ELM) proposed in Ref. [23,24]. In the large-Nlimit, the ELM mimics the IBA-shape phase diagram. Moreover, the ELM is simpler than the IBA, and it can be formulated and solved in terms of spin-1 2Pauli operators. Consequently, the ELM can naturally be implemented in a quantum platform with standard qubit registers. Furthermore, the numerical solution to the ELM is accessible classically for a wide range of particle numbers. In this sense, the ELM is a perfect test-bed for QPTs employing both classical and quantum techniques. In this paper, we exploit a variety of ML and quantum computing methods to assess the different phases of the model at finite Nand to demonstrate the usefulness of the model for a wide range of applications. We ultimately hope to prove that this model has the potential to become a new standard for QPT studies in quantum devices. 3 O(6) U(5) SU(3) Path 1 e(5) x(5) ρ=1 , χ=0 λ=1 , α=0 ρ=0 λ=0 ρ=1 , χ=− 7 /2 λ=1 , α=1/ 2 Symmetric Phase Broken Symmetry Phase Path 3 Anti−Spinodal Line Spinodal Line Path 2 Critical Line FIG. 1. Phase diagram (or Casten triangle) for the IBA Hamiltonian (3) with control parameters ρand χ. The dots at the vertices mark the three dynamical symmetries in the model (U(5), O(6) and SU(3)) and the dots in between the U(5)-O(6) and the U(5)-SU(3) lines mark the two critical point symmetries (e(5) and x(5)) in the model. In addition, the relation with the ELM phase diagram defined in terms of the control parameters λand α(4) is shown. The paper is organized as follows. In Sect. II, we provide a short revision of the usual Lipkin model, the IBA and the ELM. We specifically discuss the relation between the IBA and the ELM phase diagrams. In Sect. III, we present an ADAPT-VQE implementation for the ELM, providing benchmarks with ground-state energies for a wide range of values of the control parameters. We introduce a proposal for the implementation of the ELM in a quantum platform, together with an evaluation of the possible errors, in Sect. IV. ML techniques are applied to investigate the phase diagram of the ELM in Sect. V. Finally, Sect. VI provides the summary and conclusions. II. THE MODELS A. The Lipkin model The Lipkin-Meshkov-Glick model (LMG) [25] (Lipkin model in short) describes a set of Nparticles interacting with a long-range interaction. Each particle is allowed to be in a two-level system, upper and lower, separated by an energy gap. If each particle is considered to have spin 1/2, the complete set can be represented by a collective spin  S. Defined as  S=PN i=1 siwhere siis the spin of each individual particle. The total collective spin S=N/2 is the relevant sector for this work and it includes the ground state of the system. The original Lipkin Hamiltonian [25] can be written1as HL= (1 −λ)(S+Sz)−λ N(S++S−)2= (1 −λ)(S+Sz)−4λ NS2 x,(1) where Sν=PN i=1 si,ν for ν=x, y, z and S±=Sx±iSy. Here, λis a control parameter, and the Hamiltonian can be diagonalized in the collective basis |SMS⟩, where S=N/2 is fixed and MSruns from −N/2 to +N/2 in unit steps and the dimension of the Hilbert space is dim = N+1. In this way, the complete energy spectrum can be obtained as a function of λ. Although the Lipkin model has a very simple structure, it displays interesting symmetry properties 1The original Lipkin Hamiltonian is really HL=εJz− V 2(J2 ++J2 −)− W 2(J+J−+J−J+). Taking W=V, up to a global scale factor, it can be written as, HL= (1 −λ)Jz− 4λ NJ2 xthat is, apart from a constant factor, Eq. (1). The factor 1/N is included to scale properly the relative strengths of the one- and two-body interactions. 4 a) b) −1 2 +1 2 N s t N FIG. 2. Lipkin model in terms of fermion operators (a) and, equivalently, in terms of scalar boson operators (b). and phase transitions. Its phase diagram has been extensively studied [26,27]. It is known that, in the form given by Eq. (1), the LMG presents two phases in the large-Nlimit. These symmetric and non-symmetric phases are separated by a second-order shape phase transition at the critical point, λc= 1/5. We note that the Lipkin model has become a standard benchmark and test-bed for implementation of quantum algorithms focused on nuclear structure [28–33]. B. The interacting boson approximation The interacting boson approximation (IBA) [14] is a successful and widely employed nuclear structure model valid to describe low-lying levels of medium and heavy nuclei. The model is formulated in a second quantization formalism that includes a scalar boson (sboson, L= 0) and five quadrupole bosons (dµboson, L= 2). The IBA has u(6) as dynamical algebra and three specific situations, called dynamical symmetries, can be distinguished for which analytical solutions to the complex nuclear many-body problem are obtained. These dynamical symmetries are named: U(5), SU(3) and O(6). A simplified form of the IBA Hamiltonian is the so-called Consistent-Q Hamiltonian (CQH) [34,35], HCQH =ϵ nd−κ Q(χ)·Q(χ),(2) where ndis the d−boson number operator, Q(χ)=s†˜ d+d†s+χ(d†˜ d)(2) is the