Toward nuclear physics on a spin quantum simulator by detecting quantum phase transitions in the Agassi model

Abstract

A central challenge in nuclear physics is understanding quantum many-body systems governed by the strong nuclear force. The inherent complexity of these systems, combined with the limitations of classical computational methods, underscores the need for new approaches to study nuclear structure and dynamics. Here, we demonstrate that a spin-based digital quantum simulator using nuclear magnetic resonance, where nuclear spins simulate interacting fermions, offers a powerful tool to address this challenge. As a first step, we experimentally simulate the Agassi model, which encapsulates the interplay between collective and single-particle behaviors in finite nuclei. By representing nucleons as both bosons (nucleon pairs) and fermions (individual unpaired nucleons), the Agassi model captures highly non-linear interactions and is particularly suited for studying nuclear phase transitions, such as those between spherical and deformed shapes. We experimentally measure the correlation function as an order parameter during the evolution of the many-body system, successfully detecting a quantum phase transition. Specifically, we observe a sharp transition between the symmetric phase and the broken symmetry phase. This work underscores the potential of quantum simulation as a transformative tool in nuclear physics, particularly for exploring complex quantum many-body systems with applications in nuclear structure and reaction dynamics.

Graphical abstract

Keywords

quantum simulation / quantum phase transition / quantum many-body system

1 Introduction

In nuclear physics, the atomic nucleus consists of protons and neutrons (nucleons) bound together by the strong nuclear force. These particles interact in intricate ways, shaping the structure and properties of nuclei. The strong nuclear force, a short-range, attractive, and highly non-linear interaction, originates from quantum chromodynamics (QCD) [1, 2]. However, directly solving the QCD equations for nuclear systems is extremely challenging due to their complexity. To address this, effective models such as the shell model [3], liquid drop model [4], and interaction-based approaches [5–7] have been developed to approximate nucleon interactions and capture the essence of nuclear behavior without fully solving the QCD problem. This effort aligns with the broader domain of quantum many-body physics, which is essential for understanding the intricate interactions and collective dynamics of complex systems. The insights gained have significant implications, not only in nuclear physics but also in condensed matter physics. However, the inherent high dimensionality of these systems — commonly referred to as the “curse of dimensionality” — and the presence of strong correlations pose major challenges for both theoretical and experimental studies [8, 9]. To overcome these difficulties, a range of computational methods has been developed, including the Hartree−Fock method [10, 11], coupled-cluster theory [12, 13], variational Monte Carlo [14–16], Green’s function Monte Carlo [17], and the density matrix renormalization group [18–20]. These methods provide valuable insights into numerically solving key models in nuclear physics, such as the Lipkin−Meshkov−Glick (LMG) model [21], the Agassi model [22], and the neutron−proton pairing model [23].

Despite their significant successes, classical approaches inevitably face the challenge of “Hilbert space explosion” when simulating large systems [24]. The quantum states of a nucleus are described within a high-dimensional Hilbert space that grows exponentially with the number of nucleons, rendering classical methods impractical for large-scale simulations. In contrast, digital quantum simulators (DQSs) [25–29] naturally manage the exponential scaling of the Hilbert space. The complexity of representing and evolving a quantum state scales polynomially with the number of qubits, enabling simulations of much larger systems [30]. Furthermore, DQS can simulate the exact dynamics of a many-body Hamiltonian without relying on approximations, such as basis truncation or mean-field approximations [24, 31–35]. This enables more precise investigations of strongly correlated systems [36, 37], quantum phase transitions [38–42], and nonequilibrium dynamics — processes [43] that are essential for understanding the intricate dynamics within atomic nuclei. As quantum computing hardware continues to advance and control fidelity improves, DQS is expected to surpass classical approaches comprehensively, solidifying its role as a standard tool for addressing complex challenges in nuclear physics.

In this work, we demonstrate a spin quantum simulator based on nuclear magnetic resonance (NMR) [44–48] as a step toward advancing nuclear physics. This system has been developed as a 12-qubit DQS with high-fidelity coherent control [49, 50]. As a first demonstration, we show that the NMR DQS can be employed to simulate the Agassi model in an exact manner. The Agassi model is a significant framework in nuclear physics, particularly for describing the properties of nuclei by focusing on both collective and single-particle behaviors in finite nuclear systems [22]. It integrates aspects of the interacting boson model, which emphasizes collective motion such as vibrations and rotations, and the interacting fermion model, which focuses on single-particle excitations. Additionally, the Agassi model accounts for phase transitions in nuclei, such as the transformation between spherical and deformed shapes, and can explain phenomena like the coexistence of different nuclear shapes and certain collective excitation modes [22, 51–53]. We derive a Hamiltonian from the Agassi model suitable for experimental implementation and conduct a DQS of the model using NMR techniques. Our experimental results demonstrate quantum phase transitions from the symmetric phase (SP) to the broken symmetry phase (BSP) of the Agassi model, and we identify the critical transition point by precisely tuning the interaction strengths. This work paves the way for the use of DQS in nuclear physics, offering a promising path toward revolutionizing the field by providing tools to study and predict nuclear phenomena with unprecedented accuracy and efficiency.

