Adaptively partitioned analog quantum simulation for the nonclassical free induction decay of NV centers on near-term quantum computers

The idea of simulating quantum physics with controllable quantum devices had been proposed several decades ago. With the extensive development of quantum technology, large-scale simulation, such as the analog quantum simulation tailoring an artificial Hamiltonian mimicking the system of interest, has been implemented on elaborate quantum experimental platforms. However, due to the limitations caused by the significant noises and the connectivity, analog simulation is generically infeasible on near-term quantum computing platforms. Here we propose an alternative analog simulation approach on near-term quantum devices. Our approach circumvents the limitations by adaptively partitioning the bath into several groups based on the performance of the quantum devices. We apply our approach to simulate the free induction decay of the electron spin in a diamond NV$^-$ center coupled to a huge number of nuclei and investigate the nonclassicality induced by the nuclear spin polarization. The simulation is implemented collaboratively with authentic devices and quantum simulators on IBMQ. This work sheds light on a flexible approach to simulate large-scale materials on noisy near-term quantum computers.


I. INTRODUCTION
Simulating quantum physics has long been a widelyknown challenging problem [1].One of the primary difficulties lies in the exponential growth of the Hilbert space of a large quantum system with increasing constituent components.This would require a huge amount of computer memory to store the quantum states and the quantum operations acting on them.In particular, if we are further interested in the time evolution of the quantum system, the burden imposed on the computational resource would become even heavier and rapidly exceed the computational power of conventional computers.
Instead of developing sophisticated, but inevitably approximate, classical algorithms, an alternative proposal for solving the problem of simulating quantum physics is to harness the power of quantum mechanical systems [1][2][3][4][5], underpinned by the intuitive idea that Nature itself ultimately behaves quantum mechanically.An appealing approach is to directly map the Hamiltonian of a less controllable system of interest onto that of a quantum simulator consisting of well-controlled quantum systems, referred to as analog quantum simulation (AQS) [6][7][8].With the extensive development of quantum technology, AQS has been implemented with many quantum mechanical systems, including superconducting circuit [8][9][10], ultracold atoms [7,11], Rydberg atoms [12,13], and trapped ions [14][15][16].Noteworthily, these successful demonstrations of AQS are implemented on the elaborate quantum experimental platforms, which are generically inaccessible to public.
On the other hand, many programable quantum com- * hongbinchen@phys.ncku.edu.twputing platforms have emerged in recent years.They are featured by the accessibility to the public via online user interfaces, opening an avenue for the public to experience the principles of quantum mechanics.In particular, theorists are possible to design prototypical experiments running on the quantum computers to examine and demonstrate theoretical concepts.Consequently, many demonstrations of the fundamental principles of quantum mechanics have been achieved on these stateof-the-art quantum computing platforms [17][18][19][20][21][22][23][24][25].
In addition to the aforementioned demonstrations of fundamental quantum-information-theoretic principles, quantum computers are also conceived to be versatile in the simulation of open quantum system dynamics [26][27][28][29].However, the near-term quantum computers are still in an era of noisy intermediate-scale quantum (NISQ) devices [30].Irrespective of the prominent breakthroughs of quantum computers outperforming conventional computers [31,32], due to the limitations on the performance caused by the significant noises and the qubit topological connectivity, a straightforward simulation of largescale materials remains intractable.Either the simulation of a few atoms arranged in an one dimensional chain [26], hybrid quantum-classical algorithm [33][34][35], or variational quantum algorithms [36][37][38], can be efficiently implemented.There is one another approach, referred to as Trotterization [26,39,40], attainable on near-term quantum computers.This approach approximates the whole time-evolution operator by discretizing and decomposing it into a series of smaller ones according to the Suzuki-Trotter formula.The primary drawback of the Trotterization is the errors introduced during the decomposition.A quantum hardware-efficient approach capable of simulating large-scale materials in an AQS manner without the assistance from conventional computers is desirable.
On the other hand, an unambiguous demonstration of To guarantee the validity of the dipole-dipole hyperfine interaction, all 13 C nuclei lie outside a radius of 0.5 nm.Furthermore, we also assume that only the nuclei within a polarization area (yellow shadow) of radius 1 nm can be identically polarized in a controllable manner via the DNP technique.