quadrupole operator and ϵ, κ and χare the model parameters. The phase space of the IBA is usually represented in the Casten triangle (Fig. 1). Although the IBA is formulated within the context of the second quantization formalism, it is possible to develop a geometric representation of the model by utilizing the concept of coherent states and the intrinsic state formalism [36–38]. This alternative approach unveils geometric shapes reminiscent of the Bohr model [39], which are associated with the three dynamical symmetries of the IBA. Consequently, within this framework, the U(5) limit corresponds to spherical shapes, while the SU(3) limit leads to prolate axially symmetric shapes. Additionally, the O(6) limit gives rise to deformed γ−unstable shapes. Since the model presents several phases, studies of these and of the corresponding quantum phase transitions are of great interest. Sometimes, to study phase transitions, the CQH in Eq. (2) is rewritten, taking out a global scale factor, as HCQH = (1 −ρ)nd−ρ NQ(χ)·Q(χ),(3) where Nis the total boson number of the system. The specific values of ρand χthat produce the three IBA dynamical symmetries are given in Fig. 1. Excellent reviews presenting the IBA in relation with its phase diagram and the corresponding shape phase transitions are Refs. [40–44]. However, it is important to recognize that QPTs manifest themselves genuinely in macroscopic systems. Consequently, to study them without using mean-field methods, one needs to consider a significantly large number of bosons, which is not always computationally feasible. A possible solution is to look for a simpler model with a similar phase diagram, as the one presented in the next subsection. C. The extended Lipkin model The Extended Lipkin Model (ELM) [23] is a simple two-level model designed to mimic the IBA phase diagram. Although the Lipkin model was originally formulated in terms of fermions or spin operators, as discussed above, it can be alternatively formulated in terms of scalar bosons using the Schwinger representation [45] (see Fig. 2). Using this boson formulation, the ELM was introduced as a two-level model constructed on two types of L= 0 bosons, s 5 and t, with Hamiltonian H= (1 −λ)nt−λ NQ(α)·Q(α),(4) where the operators nt, which corresponds to the number of t−bosons, and Qαare defined as nt=t†t, Q(α)= (s†t+t†s) + α(t†t).(5) λand αare two independent control parameters that play the role of ρand χin the IBA Hamiltonian, Eq. (3). The total number of bosons N=ns+ntis a conserved quantity. We have deliberately written the ELM two-level boson Hamiltonian in the form of Eq. (4) so that it resembles the QCH of Eq. (3). This way, it becomes clear that sand tplay the role of the sand dbosons of the IBA respectively. Clearly, the difference resides in the quadrupole character of the dboson that leads to the u(6) dynamical algebra of the IBA, while the ELM is described by a u(2) dynamical one. Despite this important difference, the ELM captures the main features of the phase diagram of the IBA [23]. Using mean-field techniques similar to those used in the IBA [40–44], one can obtain the energy per particle surface for the ELM of Eq. (4) as a function of the control parameters, λand α, and a variational parameter, β, E(λ, α;β) N=β2 (1 + β2)2β2(1 −(1 + α2)λ)−β4αλ + (1 −5λ).(6) This allows us to obtain the classical limit of the Hamiltonian and to produce a phase diagram of the ELM model. The connection between the Hamiltonian in Eq. (4) and its form in terms of spin operators can be obtained by making the inverse Schwinger transformation, defining the collective spin  Sas: S+=t†s, S−=s†t, Sz=1 2(t†t−s†s).(7) Using this fermionic formulation, given a number of spin-1 2particles, the sector containing the ground state is given by a collective spin S=N/2 and the model Hamiltonian is written as, HEL =HL−λ Nα2(S+Sz)2−2αSx(S+Sz)+(S+Sz)Sx,(8) in such a way that for α= 0 the usual Lipkin Hamiltonian, HLin Eq. (1), is recovered. This Hamiltonian can also be easily diagonalized in a basis given by |SMs⟩with S=N/2. D. The phase diagram of the IBA and the ELM We employ Eq. (6) to determine the phase diagram of the ELM Hamiltonian in Eq. (4) (or Eq. (8) in the fermionic formulation). The corresponding ELM phase diagram mimics the IBA phase diagram depicted in Fig. 1for the IBA Hamiltonian in Eq. (3). As a matter of fact, the connection between the parameters of both models is rather simple, ρ→λ, χ → −r7 2α. (9) In addition to the critical line displayed by the solid curve, λc=1 5 + α2,(10) we show the spinodal (dotted) λsand antispinodal (dashed) λas lines, given by the expressions λs=α2−√α4+ 10α2+ 16 + 6 α2+ 10 , λas = 1/5.