2 Agassi model

Compared to real nuclear systems, the Agassi model is a schematic framework designed to capture the interplay between collective and single-particle behaviors in finite nuclear systems. It assumes a two-level quantum system where fermions can occupy these energy levels, with interactions occurring both within and between the levels. Thus, the Agassi model describes a two-level system consisting of N interacting fermions.

2.1 Hamiltonian

In this work, we focus on systems with an even number of particles, where the degeneracy \mathrm{\Omega } of each level is also even. A single-particle state in this space is denoted by |ϵ,m⟩, where m is the magnetic quantum number, taking values m=\pm 1,\pm 2,\pm 3, \dots ,\pm j (with j=\frac{\mathrm{\Omega }}{2}), and ϵ=\pm 1 specifies the energy level. By convention, ϵ=1 is assigned to the upper energy level, while ϵ=-1 corresponds to the lower energy level. The Agassi Hamiltonian for a system of 2\mathrm{\Omega } particles is expressed as

(1)H=\xi J^0-g{\displaystyle \sum _{ϵ,{ϵ}^{\prime }=-1,1}{A}_{ϵ}^{+}{A}_{{ϵ}^{\prime }}-{\displaystyle \frac{V}{2}}[(J^{+}{)}^2+(J^{-}{)}^2],}

where \xi J^0 represents the unperturbed Hamiltonian, with J^0 denoting the average particle number in the system and \xi denoting the single-particle energy. This term characterizes the level splitting in the two-level system. The parameter g specifies the strength of the pairing interaction between fermions, with the pair creation operator A_{ϵ}^{+} and the pair annihilation operator A_{ϵ}. Similarly, V defines the strength of the monopole interaction, where J^{+} and J^{-} act as the particle−hole creation and annihilation operators, respectively. This monopole interaction corresponds to collective excitations in the system and, when combined with pairing interactions, serves as a model for quadrupole interactions. A schematic representation of the model is provided in Fig.1(a).

More specifically, the operators in Eq. (1) are defined as

(2)\begin{array}{rl}{J}^0=& \frac{1}{2}\sum _m(c_{1,m}^{+}{c}_{1,m}^{}-c_{-1,m}^{+}{c}_{-1,m}^{}),\\ J^{+}=& (J^{-}{)}^{†}=\sum _m{c}_{1,m}^{+}{c}_{-1,m}^{},\\ A_{ϵ}^{+}=& (A_{ϵ}{)}^{†}=\sum _{m=1}^j{c}_{ϵ,m}^{+}{c}_{ϵ,-m}^{+},\\ N_{ϵ}=& \sum _{m=-j}^j{c}_{ϵ,m}^{+}{c}_{ϵ,m}^{},N=N_1+N_{-1},\end{array}

where c_{ϵ,m}^{+} and c_{ϵ,m} are the fermion creation and annihilation operators for the single-particle state |ϵ,m⟩ with c_{ϵ,m}^{+}=(c_{ϵ,m}{)}^{†}, respectively. The operator J^0 represents the average particle number in the system. The operator J^{+} is particle−hole creation operator for the state with a fixed m, which creates a particle−hole pair by exciting particles from a filled state to an empty one, such as from the lower level to the upper level, whereas J^{-} performs the reverse process as the particle-hole annihilation operator. The pair creation operator A_{ϵ}^{+} creates a pair of particles in level ϵ, while A_{ϵ} acts as the pair annihilation operator for states | ϵ,\pm m⟩. A schematic representation of these operators is provided in Fig.1(b). In addition, N_{ϵ} is the number operator, counting the number of fermions occupying the level ϵ, with N representing the total particle number.

2.2 The case of j = 1

Now, let us consider a simplified case of the Agassi model with j=1, where two fermions occupy each level. Under these conditions, we map the fermion operators of the system to spin operators via the Jordan−Wigner transformation [54]. The fermion creation and annihilation operators are transformed into spin raising and lowering operators on a one-dimensional spin chain. A schematic representation of this transformation is provided in Fig.1(c). The fermion annihilation operator is then expressed as

(3)c_i=({\otimes }_1^{i-1}I)\otimes {\sigma }^{-}\otimes ({\otimes }_{i+1}^N{\sigma }^z),

and the creation operator is represented as c_i^{+} =(c_i{)}^{†}. Here, N is the number of particles, I is the 2\times 2 identity matrix, {\sigma }^x, {\sigma }^y, and {\sigma }^z are the Pauli matrices, and {\sigma }^{\pm }=\frac{1}{2}({\sigma }^x\pm \mathrm{i}{\sigma }^y) are the spin creation and annihilation operators.