In this work we propose an alternative analog simulation approach capable of simulating large-scale materials on near-term quantum computing platforms.Our approach circumvents the limitations on the performance by adaptively partitioning the bath into several groups based on the performance of the quantum devices.We apply our approach to simulate the FID process of the electron spin of an NV − center in diamond lattice and perform the simulation on IBM Quantum computers (IBMQ) [63].
To do this, we first design a quantum circuit implementing the total Hamiltonian of an NV − center coupled to a huge nuclear spin bath.Additionally, to realize the effects of various nuclear spin polarizations, we also design a family of polarization oracles accompanied with ancillary qubits.In order to adequately divide the nuclear spin bath into smaller groups fitting into the performance of the quantum devices, we test their capabilities by a series of preliminary examinations with a few number of nuclei.Based on their performance, we can simulate the FID process either in an collaboration with authentic device and quantum simulator, or fully on quantum simulator of IBMQ.With this adaptive partition approach, we can reproduce the nonclassical FID process in the presence of a transversely polarized nuclear spin bath and estimate the corresponding CHER.We consider a single negatively charged nitrogenvacancy (NV − ) center in diamond lattice consisting of a substitutional nitrogen (N) and a vacancy (V) in an adjacent lattice site, as shown in Fig. 1(a).The axis joining V and N defines an intrinsic z-axis for the electron spin.There are totally six electrons confined in the V site, forming a complicated electron spin configuration.The ground state of the electron spin is a spin triplet state with S = 1. Figure 1(b) shows the energy level structure of the electron spin ground state.There is a zero-field splitting D/2π = 2.87 GHz between the sublevels m S = 0 and m S = ±1.In the absence of the external magnetic field, the two sublevels m S = ±1 degenerate; while the degeneracy will be lifted due to the Zeeman effect by applying an external magnetic field B. For simplicity, we assume that the external magnetic field B = B z e z is aligned with the z-axis.Due to the Zeeman splitting, we can selectively excite the two different spin transitions |0 ↔ | ± 1 with microwave (MW) pulses at an appropriate frequency.Therefore, the free Hamiltonian of the electron spin triplet is given by where γ e /2π = 2.8025 MHz/G is the electron gyromagnetic ratio.
The diamond lattice sites are mostly occupied by the spinless 12 C nuclei [light gray spheres in Fig. 1(c)], which have negligible effects on the electron spin dynamics.The electron spin decoherence is mainly caused by the randomly distributed 13 C isotopes of natural abundance about 1.1% [dark gray spheres in Fig. 1(c)] with nuclear spin J = 1/2.Then the free Hamiltonian of the nuclear spin bath consisting of 13 C isotopes indexed by k is given by with γ C /2π = 1.0704 kHz/G being the gyromagnetic ratio of the 13 C nuclei.
The coupling between the electron spin and the 13 C nuclear spin bath is given by the hyperfine interaction with interaction Hamiltonian expressed as Since the electron wavefunction is tightly confined in the V site, the Fermi contact risen by the overlap with the electron wavefunction becomes negligible for nuclei farther away than 0.5 nm from the NV − center.In our simulation, we post-select a randomly generated configuration with all 13 C nuclei lying outside a radius of 0.5 nm, as schematically shown in Fig. 1(c).Therefore, the hyperfine interaction (3) is caused by the dipole-dipole interaction and the hyperfine coefficients are given by with µ 0 the magnetic permeability of vacuum, r (k) the displacement vector toward the k th nucleus, and e (k) the unit vector of r (k) .Moreover, it is worthwhile to note that the dilute 13 C nuclear spin bath leads to a relaxation time T 1 of the electron spin in the order of milliseconds [64,65] and a dephasing time T * 2 of microseconds [66][67][68].Due to this experimentally measured three-order of magnitude difference between T 1 and T * 2 , the electron spin dynamics can be well approximated by pure dephasing, on the time scale under study.Therefore, it is relevant for us to neglect the terms proportional to S x and S y in Eq. ( 3) and consider only the S z component phenomenologically.Then the total Hamiltonian can be expressed as (5) and only the three hyperfine coefficients Additionally, it is critical to note that the total Hamiltonian (5) can be expressed in a block diagonal form with respect to the electron spin basis as where Finally, the total unitary time evolution operator is also block diagonal with respect to the electron spin basis with conditional evolution operators U m S (t) = exp(−i H m S t).