(11) We provide a schematic representation of the phase diagram in Fig. 3. The spinodal and antispinodal lines in this diagram define the region where the energy surface presents two minima: a spherical one and a deformed one, 6 β β β β ββ λ θ β β FIG. 3. Zoom around the critical line of the ELM phase diagram for the Hamiltonian Eq. (4) in the plane of the control parameters (λ, α). The value of αis directly connected with θthrough α=p2/3 sin(π/3−θ). For α= 0, the transition is second-order, whereas it is first-order otherwise. We show the critical line of Eq. (10) as a solid line, and the spinodal and anti-spinodal lines of Eq. (11) in dotted and dashed lines, respectively. We provide schematic representations of the energy surfaces as a function of the deformation parameter, β, in the different regions of the phase diagram. indicating a coexistence of both phases. In the symmetric phase, there is a single energy minimum at the origin (β= 0). When the spinodal line is reached, a second, non-symmetric minimum develops for β= 0. At the critical line, the symmetric and the non-symmetric minima are degenerate. Then, at the anti-spinodal line the symmetric minimum at β= 0 disappears to become an inflection point. From there on, the β= 0 point becomes a maximum and there are only non-symmetric deformed minima. The energy functional given by Eq. (6) is a case of the cusp catastrophe whose germ is x4[46]. For α= 0, which corresponds to the usual Lipkin model, there is an isolated point of second-order phase transition, with a β4behavior at the origin. In that case, the spinodal, anti-spinodal and critical points coincide at the critical value, λc= 1/5. For α= 0, the phase transition changes its character to first order with the critical point given by Eq. (10). Here, both minima (one spherical and one deformed) are degenerate and there is a coexistence region around the critical point, λc, between λas and λs. Moreover, for λ>λas, two non-symmetric minima coexist. E. ELM numerical calculations Having established the similarity of the IBA and the ELM models at the mean field level, we now proceed to present numerical ELM calculations for a finite particle number, N. We aim to show that precursors of the phase transitions are clearly observed even for small systems. All calculations are done for N= 6. We select three distinct paths that represent various regions of the phase diagram. Path 1 follows the α= 0 line, corresponding to the standard Lipkin model, which exhibits a second-order phase transition at the mean-field level. Path 2 traverses the α= 1/√2 line, associated with a first-order phase transition. Finally, path 3 keeps λ= 1 while varying αas the control parameter. Figures 4(a), 4(b), and 4(c) illustrate the correlation energies along these paths in the ELM phase diagram. In the large-Nlimit, the ground state energy per particle displays a change in slope at the critical point, λcof Eq. (10). This transition is gradual for α= 0 (path 1) due to the second-order nature of the phase transition. For α= 0, the transition is more abrupt, particularly for α= 1/√2 (path 2), as a result of a first-order phase transition. Despite the finite size effects in our simulations for a small system with N= 6, Fig. 4demonstrates precursors of a QPT in the ground state energy around λcfor paths 1 and 2, depicted in panels (a) and (b). Notably, the slope change near λcis smooth for path 1 (panel a), indicative of a second-order phase transition, and more abrupt for path 2 (panel b), suggesting a first-order phase transition. In contrast, no indication of a QPT is observed for path 3 (panel c). It is worth noting that the apparent crossings observed for some excited states in panel (c) are not genuine, 7 0 0.2 0.4 0.6 0.8 Control parameter -2 -1.5 -1 -0.5 0 0.5 1 Energy per particle (arbitrary units) 0 0.2 0.4 0.6 0.8 Control parameter 0 0.2 0.4 0.6 0.8 1 Control parameter Ground state ADAPT-VQE Ground state Excited states a) b) c) λλα α = 1/√2 λ = 1 Path 2 Path 3 Path 1 α = 0 ×√2 FIG. 4. Correlation energy diagram for the ELM for three different paths: a) path 1, α= 0 and λchanging from 0 to 1 (second order phase transition), b) path 2, for α= 1/√2 and λchanging from 0 to 1 (first order phase transition), and c) path 3, for λ= 1 and αchanging from 0 to 1/√2 (no phase transition). Calculations are performed for N= 6, and the represented magnitudes are per particle (E/N). Different line styles are used for different levels (full line for the ground state and broken lines for the excited states). Full (orange) dots are the results for the ground-state energy obtained with the ADAPT-VQE approximate calculations (see Section III). but rather avoided crossings, as the mechanism of level repulsion is at play in this scenario. In addition to the energy of the system, there are other physical properties that reflect the change of phase. For the ELM, the expectation value of (Sz+S) in the ground state can be considered as an order parameter that, in the large-Nlimit, is zero in the symmetric phase, and different from zero in the broken one. Panels a), b) and c) in Fig. 5 represent this order parameter, normalized to N, for N= 6 and the same three paths discussed in the preceding figure. For finite particle number, N, the sudden change of behaviour in this order parameter is not exactly at the mean-field