Considering the case of N=4, the system with four fermions is mapped to a one-dimensional spin chain. By relabeling the original fermion operators c_{1,m} as

(4)c_{1,1}\to c_1, c_{1,-1}\to c_2,c_{-1,1}\to c_3,c_{-1,-1}\to c_4,

the operators in the original Hamiltonian in Eq. (1) are now rewritten as: J^0=(1 /4 )( {\sigma }_1^z +{\sigma }_2^z-{\sigma }_3^z-{\sigma }_4^z), J^{+}=(J^{-}{)}^{†}=- {\sigma }_2^{+} {\sigma }_3^z {\sigma }_4^{-}-{\sigma }_1^{+}{\sigma }_2^z{\sigma }_3^{-}, A_1^{+}=( A_1{)}^{†}= {\sigma }_1^{+} {\sigma }_2^{+}, and A_{-1}^{+}=(A_{-1}{)}^{†}={\sigma }_3^{+}{\sigma }_4^{+}, respectively. Here, {\sigma }_i means the expanded Pauli operator of the i-th spin.

Therefore, the Agassi Hamiltonian in Eq. (1) for j=1 is decomposed into three distinct components, which are explicitly expressed in terms of spin operators as follows:

(5)\begin{array}{rl}H=& H_1+H_2+H_3,\\ H_1=& \frac{\xi -g}{4}({\sigma }_1^z+{\sigma }_2^z)-\frac{\xi +g}{4}({\sigma }_3^z+{\sigma }_4^z),\\ H_2=& -\frac{g}{4}({\sigma }_1^z{\sigma }_2^z+{\sigma }_3^z{\sigma }_4^z),\\ H_3=& -(g+V)({\sigma }_1^{+}{\sigma }_2^{+}{\sigma }_3^{-}{\sigma }_4^{-}+{\sigma }_1^{-}{\sigma }_2^{-}{\sigma }_3^{+}{\sigma }_4^{+}).\end{array}

Here, H_1 and H_2 depend only on {\sigma }^z. It is clearly that

(6)[H_1 ,H_2]=0, [H_2,H_3]=0 ,[H_1,H_3]\ne 0.

Moreover, a spin DQS is particularly well-suited to simulate this Hamiltonian and its evolution using the Trotter-Suzuki formula [26, 55, 56]. Quantum phase transitions can occur and be observed by tuning the system parameters.

2.3 Phase diagram

The Agassi model exhibits three distinct phases: a SP and two BSPs, corresponding to the spherical, deformed, and superfluid Bardeen−Cooper−Schrieffer (BCS) phases [57]. The model incorporates both pairing interactions and monopole interactions, with their respective strengths g and V, which simulates the complex quadrupole interaction of pairing-plus-quadrupole model. By tuning these two system parameters, the interplay between pairing and monopole interactions gives rise to competition, ultimately driving phase-transition behavior in the system.

The phase diagram of the Agassi model is presented in Fig.2. When the pairing interaction is dominant, the system forms a strong pairing state, where all fermions exhibit behavior analogous to that of a conventional superconducting state. This is referred to as the BCS phase. In contrast, when the monopole interaction is stronger, the system enters the deformed phase, characterized by deformation properties. In this phase, the system breaks spherical symmetry, giving rise to collective excitations such as quadrupole vibrations or rotational modes. For the case of j=1, the system undergoes a one-dimensional phase transition that depends solely on the parameter g+V. Specifically, the system resides in the SP when and transitions to the BSP when g+V>1 [53].

3 NMR experiment

3.1 Experimental setup

In the experiment, we use the NMR DQS to simulate the Agassi model with j=1. Four qubits are required to encode the occupancy of four fermions. The sample consists of ¹³C-labeled trans-crotonic acid, dissolved in d6-acetone. The four ¹³C nuclear spins serve as qubits, corresponding to the fermionic spin states c_1, c_2, c_3, and c_4 in Eq. (4), respectively. All experiments are conducted at room temperature on a Bruker 600 MHz NMR spectrometer, with the internal Hamiltonian given by

(7)

where {\omega }_i/(2\pi ) represents the Larmor frequency of the i-th spin, and J_{ij} denotes the scalar coupling constant between spins i and j. These parameters describe the interaction between the nuclear spins and the external magnetic field, as well as the coupling strength between the spins. We determine both parameters using standard NMR spectroscopy techniques, with specific values provided in Fig.3(a). In NMR, single-qubit rotations are achieved using transverse radio-frequency pulses, while two-qubit gates are implemented through the free evolution of the interaction terms in H_{\text{NMR}}. These operations, in combination, enable the realization of arbitrary unitary evolutions [58, 59].