B. FID process of electron spin
The free-induction-decay (FID) process of the electron spin is a pure dephasing dynamics caused by the 13 C nuclear spin bath.The initial state of total system is assumed to be a direct product of all constituent componets where is the initial state of the k th nuclear spin with polarization p (k) , and I (k) and σ(k) are the identity and the Pauli operators, respectively, acting on the k th nuclear spin Hilbert space.In a conventional FID experiment, the electron spin will be first optically polarized to |0 by a 532-nm green laser, and a subsequent π/2 MW pulse will set the electron spin state to a superposition state Therefore, in our simulation, the electron spin is described in a qubit manifold with Hilbert space spanned by the two sublevels m S = 0 and m S = 1.
Once the electron spin state is set to ρ NV (0), the hyperfine interaction in Eq. ( 5) is turned on and the time evolution of the total system is governed by the block diagonal unitary operator where 1 | is the axis of nuclear spin precession.
The electron spin reduced density matrix ρ NV (t) = Tr C U T (t)ρ T (0) U † T (t) is obtained by tracing over the 13 C nuclear spin bath from the total system, and the electron spin pure dephasing dynamics is characterized by the dephasing factor Moreover, since we are paying particular attention to the pure dephasing dynamics caused by the 13 C nuclear spin bath, it is clear that the leading factor exp[i(D + γ e B z )t] plays no role in describing the profile of φ(t) but merely introducing a rapidly rotating phase.Consequently, for our purpose, we can neglect the leading factor.Finally, with the help of the prescription ( u • σ and the orthogonality of the identity and the Pauli operators Trσ j σk = 2δ jk , the dephasing factor can be expressed analytically as C. Nuclear spin polarization Equation (11) suggests that one is possible to manipulate the dynamical behavior of the electron spin by engineering the polarization p (k) and the precession axis u (k) of the nuclear spin bath.One of the extensively developed techniques engineering the nuclear spin bath is the dynamical nuclear polarization (DNP) [69][70][71][72][73][74][75][76][77][78][79], which utilizes the hyperfine interaction and the resonance between the electron spin and the nuclei to transfer the electron spin polarization to the surrounding nuclear spins, achieving a hyperpolarized nuclear spin bath.
On the other hand, since the underlying mechanism of the DNP relies on the hyperfine interaction between the electron spin and the nuclei, which attenuates rapidly with increasing displacement, as can be seen from Eq. ( 4), it is generically infeasible to polarize the whole nuclear spin bath.Therefore, we assume that only the nuclei within a polarization area of radius 1 nm [yellow shadow in Fig. 1(c)] can be polarized with identical polarization p; otherwise p = 0 for r (k) ≥ 1 nm.

III. DYNAMICAL PROCESS NONCLASSICALITY
From the above discussion, we acquire the fact that the decoherence of the electron spin is caused by the hyperfine interaction to the 13 C nuclear spin bath.In fact, this phenomenon of decoherence is ubiquitous in any quantum systems, as they are impossible to be fully isolated from their environments, and the inevitable interactions to their environments constitute the origin of decoherence [80][81][82][83][84]. From the quantum-information-theoretic perspective, the interactions will establish complicated correlations between them; while the correlations are subject to the destructions risen by the fluctuations in the huge environments, rendering themselves fragile and transient.
Consequently, we have proposed to classify a dynamical process according to the witness of the nonclassical correlations between the primary system and its environments, and to characterize the nonclassicality with the technique of canonical Hamiltonian ensemble representation (CHER) [59][60][61].Our definition of process nonclassicality is constructed based on the possibility to explain a dynamical process in an ensemble-averaged manner.The mathematical tool of fundamental importance in our definition is the Hamiltonian ensemble (HE) {(p λ , H λ )} λ , which consists of a collection of traceless Hermitian operators H λ ∈ su(n) associated with a probability p λ of occurrence [85,86].For a given HE, it will give rise to an ensemble-averaged dynamics expressed as where U λ (t) = exp(−i H λ t) is the unitary time-evolution operator generated by the member Hamiltonian operator H λ .A particularly intriguing example considers a single qubit subject to spectral disorder with the HE given by {(p(ω), ωσ z /2)} ω , where p(ω) can be any probability distribution function, then the ensemble-averaged dynamics describes pure dephasing: with the dephasing factor φ(t) = p(ω) exp(−iωt)dω being the Fourier transform of p(ω).