critical value of the control parameter. Also for this observable, precursors for the phase transition are clearly observed in the ground state for paths 1 and 2 in the vicinity of λc, Eq. (10). In contrast, we do not see change in behavior along path 3, where the system is always in the broken phase. In addition, in Fig. 5the change in the order parameter at the critical point is more abrupt for path 2 than for path 1. This is due to the fact that, in the large-Nlimit, the transition in path 1 is of second order (continuous change in the order parameter at the critical point), while in path 2, it is of first order (discontinuous change in the order parameter at the critical point). III. ADAPT-VQE ALGORITHM FOR THE ELM We have so far discussed the phase diagram of the ELM. We have illustrated the richness of the phase diagram of this model, which is reflected in the properties of even relatively small systems. We now turn to discuss how simulations for such small systems may potentially be performed in quantum devices. A. The ADAPT-VQE algorithm The ADAPT-VQE algorithm [47,48] is a method within the class of Variational Quantum Eigensolver (VQE) approaches [49,50]. VQEs aim at solving variational problems, such as the ground state of quantum many-body 8 0 0.2 0.4 0.6 0.8 Control parameter 0 0.1 0.2 0.3 0.4 0.5 0.6 (<Sz>gs+ N/2)/N 0 0.2 0.4 0.6 0.8 Control parameter 0 0.2 0.4 0.6 0.8 1 Control parameter a) b) c) λλα α = 1/√2 λ = 1 Path 2 Path 3 Path 1 α = 0 ×√2 FIG. 5. Expectation value of Sz+N/2, normalized to N, for the ELM ground state along the three paths discussed in the text as a function of the control parameter. Calculations are done for N= 6. systems, by parametrizing their wave functions in terms of quantum circuits, and minimizing the corresponding energies employing classical algorithms. The hope is that such hybrid methods may provide a quantum advantage over classical simulations [49,50] ADAPT-VQE was originally devised to calculate bond dissociation curves in molecules [47], but has since been employed to study other physical systems like the Lipkin model [48] or nuclei [51]. The methods is iterative and starts from an initial reference trial state, |ref⟩. This state is updated by means of encoded operators in a previously-defined operator pool. Thus, at iteration n, the resulting quantum state |n⟩is given by a product |n⟩=eiθnAn|n−1⟩= n Y k=1 eiθkAk|ref⟩.(12) Here, θkwith k= 1,··· , n are nvariational parameters associated to the noperators, Ak. The optimal values of these parameters are obtained by minimization of the energy surface resulting from the expectation value of this state with the Hamiltonian. At iteration n−1, the algorithm must decide which operator to choose at iteration n. The gradient at θk= 0 can be written in a compact expression as [47] ∂E(n) ∂θkθk=0 =i⟨n|[H, Ak]|n⟩.(13) The algorithm selects the operator Anthat produces the largest gradient, and thus is expected to have the optimal descent in the energy surface. Compared to approaches without adaptive gradients, this should allow for a faster approach to the real energy minimum, potentially avoiding plateaus [52]. We note that the operators Akare selected from a common pool, and may be repeated along the iterative minimization. Within ADAPT, quantum circuits can be used to construct the states and measure the commutators. The variational parameters are optimized at each iteration minimizing the total energy surface, on a classical computer, aided by measurements in the quantum computer [51]. For the purpose of evaluating its performance, we do not operate explicitly with quantum circuits, but rather simulate the evolution of the algorithm on a classical computer. We intend to show that the formalism developed in Ref. [48] for the standard Lipkin model is also valid for the ELM. We note this is indeed the case, even though the ELM has a more complex phase diagram including first- and 9 second-order phase transitions. In order to apply the method, we need to express the collective spins appearing in the ELM Hamiltonian, Eqs. (1) and (8), in terms of individual spins, S+=X k a† k,+ak,−≡X k σk +,(14) S−=X k a† k,−ak,+≡X k σk −,(15) Sz=1 2X k (a† k,+ak,+−a† k,−ak,−)≡1 2X k σk z.(16) where σk ±≡(σk x±iσk y)/2. Here, σk x,σk y, and σk zare the Pauli matrices for the k-th site. a† k,±and ak,±are the creation and annihilation operators for a particle in the k−th position with spin projection ±corresponding to the upper or lower levels, respectively. Because the Lipkin model only allows vertical excitations, one can treat the upper and lower states within the same site as independent qubit projections. The model can be interpreted in the context of quantum information processing [53] without explicitly resorting to the Jordan-Wigner transformation [54], which reduces the size of the Hilbert space by a factor of 2. In this approach, the standard Lipkin model Hamiltonain is written as, HL= (1 −λ) S+1 2X i σi z!