3.2 Initialization and evolution

First, we set the parameter \xi to a large value (\xi \gg g, V), where the Hamiltonian is dominated by H_1. The system is initially prepared in the polarized state | {\varphi }_0⟩=|\downarrow \downarrow \uparrow \uparrow ⟩, which is the ground state of H_1, corresponding to the projection of the minimum angular momentum J^0=-1. We then perform a quantum quench by suddenly reducing \xi to a smaller value (\xi =1). After quenching, the state is an out-of-equilibrium state governed by the Hamiltonian H, which undergoes evolution driven by H. In NMR, the entire system is initially prepared into | 0000⟩ from thermal equilibrium using the spatial averaging method [60–62]. Subsequently, a 1 ms R_y(\pi ) pulse (a \pi rotation about the y-axis) is applied to the first two qubits, creating the state |{\varphi }_0⟩. Experimental details for the initialization step can be found in the Appendix.

The system then evolves under the Agassi Hamiltonian H in Eq. (5), with the evolution operator expressed as U(t)= {\mathrm{e}}^{-\mathrm{i}Ht}. Since the three terms in this Hamiltonian satisfy the commutation relation in Eq. (6), we use the following Trotter-Suzuki decomposition to implement its evolution:

(8)U(t)\simeq [{\mathrm{e}}^{-\mathrm{i}(H_1+H_2)\mathrm{\Delta }t}{\mathrm{e}}^{-\mathrm{i}{H}_3\mathrm{\Delta }t}{]}^M,

where M is the total number of decomposition steps and \mathrm{\Delta } t=t/M is the time interval for each step. In the experiment, we engineer the evolution of the four-body interactions using gradient-based optimization to generate shaped pulses [63–65], which significantly enhance control accuracy and shorten the pulse duration. The duration of the shaped pulse is 40\text{ms}, with a fidelity exceeding 0.995.

3.3 Correlation function as the probe

As mentioned above, the system undergoes a one-dimensional phase transition that depends solely on the parameter g+V. Specifically, the phase transition occurs at g+V=1, where the system transitions from the SP to the BSP by crossing the critical point [53]. The order parameter used to characterize this quantum phase transition should be as sharp as possible at the critical point. Given that the Agassi model involves highly nonlinear terms, the two-point correlation function C, which focuses on equilibrium properties and spatial correlations within the system, is defined as

(9)C_{i,j}^z=⟨{\sigma }_i^z{\sigma }_j^z⟩ -⟨ {\sigma }_i^z ⟩⟨{\sigma }_j^z⟩ ,

and serves as an excellent probe to characterize the phase transition in the Agassi model. Here, i and j denote different qubits, and we measure the correlation function C(t)=C_{1,2}^z(t) between C_1 and C_2 at different times t. In the experiment, by applying \pi /2 readout pulses on C_1 or C_2, we can obtain ⟨{\sigma }_1^z{\sigma }_2^z⟩, ⟨{\sigma }_1^z⟩, and ⟨{\sigma }_2^z⟩, as shown in Fig.3(b).

In Fig.4, we vary the number of decomposition steps M from 1 to 10, and numerically simulate the values of correlation function C(t) in Fig.4(a). We set g=V=0.5, so the quantum phase transition occurs at t=1. With the number of decomposition steps M increases, the numerical curve converges toward the theoretical result. At M=5, C(t) is already very clear to characterize the phase transition, indicating that correlation function is a good order parameter for investigating dynamics of the Agassi model. In Fig.4(b), we plot the survival probability |⟨{\varphi }_t|{\varphi }_0⟩{|}^2 with the parameter set as g=V=0.5 (g+V=1), where |{\varphi }_t⟩ =U(t )|{\varphi }_0⟩ is the final state after the evolution. The plot shows that the system is not trivial and undergoes drastic changes over the time interval.Towards the end of (g+V )t, the survival probability approaches 1, which may be related to the critical point of phase transition, as verified in Fig.4(a).

3.4 Experimental results

In the experiment, we measure the expectation values of ⟨{\sigma }_1^z{\sigma }_2^z⟩, ⟨{\sigma }_1^z⟩, and ⟨{\sigma }_2^z⟩ at different times t. The NMR readout techniques are detailed in the Appendix. These results, along with Eq. (9), enable the calculation of the correlation function C(t), which serves as a probe to detect the phase transition and its dependence on (g+V) t. As shown in Fig.5, we present the experimental results for three cases: SP, critical point, and BSP. The upper panel shows the measured quantities ⟨{\sigma }_1^z⟩, ⟨{\sigma }_2^z⟩, and ⟨{\sigma }_1^z{\sigma }_2^z⟩, while the lower panel displays the correlation function C(t) derived from these measurements.