Crucially, it has been shown that [59], if a primary system and its environments remain at all times classically correlated without establishing nonclassical correlations during their interactions, then the reduced system dynamics E t can be explained in terms of a HE in the sense of ensemble-averaged dynamics (12).Namely, the incoherent dynamical behavior can be conceived as a results of the consumption of classical correlations.On the contrary, if nonclassical correlations emerge during the interactions, then one may fail to construct a HE with legitimate probability distribution function p λ , and necessarily appeals to a quasi-distribution ℘ λ with negative values instead.Consequently, the quasi-distribution ℘ λ , referred to as the CHER, can be used to characterize the nonclassicality of a dynamics E t [59][60][61].
Considering the FID process governed by the unitary operator (9), the electron spin undergoes a pure dephasing dynamics characterized by the dephasing factor (11).In view of Eq. ( 13), the corresponding CHER ℘(ω) of the electron spin FID is determined by the inverse Fourier transform It is interesting to note that the electron spin FID has shown to be nonclassical when the 13 C nuclear spin bath is transversely polarized; moreover, the degree of nonclassicality will become stronger with increasing polarization and magnetic field [62].
Now we proceed to design the quantum circuit model implementing the analog quantum simulation (AQS) for NV − center coupling to the whole 13 C nuclear spin bath.The purpose of the AQS is to tailor an artificial Hamiltonian with controllable quantum systems mimicking the one of interest.We therefore design a quantum circuit by mapping the total unitary time evolution operator (15) into quantum gates, where the factor exp[−i(D + γ e B z )t] has been neglected from Eq. (9).
It is crucial to note that the hyperfine interaction in Eq. ( 5) gives rise to an intrinsic conditional operation conditioned on the electron spin state.This can be realized by the controlled-U gates on IBMQ after the following manipulation of Eq. ( 15): where I (NV) is the identity operator acting on the qubit playing the role of electron spin.Then the above unitary operator can be realized with quantum gates as: The second term denotes a series of identical and independent R z (Ω 0 t) rotations, with matrix representation on the qubits playing the role of 13 C nuclear spins, followed by the controlled-U gates conditioned on the electron qubit denoted by the first term.They can be realized by the circuit on IBMQ; meanwhile, the gate parameters can be determined according to the Hamiltonian (6) as following: Further details are shown in Appendix A. Consequently, the total unitary time evolution operator ( 15) can be realized with the AQS circuit succinctly shown below:

B. State preparation and polarization oracle
Once the total unitary time evolution operator (15) has been realized with quantum circuit, following the discussions in Sec.II, the next step is to prepare the initial state as given in Eq. ( 8) according to the FID experiments.
The qubit initial state on IBMQ is preset to |0 .An Hadamard gate realizes the effect of a π/2 MW pulse setting the electron spin state to (|0 + |1 )/ √ 2, as shown in Fig. 2. On the other hand, a single-qubit gate on nucleus qubit is insufficient to realize various nuclear spin states, particularly those of mixed states.To do this, we design the polarization oracle P (k) acting on the k th nucleus qubit associated with an additional ancilla qubit, as shown in Fig. 2.After the operation of an appropriate P (k) , tracing out the ancilla qubit leaves the nucleus qubit in the state ρ (k) = [ I (k) + p (k) • σ(k) ]/2 with a corresponding polarization vector p (k) .Table I shows a family of polarization oracles P (k) and the corresponding polarization vectors p (k) .Therefore, we can manipulate individual nucleus qubit state and realize a nuclear spin bath of experimental condition schematically shown in Fig. 1(c).