−λ 2N X i<j (σi xσj x−σi yσj y) + X i<j (σi xσj x+σi yσj y) ,(17) which leads to the ELM Hamiltonian HEL =HL−λ Nα2 S2+1 4X i<j σi zσj z+SX i σi z  −α 2SX i σi x+1 2X i<j σi xσj z+1 2X i<j σi zσj x .(18) In order to keep the circuits as shallow as possible, we restrict the operator pool to one- and two-body operators that act on particles in at most two pairs of levels. The specific form of the one-body operators that we include in the pool are: Gk +=σk ++σk −=σk x,(19) Gk −=−i(σk +−σk −) = σk y,(20) Gk 0=σk z,(21) while for the two-body operators, with j < k, our pool contains the operator structures: Tjk +=σj +σk ++σj −σk −=1 2(σj xσk x−σj yσk y),(22) Tjk −=−i(σj +σk +−σj −σk −) = 1 2(σj xσk y+σj yσk x),(23) Ujk +=σj +σk −+σj −σk +=1 2(σj xσk x+σj yσk y),(24) Ujk −=−i(σj +σk −−σj −σk +) = 1 2(σj yσk x−σj xσk y),(25) Vjk += (σj ++σj −)σk z=σj xσk z,(26) Vjk −=−i(σj +−σj −)σk z=σj yσk z,(27) Vjk 0=σj zσk z.(28) Note that this is the same pool of operators used for the standard Lipkin model in Ref. [48]. 16 0.01 steps, yielding a grid of 10201 data points. Each evolution is computed for a conveniently adimensionalized time t∈[0,4] with steps of size 0.04. Then, we label our simulations as either 0, if they correspond to the symmetric phase, or 1, if they correspond to the broken-symmetry phase. We remove the simulations of the points corresponding to the three paths mentioned in Sect. II D from the dataset. These, alongside a randomly selected 20% of the remaining points, are kept apart for testing purposes. The remaining data is then used for training and validation of the training. The cross-validation strategy employs a randomly selected 20% of the non-test data at each iteration (epoch) of the training. This validation data is used to confirm that there is no overfitting, sudden forgetting or other anomalies during the training process, as well as to optimize the hyperparameters of the CNN, such as the number of layers and the number of neurons per layer. The test dataset is used exclusively to compute the results reported here. First, we use Sz(t) in Eq. (38) to train the CNN with the labeled data. The output of the network is not a discrete classification; instead, it provides the probability, PBS, for the corresponding point in the phase diagram to belong to the broken symmetry (BS) phase. Ideally, this probability should be close to 0 in the symmetric phase, close to 1 in the broken phase, and in between those values when close to a critical point. Once the CNN has been trained, the performance is analyzed with the dataset that was not considered in the training. Figure 10 depicts the probability PBS for the testing dataset using the evolution of the operator Sz(t), a system size N= 6, an initial state |Φ0⟩= 1/√3(|mz=−N/2⟩+|mz= 0⟩+|mz=N/2⟩) and control parameters α∈[0,1/√2] and λ∈[0,1]. We show in panels a), b), and c) the results for the three relevant paths, considering the exact evolution for training and predicting. Other than the inaccuracy at the extremes of paths 1 and 3, for values λ= 1 and α= 0, respectively, the rest of the predictions are precise and mostly with complete certainty. Paths 1 and 2 show a very accurate prediction of the critical point (marked with the red dotted line), as PBS suddenly jumps from 0 to 1 precisely in this area, despite working with a relatively small system size. Panels d), e) and f) (including the insets which provide a zoomed view) give a more general view of the quantum phase diagram classification (using also Sz(t) and N= 6) across the entire parameter space, which is, once again, very precise. In panel d), the system is trained with the exact evolution operator (U). Symmetric and broken phases are well determined and the critical line (yellow line) is nicely reproduced, although a small part of the BS phase is predicted as symmetric (with low certainty). In panels e) and f), the Trotter evolution is considered, with nT= 4 and nT= 1, respectively. It is remarkable that these results are of the same quality as the exact ones, specially for nT= 1. This fact is particularly relevant as it means that it is not necessary to correct the Trotter infidelities, saving computational resources and, thus, reducing the total infidelity incurred by lowering the number of required quantum gates. Second, we use the correlation operator of Eq. (37) to train the CNN to predict the phase diagram. This option is motivated by the results obtained in Refs. [21,22]. Choosing different correlation functions can yield slightly different results and, due to the nature of deep learning algorithms, it is usually not possible to intuitively guess which one may be the best option. We stress that, while the following results were achieved using the Cz(1,2, t) correlation function, analogous results were achieved with all possible combinations of i, j. The correlations Cy(i, j, t) and Cx(i, j, t) might also be viable probes. We follow the same procedure just described for Sz, replacing its evolution for the time evolution of Cz(1,2, t) as input of the CNN. In addition, the