In Fig.5(a), we set g=V=0.2, where the system is in the SP. The correlation function C(t) oscillates with time t, with the maximum amplitude remaining below 0.5, which is indicative of the system being in the SP. In Fig.5(b), we set g=V=0.5, which corresponds to the critical point of the model. In the experiment, the correlation function C(t) reaches a maximum amplitude of 1 without oscillations. In Fig.5(c), we set g=V=0.9, where the system is in the BSP. The correlation function C(t) reaches a maximum amplitude of 1, followed by ongoing oscillations. This result signifies that the system is in the BSP. Moreover, increasing the interaction parameters further enhances the oscillation amplitude, although the maximum amplitude remains capped at 1.

To illustrate the quantum phase transition of the Agassi model more clearly, we set the parameter g+V within the range (0,2 ] and measure the time evolution of the correlation function at intervals of 0.2. We plot the maximum amplitude of the correlation function C(t) as a function of g+V, with the experimental result shown in Fig.6. At the critical point g+V=1, the maximum amplitude of C(t) reaches 1 and remains around this value for larger interaction parameters. This is a clear signature of the phase diagram of the Agassi model, distinguishing the SP before the critical point and the BSP beyond it. The experimental results demonstrate that the characteristics of these quantum phases can be directly discerned from the time evolution of the correlation function C(t). More experimental detais can be found in the Appendix.

3.5 OTOC probe

Additionally, we note the growing interest in out-of-time-ordered correlators (OTOC), denoted as \mathfrak{C}(t), a technique that has gained widespread attention across various interdisciplinary fields. OTOC is a powerful tool for quantifying quantum information scrambling and dynamical sensitivity to local perturbations. It has been successfully applied on the NMR platform for observing dynamical quantum phase transitions [66].

It measures the non-commutativity between two operators, V(0) and W(0), in time evolution, and is defined as

(10)\mathfrak{C}(t)=⟨[W (t),V( 0){]}^2⟩,

where W(t) ={\mathrm{e}}^{\mathrm{i}Ht}W(0){\mathrm{e}}^{-\mathrm{i}Ht}. This time-dependent quantity captures operator spreading and information propagation, providing a useful tool for studying dynamical quantum phase transitions. Here, we select two operators from the single-qubit Pauli basis {\sigma }_i^k with i=1,\dots ,4 and k=x,y,z as W(0) and V(0) to measure the OTOC. We compare the results of the OTOC and the correlation function as probes, and present the numerical results in Fig.7. All selections of the operator combination W(0) and V(0) are simulated numerically, and their results fall into only four distinct categories. We choose four different combinations to entirely show the four distinct outcomes presented in the figure. As shown in the figure, the results obtained by the OTOC probe selection combination W(0)= {\sigma }_1^z and V(0)={\sigma }_2^z are completely consistent with the results theoretically calculated by our C(t), and that combination is the optimal selection that displays the sharpest phase transition in Fig.7(b). Notably, the OTOC probe does not provide a more significant characterization of the phase transition in the Agassi model. When the two operators are both {\sigma }_z, both probes exhibit the same behavior. This is why we choose the correlation function C(t) as the order parameter to detect the quantum phase transition of the Agassi model in our experiment, the consistency with the optimal combination suggests that, within our system, the correlation function C(t) serves as the most effective and reliable probe for detecting phase transitions.

4 Discussion

In this work, as a step toward simulating nuclear physics on a quantum processor, we perform the DQS of the Agassi model with N=4 sites using nuclear spins. The implementation uses a Jordan−Wigner spin-mapping approach within the framework of NMR, showcasing the feasibility of this method for solving quantum many-body systems. Through DQS, we successfully obtain the phase transition diagram of this nuclear physics model, providing experimental validation of its theoretical predictions. By systematically varying the interaction strengths g and V, we obtain the evolution dynamics and successfully map the quantum phase transition curve from SP to BSP. Notably, we experimentally confirm that the phase transition occurs at g+V=1, consistent with theoretical expectations and numerical simulation results.

This work underscores the significant potential of DQS in addressing complex quantum many-body problems. It provides a more effective solution to such systems than classical computational methods, overcoming the inherent high-dimensional challenges in many-body systems. For nuclear physics, DQS offers a transformative platform for simulating and probing models that capture intricate interplays. This represents a key step toward understanding phenomena such as nuclear structure, phase transitions, and reaction dynamics, paving the way for studying larger and more sophisticated nuclear physics models.