At the end of the AQS circuit, the quantum state tomography (QST) is applied to probe the state of the FIG. 2. The overall quantum circuit implementing the AQS for NV − center coupling to the whole 13 C nuclear spin bath.To prepare the qubit initial states satisfying the experimental condition, each qubit will go through a stage of state preparation.The Hadamard gate on the electron qubit sets the qubit state to (|0 + |1 )/ √ 2, reflecting the effect of a π/2 MW pulse.While the mixed state of the nucleus qubit can be realized by a polarization oracle P (k) acting on the k th nucleus qubit associated with an additional ancilla qubit.The desired nuclear spin polarization can be achieved by the polarization oracles listed in Table I.At the end of the electron qubit, the QST is applied to construct the time evolution of the dephasing factor φ(t) along a time sequence.(0,0,1) (0,0,0) (0, 0, cos θ) (1,0,0) (sin θ1 sin θ2, 0, cos θ1) electron qubit.Additionally, since we are aiming at simulating the electron spin pure dephasing characterized by the dephasing factor (11), its time evolution can be constructed by measuring σx and σy along a time sequence according to φ(t) = σx t − i σy t .Finally, the overall layout of the circuit is shown in Fig. 2.

V. PRELIMINARY EXAMINATIONS
To perform the AQS circuit on IBMQ [63], we have to map the circuit onto the qubits of the quantum devices.However, due to the qubit topological connectivity, it is obviously infeasible to map the whole circuit simulating hundreds of nucleus qubits onto IBMQ devices.
To verify the validity of the circuit, as well as to benchmark the performance of the IBMQ devices for later purpose, we first perform two prototypical circuits simulating the effects of three and six 13 C nuclei on FIG. 3. The qubits we used in the simulation on the (a) ibm auckland and (b) ibm washington quantum devices.The red, dark gray, and orange qubits play the role of the electron spins, the nuclear spins, and the ancilla qubits controlling the nuclear spin polarizations, respectively.ibm auckland and ibm washington, respectively.The qubits used and the labels on IBMQ devices are shown in Fig. 3.The red qubits play the role of the electron FIG. 4. The AQS results for three nuclei obtained from ibm auckland.We demonstrate the results for two polarizations, p (k) = (0, 0, 0) (upper panels) and p (k) = (0, 0, 1) (lower panels), at various values of the magnetic field.The simulation results for p (k) = (0, 0, 1) fit the theoretical calculations well since the polarization corresponds to the preset qubit state |0 without additional operation.On the other hand, to prepare the nuclear spin polarization p (k) = (0, 0, 0) requires a CNOT gate, resulting in obvious discrepancies, particularly the erroneous imaginary part Im[φ(t)].FIG. 5.The AQS results for six nuclei obtained from ibm washington.We demonstrate the results for two polarizations, p (k) = (0, 0, 0) (upper panels) and p (k) = (0, 0, 1) (lower panels), at various values of the magnetic field.Due to the limitation imposed by the topological connectivity of IBMQ devices, nucleus qubits exceeding three will lie at farther positions away from the electronic qubit, resulting in a rapidly increasing number of CNOT gates.This not only enhances the noise, but also deepens the circuit, rendering the simulation unreliable.
spin, and the dark gray and orange qubits denote the nucleus and the ancilla qubits controlling the nuclear spin polarizations, respectively.
Figures 4 and 5 show the results of the prototypical simulations of three and six nuclei, respectively.We demonstrate the results of two polarizations, i.e., p (k) = (0, 0, 0) and (0, 0, 1), at various values of the magnetic field.In Fig. 4, the results obtained from ibm auckland for p (k) = (0, 0, 1) (lower panels) are in good agreement with the theoretical calculations given by Eq. ( 11); while the ones for p (k) = (0, 0, 0) (upper panels) show discrepancies, particularly the erroneous imaginary part Im[φ(t)].These discrepancies can be understood by the polarization oracles listed in Table I.Polarization p (k) = (0, 0, 1) corresponds to the preset qubit state |0 without additional operation.However, the ones for p (k) = (0, 0, 0) requires a CNOT gate coupling to an ancilla qubit for each nucleus qubit, which constitutes one of the primary sources of noise on IBMQ devices.