initial state |Φ0⟩=| ↑↓↑↓↑↓ ⟩ was arbitrarily chosen. Note that, since σi zacts on a particular qubit, one cannot compute the correlation function in the collective spin basis. The results for this second choice of correlator are shown in Fig. 11. We reach the same conclusions as for Sz(t), but with an improvement in the ability of the CNN to correctly classify the quantum phase of the system and precisely identify the critical point. For the three selected paths, the CNN only shows uncertainty for the points closest to the critical points in paths 1 and 2 (λ= 0.20 and λ= 0.18), which is the best possible guess. This ML approach to phase classification has another advantage, which is its robustness against errors introduced before the training. This is generally true for properly constructed CNNs [69], but they become particularly robust for data such as the one at hand. The evolution of Cz(1,2, t) at each (λ, α) value has a particular shape, and the CNN learns the features of these shapes to classify them. The variation of the parameter λhas a very prominent effect on the shape of this evolution, while the parameter αhas a relatively small effect in comparison. With the addition of noise, data points at different values of αfor the same value of λbecome difficult to distinguish, but the CNN is still able to, overall, differentiate points at different values of λ. Moreover, the QPT depends mostly on λ, so even if the details on αare lost, the CNN can still classify most of the phases correctly, albeit with a shift in the exact location of the critical line. Having said that, one can train the Neural Network with the same kind of noisy data that one is expecting to predict. This allows the CNN to, in a way, learn the noise pattern, recovering a precise prediction of the QPT. Both of these issues are shown in the predictions of Fig. 12, where a random Gaussian noise with a standard deviation σ= 0.2 was added to the input data. Panel a) shows the results for a CNN trained with noiseless data, when predicting the phases of noisy data. We find that the predicted phase transition shifts, and the predictions of the symmetric phase enter the area of the BS phase, but the overall phase classification remains quite accurate. In contrast, panel b) shows a CNN trained on noisy data. Here, the misclassification issue is fixed and the phase transition is once again accurately predicted. We note, however, that infidelities in quantum gates and other sources of errors in quantum simulators are 17 0.0 0.2 0.4 0.6 0.8 1.0 λ 0.00 0.25 0.50 0.75 1.00 PBS Path 1 ( α = 0) 0.0 0.2 0.4 0.6 0.8 1.0 λ 0.00 0.25 0.50 0.75 1.00 PBS Path 2 ( α = 1 / p 2) 0.0 0.2 0.4 0.6 0.8 1.0 α 0.00 0.25 0.50 0.75 1.00 PBS Path 3 ( λ = 1) a ) b ) c ) Path 1 Path 2 Path 3 d ) L 1 L 2 L 3 U Path 1 Path 2 Path 3 e ) L 1 L 2 L 3 UT ( nT = 4) Path 1 Path 2 Path 3 f ) L 1 L 2 L 3 UT ( nT = 1) 0.0 0.2 0.4 0.6 0.8 1.0 PBS L 1 → λ = 0 , α = 0 ; L 2 → λ = 1 , α = 0 ; L 3 → λ = 1 , α = 1 / p 2 FIG. 11. Same as Fig. 10, but using the function Cz(1,2, t) computed in the spin basis for a system of size N= 6, with initial state |Φ0⟩=| ↑↓↑↓↑↓ ⟩. biased, unlike random noise. It remains unclear whether the infidelities of a real quantum simulator would affect the predictions of the CNN, although it is safe to assume that significantly smaller errors will have a significantly smaller impact. B. Supervised classification with partial information The main and most obvious disadvantage of supervised ML algorithms is that they require computing or labeling the data with prior knowledge of the correct classification. For that reason, one of the main goals of many supervised ML algorithms is to achieve a good enough generalization using a reduced dataset. We explore the generalization ability of our setup as follows. First, we train our CNN only with data corresponding to the Lipkin model of Eq. (1), that is, with α= 0 in the ELM Hamiltonian (8). Then, we try to predict the quantum phase diagram for the ELM, with α= 0. Note that the regular Lipkin Hamiltonian exhibits only a second-order QPT, while the ELM presents a first-order QPT for values of α= 0 [23]. Figure 13 shows the predicted quantum phase diagram using this approach. Since the CNN had no information about points with α= 0, we may expect the predictions to become inaccurate as we move towards large values of α. However, the CNN predictions remain surprisingly accurate for large values of α. We take this as a sign of the generalization capabilities of CNN’s. We stress that these results are not trivial. We are able to predict a first-order QPT in a fundamentally different Hamiltonian, with only data from the second-order QPT of the known Lipkin model. The prediction with only partial information has a slightly smaller categorical accuracy (98.86%) when predicting the quantum phase of the system, compared to the results of Fig. 11 (99.85%). In other words, using the complete quantum phase space for training provides marginally more accurate predictions. The difference is, however, not easily observed in Fig. 13. 