References

[1] R. Machleidt and D. R. Entem, Chiral effective field theory and nuclear forces, Phys. Rep. 503(1), 1 (2011)

[2] L. Meng, B. Wang, G. J. Wang, and S. L. Zhu, Chiral perturbation theory for heavy hadrons and chiral effective field theory for heavy hadronic molecules, Phys. Rep. 1019, 1 (2023)

[3] E. Caurier, G. Martınez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, The shell model as a unified view of nuclear structure, Rev. Mod. Phys. 77(2), 427 (2005)

[4] V. M. Kolomietz and S. Shlomo, Nuclear fermi-liquid drop model, Phys. Rep. 390(3), 133 (2004)

[5] E. Epelbaum, H. W. Hammer, and U. G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys. 81(4), 1773 (2009)

[6] N. Ishii, S. Aoki, and T. Hatsuda, Nuclear force from lattice QCD, Phys. Rev. Lett. 99(2), 022001 (2007)

[7] K. Hebeler, Three-nucleon forces: Implementation and applications to atomic nuclei and dense matter, Phys. Rep. 890, 1 (2021)

[8] A. Rocchetto, E. Grant, S. Strelchuk, G. Carleo, and S. Severini, Learning hard quantum distributions with variational autoencoders, npj Quantum Inf. 4, 28 (2018)

[9] G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355(6325), 602 (2017)

[10] Y. Aboussir, J. M. Pearson, A. K. Dutta, and F. Tondeur, Nuclear mass formula via an approximation to the Hartree–Fock method, At. Data Nucl. Data Tables 61(1), 127 (1995)

[11] A. Tichai, J. Langhammer, S. Binder, and R. Roth, Hartree–Fock many-body perturbation theory for nuclear ground-states, Phys. Lett. B 756, 283 (2016)

[12] R. F. Bishop, An overview of coupled cluster theory and its applications in physics, Theor. Chim. Acta 80(2−3), 95 (1991)

[13] D. J. Dean and M. Hjorth-Jensen, Coupled-cluster approach to nuclear physics, Phys. Rev. C 69(5), 054320 (2004)

[14] J. W. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Quantum Monte Carlo methods for nuclear physics, Rev. Mod. Phys. 87(3), 1067 (2015)

[15] C. Adams, G. Carleo, A. Lovato, and N. Rocco, Variational Monte Carlo calculations of A≤ 4 nuclei with an artificial neural-network correlator ansatz, Phys. Rev. Lett. 127(2), 022502 (2021)

[16] C. Özen, Y. Alhassid, and H. Nakada, Crossover from vibrational to rotational collectivity in heavy nuclei in the shell-model Monte Carlo approach, Phys. Rev. Lett. 110, 042502 (2012)

[17] D. M. Arnow, M. H. Kalos, M. A. Lee, and K. E. Schmidt, Green’s function Monte Carlo for few fermion problems, J. Chem. Phys. 77(11), 5562 (1982)

[18] U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77(1), 259 (2005)

[19] F. Verstraete, T. Nishino, U. Schollwöck, M. C. Ban˜uls, G. K. Chan, and M. E. Stoudenmire, Density matrix renormalization group, 30 years on, Nat. Rev. Phys. 5(5), 273 (2023)

[20] E. D. Jurgenson, P. Navratil, and R. J. Furnstahl, Evolving nuclear many-body forces with the similarity renormalization group, Phys. Rev. C 83(3), 034301 (2011)

[21] H. Lipkin, N. Meshkov, and A. Glick, Validity of many-body approximation methods for a solvable model (i): Exact solutions and perturbation theory, Nucl. Phys. 62(2), 188 (1965)

[22] D. Agassi, Validity of the BCS and RPA approximations in the pairing-plus-monopole solvable model, Nucl. Phys. A. 116(1), 49 (1968)

[23] S. Frauendorf and A. O. Macchiavelli, Overview of neutron–proton pairing, Prog. Part. Nucl. Phys. 78, 24 (2014)

[24] B. Fauseweh, Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges, Nat. Commun. 15(1), 2123 (2024)

[25] R. P. Feynman, Simulating physics with computers, in: Feynman and Computation, CRC Press, 2018, pp 133–153

[26] S. Lloyd, Universal quantum simulators, Science 273(5278), 1073 (1996)

[27] U. L. Heras, A. Mezzacapo, L. Lamata, S. Filipp, A. Wallraff, and E. Solano, Digital quantum simulation of spin systems in superconducting circuits, Phys. Rev. Lett. 112(20), 200501 (2014)

[28] I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86(1), 153 (2014)