Additionally, we have also increased the number of nuclei to six and shown the results in Fig. 5.It can be seen that the results obtained from ibm washington de-viate even more significantly from the theoretical calculations.The reason for this enhanced deviation can be understood from the topological connectivity of IBMQ devices.As shown in Fig. 3(b), an electron qubit can at most physically connect to three nucleus qubits, each of which is appended an additional ancilla qubit.An increasing number of nucleus qubits will lie at farther positions away from the electronic qubit, resulting in a rapidly increasing number of CNOT gates in the backend implementation, as well as the detrimental noises.Furthermore, an increasing number of CNOT gates also implies a deeper circuit, which requires a longer execution time approaching, or even exceeding, the life time of physical qubits, rendering the results unreliable.
Finally, we have also performed the AQS for ten nuclei on ibmq qasm simulator.We find that the results from the simulator fit the theoretical calculations very well besides tiny errors due to the approximations introduced by classical simulation algorithm; whereas, this simulator has a limited computational capability and can simulate the effects of at most ten nucleus-ancilla qubit pairs in a single task.The results and further discussions are shown in Appendix B.

VI. ADAPTIVELY PARTITIONED AQS
From the previous preliminary examinations, it can be seen that the number of nuclei simulated in a single task is very limited, far from simulating large-scale materials in an AQS manner.To circumvent the limitations of near-term quantum devices, we design a simulation algorithm by adaptively dividing the nuclear spin bath into several groups, each of which fits into the performance of the quantum devices.
In our simulation, we first generate a nuclear spin configuration of natural abundance about 1.1%, consisting of 520 13 C nuclei randomly distributed over the diamond lattice sites.To ensure the validity of the dipole-dipole interaction described by Eq. ( 4), we have also verified that all nuclei are farther away than 0.5 nm from the electron spin.Moreover, in our configuration, there are ten nuclei lying within the polarization area to be polarized to a specific polarization vector; while the other nuclei outside the polarization area will be set to be unpolarized with p (k) = (0, 0, 0).
The simulations are implemented in a collaboration with the authentic device ibm auckland and the simulator ibmq qasm simulator on IBMQ.Based on their performance examined in Sec.V, the ten nuclei to be polarized are divided into four groups; each group consists of at most three nuclei and can be simulated in a single task on ibm auckland.Additionally, the outer maximally mixed nuclei are divided into larger groups, wherein ten nuclei can be simulated in a single task on ibmq qasm simulator groupwisely.Finally, according to Eq. ( 11), the desired dephasing factor φ(t) accounting for 520 nuclei is given by the product of the results of all groups, and the corresponding CHER ℘(ω) can be estimated according to the inverse Fourier transform (14).
We first show the results in Fig. 6 for an unpolarized nuclear spin bath, i.e., p (k) = (0, 0, 0) for both the ten nuclei simulated on ibm auckland and the outer nuclei on ibmq qasm simulator, denoted by the colored dots.As expected from the upper panels of Fig. 4, the polarization oracle implementing p (k) = (0, 0, 0) on ibm auckland leads to significant errors in Fig. 6, particularly in the beginning of the time evolution.As a comparative study, we also demonstrate a counterpart fully performed on ibmq qasm simulator, denoted by the colored circles.Although the simulator gives better results than those of collaborative simulation, the errors now become visible in the imaginary parts, due to the amplification caused by the production over all groups of nuclei.
In the lower panels of Fig. 6, we show the corresponding CHER ℘(ω) at various values of the magnetic field.The theoretical calculations show that the CHER should be positive in the case of unpolarized nuclear spin bath, whereas the errors caused by ibm auckland result in negative wings.In view of the physical meaning of the negativity as a witness of nonclassical system-environment correlations [59], the origin of the negative wings can be understood from the polarization oracle entangling the nucleus-ancilla qubit pairs.We speculate that it would be the cross-talk between the nucleus-ancilla qubit pairs gives rise to nontrivial relative phase between pairs, which in turn comes into play in the dynamics of the electron spin qubit and is captured by the negativity in the CHERs.On the other hand, the results fully given by ibmq qasm simulator reproduce the central peak very well.However, the algorithmic errors give rise to irregularly wavy wings on both sides of the central peak.