18 a) b) Path 1 Path 2 Path 3 L 1 L 2 L 3 Path 1 Path 2 Path 3 L 1 L 2 L 3 FIG. 12. Effect of noise. Phase probabilities, PBS , for the testing dataset as predicted by the CNN using Cz(1,2, t), a system size N= 6, an initial state |Φ0⟩=| ↑↓↑↓↑↓ ⟩ and using the trotterized evolution UT(nT= 1). Panel a): results with a CNN trained on noiseless data and tested on noisy data (see text). Panel b): results with a CNN trained and tested on noisy data (see text). Path 1 Path 2 Path 3 L 1 L 2 L 3 FIG. 13. The same as in Fig. 11, but training the CNN only with points within path 1 (L1L2), i.e. α= 0. C. Unsupervised classification Finally, we are interested in correctly predicting the phase diagram of the system without any prior theoretical knowledge about it. To achieve this, the use of unsupervised methods, which do not employ labeled training data, becomes necessary. Of course, with this approach, new challenges arise. The lack of labeled data for training means that these algorithms rely on the definition of some sort of similarity between the data. As will be shown below, unsupervised ML will allow to distinguish points near the critical area from those that lay well apart. This will provide a way to deduce the quantum phase diagram, but requires some previous knowledge of 19 the model at hand, i.e., one needs to know where different phases exist. Of course, for more complex quantum phase diagrams with more than a single QPT, a different approach is likely to be necessary. There are many possible unsupervised ML algorithms that could be used, namely, clustering, factor analysis, principal component analysis, and embedding techniques. Analogously, different observables have been studied as input for the unsupervised algorithms, concluding that the evolution of the functions Cz(1,2, t) and Sz(t), which were used in the supervised ML, are unsuitable for the unsupervised cases, being impossible to obtain conclusive results. This was expected as part of the limitations of unsupervised ML, which cannot easily learn specific patterns or features to classify. We consider a different alternative and use as input the different correlation functions that arise from all possible combinations of i, j, and νin Eq. (37). This means that, in a system of size N= 6, for a single data point, i.e. a single value of αand λfor the Hamiltonian, 45 correlation functions are computed (note that Cν(i, i) = 0 and Cν(i, j) = Cν(j, i)). We use these correlations for a clustering algorithm known as Fuzzy C-means [70,71], which is a variation of the popular K-means clustering algorithm [72]. K-means [72] is a general-purpose clustering algorithm in which, given a set of nsamples  X={xi, . . . , xn}and a number of clusters K, cluster centroids  C={c1, . . . ,cK}are computed to minimize the objective function, f( C) = n X i k X j wi,j(||xi−cj||)2,(39) where wi,j ∈ {0,1}is a partition matrix that tells us whether sample ibelongs to cluster j(wi,k = 1) or not (wi,k = 0). We implicitly assume that xiand cjcorrespond to multidimensional vectors. Once the number of clusters has been fixed, cluster centroids are initialized, commonly either randomly or through some initialization function that may speed up the convergence, and each sample is assigned to the closest centroid. Then, the mean position of the samples assigned to each cluster defines the position of the cluster centroid. The samples are once again assigned to the new closest centroids and the process is repeated until convergence is reached. Fuzzy C-means is very similar to the K-means algorithm but the partition function wi,j ∈[0,1] is given by wi,j =1 Pk j′||xi−cj|| ||xi−cj′|| 2 m−1 ,(40) where m∈(0,∞) is a parameter that controls the “fuzziness” of the clusters. These weights are a measure of the degree of membership of the sample ito the cluster jand now appear in the partition function, Eq. (39), to the power of m, as (wi,j)m. Note that, contrary to K-means clustering, each sample can now be partially a member of several clusters simultaneously, to a certain degree given by the weight wi,j . In the limit m→1, the membership converges to either 0 or 1 and we recover the K-means algorithm. The bigger the value of the fuzziness parameter m, the larger the overlap between clusters and, as a matter of fact, in the limit m→ ∞, all samples belong to all clusters equally. A benefit of this fuzzy approach is that it allows to distinguish the points that are more representative of each cluster and those that are in-between. In Fig. 14, the value of the cluster membership for each point of the phase diagram is presented, using the Fuzzy C-means algorithm with k= 2 clusters and a fuzziness parameter m= 4. Since k= 2, wi,1= 1 −wi,2. Note that this algorithm is not capturing the quantum phases of the system, but whether a point is close to the critical area or the phase is well defined, without distinguishing between the symmetric or the broken phase. Thanks