[29] B. Cheng, X. H. Deng, X. Gu, Y. He, G. Hu, P. Huang, J. Li, B. C. Lin, D. Lu, Y. Lu, C. Qiu, H. Wang, T. Xin, S. Yu, M. H. Yung, J. Zeng, S. Zhang, Y. Zhong, X. Peng, F. Nori, and D. Yu, Noisy intermediate-scale quantum computers, Front. Phys. (Beijing) 18(2), 21308 (2023)

[30] F. Tacchino, A. Chiesa, S. Carretta, and D. Gerace, Quantum computers as universal quantum simulators: State-of-the-art and perspectives, Adv. Quantum Technol. 3(3), 1900052 (2020)

[31] J. Dukelsky, J. E. Garcıa-Ramos, J. Arias, P. Perez-Fernandez, and P. Schuck, Number conserving particle-hole RPA for superfluid nuclei, Phys. Lett. B 795, 537 (2019)

[32] R. Sweke, M. Sanz, I. Sinayskiy, F. Petruccione, and E. Solano, Digital quantum simulation of many-body non-Markovian dynamics, Phys. Rev. A 94(2), 022317 (2016)

[33] A. Smith, M. Kim, F. Pollmann, and J. Knolle, Simulating quantum many-body dynamics on a current digital quantum computer, npj Quantum Inf. 5, 106 (2019)

[34] M. Illa, C. E. Robin, and M. J. Savage, Quantum simulations of SO(5) many-fermion systems using qudits, Phys. Rev. C 108(6), 064306 (2023)

[35] T. Nikšić, D. Vretenar, and P. Ring, Relativistic nuclear energy density functionals: Mean-field and beyond, Prog. Part. Nucl. Phys. 66(3), 519 (2011)

[36] X. Nie, X. Zhu, Y. Fan, X. Long, H. Liu, K. Huang, C. Xi, L. Che, Y. Zheng, Y. Feng, X. Yang, and D. Lu, Self-consistent determination of single-impurity Anderson model using hybrid quantum-classical approach on a spin quantum simulator, Phys. Rev. Lett. 133(14), 140602 (2024)

[37] T. Xin, Y. Li, Y. Fan, X. Zhu, Y. Zhang, X. Nie, J. Li, Q. Liu, and D. Lu, Quantum phases of three-dimensional chiral topological insulators on a spin quantum simulator, Phys. Rev. Lett. 125(9), 090502 (2020)

[38] P. Cejnar, J. Jolie, and R. F. Casten, Quantum phase transitions in the shapes of atomic nuclei, Rev. Mod. Phys. 82(3), 2155 (2010)

[39] R. Boyack, H. Yerzhakov, and J. Maciejko, Quantum phase transitions in Dirac fermion systems, Eur. Phys. J. Spec. Top. 230(4), 979 (2021)

[40] S. Paschen and Q. Si, Quantum phases driven by strong correlations, Nat. Rev. Phys. 3(1), 9 (2020)

[41] Y. Wang, W. Ji, Z. Chai, Y. Guo, M. Wang, X. Ye, P. Yu, L. Zhang, X. Qin, P. Wang, F. Shi, X. Rong, D. Lu, X. J. Liu, and J. Du, Experimental observation of dynamical bulk-surface correspondence in momentum space for topological phases, Phys. Rev. A 100(5), 052328 (2019)

[42] J. Niu, Y. Li, L. Zhang, J. Zhang, J. Chu, J. Huang, W. Huang, L. Nie, J. Qiu, X. Sun, Z. Tao, W. Wei, J. Zhang, Y. Zhou, Y. Chen, L. Hu, Y. Liu, S. Liu, Y. Zhong, D. Lu, and D. Yu, Demonstrating path-independent anyonic braiding on a modular superconducting quantum processor, Phys. Rev. Lett. 132(2), 020601 (2024)

[43] N. Defenu, A. Lerose, and S. Pappalardi, Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions, Phys. Rep. 1074, 1 (2024)

[44] D. Lu,A. Brodutch,J. Park,H. Katiyar,T. Jochym-O’Connor,R. Laflamme, NMR quantum information processing, in: Electron Spin Resonance (ESR) Based Quantum Computing, Springer, 2016, p. 193

[45] C. Qiu, X. Nie, and D. Lu, Quantum simulations with nuclear magnetic resonance system, Chin. Phys. B 30(4), 048201 (2021)

[46] A. J. Park, E. McKay, D. Lu, and R. Laflamme, Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator, New J. Phys. 18(4), 043043 (2016)

[47] Z. Zhang, X. Long, X. Zhao, Z. Lin, K. Tang, H. Liu, X. Yang, X. Nie, J. Wu, J. Li, T. Xin, K. Li, and D. Lu, Identifying abelian and non-Abelian topological orders in the string-net model using a quantum scattering circuit, Phys. Rev. A 105(3), L030402 (2022)

[48] X. Long, W. T. He, N. N. Zhang, K. Tang, Z. Lin, H. Liu, X. Nie, G. Feng, J. Li, T. Xin, Q. Ai, and D. Lu, Entanglement-enhanced quantum metrology in colored noise by quantum Zeno effect, Phys. Rev. Lett. 129(7), 070502 (2022)

[49] D. Lu, K. Li, J. Li, H. Katiyar, A. J. Park, G. Feng, T. Xin, H. Li, G. Long, A. Brodutch, J. Baugh, B. Zeng, and R. Laflamme, Enhancing quantum control by bootstrapping a quantum processor of 12 qubits, npj Quantum Inf. 3, 45 (2017)

[50] J. Li, Z. Luo, T. Xin, H. Wang, D. Kribs, D. Lu, B. Zeng, and R. Laflamme, Experimental implementation of efficient quantum pseudorandomness on a 12-spin system, Phys. Rev. Lett. 123(3), 030502 (2019)

[51] E. Davis and W. Heiss, Random-phase approximation and broken symmetry, J. Phys. G Nucl. Phys. 12(9), 805 (1986)

[52] A. Sáiz, J. E. García-Ramos, J. M. Arias, L. Lamata, and P. Pérez-Fernández, Digital quantum simulation of an extended Agassi model: Using machine learning to disentangle its phase-diagram, Phys. Rev. C 106(6), 064322 (2022)

[53] P. Pérez-Fernández, J. M. Arias, J. E. García-Ramos, and L. Lamata, A digital quantum simulation of the Agassi model, Phys. Lett. B 829, 137133 (2022)

[54] P. Jordan and E. Wigner, Über das paulische äquivalenzverbot, Eur. Phys. J. A 47(9-10), 631 (1928)

[55] M. Suzuki, Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys. 51(2), 183 (1976)

[56] L. Lamata, A. Pérez-Fernández, M. Sanz, and E. Solano, Digital-analog quantum simulations with superconducting circuits, Adv. Phys. X 3(1), 1457981 (2018)

[57] J. E. García-Ramos, J. Dukelsky, P. Pérez-Fernández, and J. Arias, Phase diagram of an extended Agassi model, Phys. Rev. C 97(5), 054303 (2018)

[58] Z. Lin, H. Liu, K. Tang, Y. Liu, L. Che, X. Long, X. Wang, Y. Fan, K. Huang, X. Yang, T. Xin, X. Nie, and D. Lu, Hardware-efficient quantum principal component analysis for medical image recognition, Front. Phys. (Beijing) 19(5), 51202 (2024)

[59] X. Nie, X. Zhu, K. Huang, K. Tang, X. Long, Z. Lin, Y. Tian, C. Qiu, C. Xi, X. Yang, J. Li, Y. Dong, T. Xin, and D. Lu, Experimental realization of a quantum refrigerator driven by indefinite causal orders, Phys. Rev. Lett. 129(10), 100603 (2022)

[60] D. G. Cory, M. D. Price, and T. F. Havel, Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing, Physica D 120(1-2), 82 (1998)

[61] C. Xi, X. Liu, H. Liu, K. Huang, X. Long, D. Ebler, X. Nie, O. Dahlsten, and D. Lu, Experimental validation of enhanced information capacity by quantum switch in accordance with thermodynamic laws, Phys. Rev. Lett. 133(4), 040401 (2024)

[62] K. Huang, C. Xi, X. Long, H. Liu, Y. Fan, X. Wang, Y. Zheng, Y. Feng, X. Nie, and D. Lu, Experimental realization of self-contained quantum refrigeration, Phys. Rev. Lett. 132(21), 210403 (2024)

[63] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen, and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172(2), 296 (2005)

[64] G. Feng, F. H. Cho, H. Katiyar, J. Li, D. Lu, J. Baugh, and R. Laflamme, Gradient-based closed-loop quantum optimal control in a solid-state two-qubit system, Phys. Rev. A 98(5), 052341 (2018)

[65] X. Lin, J. Fan, R. Ye, M. Zhou, Y. Song, D. Lu, and N. Xu, Online optimization for optical readout of a single electron spin in diamond, Front. Phys. (Beijing) 18(2), 21301 (2023)

[66] X. F. Nie, B. B. Wei, X. Chen, Z. Zhang, X. Z. Zhao, C. D. Qiu, Y. Tian, Y. L. Ji, T. Xin, D. W. Lu, and J. Li, Experimental observation of equilibrium and dynamical quantum phase transitions via out-of-time-ordered correlators, Phys. Rev. Lett. 124(25), 250601 (2020)

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