to this kind of soft clustering, we can identify the points that have a higher (or lower) degree of membership as the points closest to the QPT of the system. In fact, the dark green line in Fig. 14 corresponds to the maximum value of the degree of membership for each value of α, which is close to the theoretical critical points for each value of α(light green line). Notably, for a system with a single QPT and two phases, this prediction of the QPT line allows for a relatively precise reconstruction of the quantum phase diagram, even if the unsupervised ML algorithm is unable to explicitly distinguish the corresponding quantum phases. In general, predicting the location of the QPT with unsupervised ML methods can become a difficult task, since the system is expected to behave very similarly when measured on two points if both of them are near the critical point, even though they may correspond to different phases. This is specially true for small system sizes, like the ones used on the supervised ML methods before, where due to finite size effects the QPT becomes smooth instead of sharp. 20 FIG. 14. Cluster 1 membership for the phase diagram of the ELM, computed through the Fuzzy C-means algorithm for a system size of N= 6 (see text). The light green line λc= 1/(5 + α2) corresponds to the theoretical critical points. The dark green line corresponds to the maximum value of the membership for each αvalue. VI. SUMMARY AND CONCLUSIONS In this work, we have explored the possibility of implementing the extended Lipkin model (ELM) in a analog-digital quantum simulator. First, we have presented the model and shown the connection with the nuclear interacting boson approximation (IBA). We have paid special attention to the connection between both models in the large-Nlimit where the phase diagram is identical in the two cases. In both models, spherical and deformed phases exist and, depending on the trajectory followed in the phase space, first- and second-order quantum phase transitions appear. Furthermore, we have used the ADAPT-VQE algorithm to calculate the ground state energy. To this end, the Jordan- Wigner mapping is not needed, because the ELM in the spin representation is directly mapped into the quantum simulator. The results are very close to the exact ones, but the convergence strongly depends on the position of the Hamiltonian in the parameter space. Next, a proposal for the implementation of the time evolution operator of the ELM on a quantum platform based on the Digital-Analog Quantum Computation (DAQC) paradigm is presented. Calculations of the expected fidelities are also presented. Finally, we present a Machine Learning study to identify the phases of the ELM. This study proposes, as proxies to identify the phase of the system, the time evolution of ⟨Sz(t)⟩and a two-point correlator Cz(1,2, t), Eq. (37). Our analysis has three different flavours. We first perform a supervised classification task with full data. Then, we carry out the same exercise with partial, noisy information. Lastly, we also use an unsupervised classification exercise based on the C-means algorithm. In all cases, the results obtained capture the position of the different phases and can predict rather precisely the critical area. These studies lay the groundwork for the implementation of more intricate nuclear structure models on quantum platforms. In particular, the use of the ADAPT-VQE method to calculate the ground state energy or even the excited states opens the possibility of dealing with nuclear systems with large Hilbert spaces [51]. The proposed supervised method for determining the phase diagram of the system (see Sec. V) requires the calculation of the evolution of a well chosen observable for an arbitrary wave function that does not correspond to the ground state of the system. This is a very relevant fact, because the calculation of the ground state in medium and large size many-body systems can be very challenging. The method nonetheless provides accurate results using very simple states and taking advantage of Machine Learning techniques. Moreover, the use of unsupervised methods (see Sect. V C) has provided very promising results allowing to locate the position of the critical area in the phase diagram with minimal biases. 21 VII. ACKNOWLEDGMENTS This work was partially supported through the projects PID2022-136228NB-C21, PID2022-136228NB-C22, PID2020- 114687GB-I00, PID2020-118758GB-I00 funded by MCIN/AEI/10.13039/50110001103 and by “ERDF A way of making Europe”; “Ram´on y Cajal” grant RYC2018-026072 funded by MCIN/AEI/10.13039/501100011033 and by “ESF Investing in your future”; the “Consolidaci´on Investigadora” Grant CNS2022-135529 funded by MCIN/AEI/10.13039/- 501100011033 and by the “European Union NextGenerationEU/PRTR” and the “Unit of Excellence Mar´ıa de Maeztu 2020-2023” award to the Institute of Cosmos Sciences, Grant CEX2019-000918-M funded by MCIN/AEI/10.13039/501- 100011033. 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