In Fig. 7, we show the effects of a z-polarized nuclear spin bath.Similarly, the colored dots denote the results simulated in a collaborative manner, where the ten polarized nuclei with p (k) = (0, 0, 1) are simulated on ibm auckland and the outer unpolarized nuclei are on ibmq qasm simulator, and the colored circles denote the counterpart fully performed on ibmq qasm simulator.As expected from the preliminary examinations, the collaborative simulations on IBMQ for p (k) = (0, 0, 1) are much better than those for p (k) = (0, 0, 0) due to the corresponding polarization oracles.Moreover, the results fully given by ibmq qasm simulator also fit the theoretical calculations very well besides the visible algorithmic errors in the imaginary parts.Furthermore, the profile of the CHER varies drastically with increasing magnetic field in this case.Several sharp peaks emerge at strong fields.Remarkably, this phenomenon has also been wellreproduced in our simulations.
It has been shown that the nonclassicality is induced by the nuclear spin precession in the presence of a transversely polarized nuclear spin bath [62].In Fig. 8, we show the simulation of the nonclassicality induced by an x-polarized nuclear spin bath at various values of the magnetic fields.The polarization oracle imple-FIG.6.The adaptively partitioned AQS results for 520 unpolarized nuclei at various values of the magnetic field (upper panels) and the corresponding CHER (lower panels).The collaborative simulations with ibm auckland and ibmq qasm simulator are denoted by the colored dots, and the ones fully given by ibmq qasm simulator are denoted by the colored circles.Due to the polarization oracle implementing p (k) = (0, 0, 0) on ibm auckland, the errors are prominent in the beginning of the time evolution.The simulator gives better results besides the amplified algorithmic errors in the imaginary part.Although the CHER in the case of unpolarized nuclear spin bath should be positive, the errors caused by implementing the entangling polarization oracle on ibm auckland give rise to negative wings.On the other hand, the results fully given by ibmq qasm simulator reproduce the central peak very well; while the algorithmic errors give rise to irregularly wavy wings on both sides of the central peak.FIG. 7. The adaptively partitioned AQS results for a z-polarized nuclear spin bath at various values of the magnetic field (upper panels) and the corresponding CHER (lower panels).The collaborative simulations with ibm auckland and ibmq qasm simulator are denoted by the colored dots, and the ones fully given by ibmq qasm simulator are denoted by the colored circles.Due to the null operation of the polarization oracle implementing p (k) = (0, 0, 1), the collaborative simulations are in good agreement with the theoretical calculations.Remarkably, the emergence of the sharp peaks in the profile of the CHER has also been well-reproduced in our simulations.menting p (k) = (1, 0, 0) requires a quantum gate on the nucleus qubit to be polarized.After the amplification of the production over all x-polarized nucleus qubits on ibm auckland, the errors in the collaborative simulations become prominent; while the overall profile remains visible.Similarly, the results fully given by ibmq qasm simulator also suffer from the amplified algorithmic errors.Remarkably, in the lower panels of Fig. 8, we can observe the emergence of the nonclassicality in terms of the negativity in the CHER ℘(ω).Although the nonclassicality is smeared at B z = 50 G due to the errors on ibm auckland, it becomes visible at stronger fields, as FIG. 8.The adaptively partitioned AQS results for an x-polarized nuclear spin bath at various values of the magnetic field (upper panels) and the corresponding CHER (lower panels).The collaborative simulations with ibm auckland and ibmq qasm simulator are denoted by the colored dots, and the ones fully given by ibmq qasm simulator are denoted by the colored circles.Due to the polarization oracle implementing p (k) = (1, 0, 0) on ibm auckland, the collaborative simulations ultimately deviate prominently from the theoretical calculations.In this case, the ones fully given by ibmq qasm simulator also suffer from the amplified algorithmic errors.Remarkably, the negativity in the CHER ℘(ω) is enhanced against the errors at stronger fields and becomes visible, as shown in the insets.This is an indicator of the nonclassicality reproduced in our simulations.
shown in the insets for B z = 100 G and 200 G.

VII. CONCLUSION
In this work we propose to simulate large-scale materials in a manner of analog quantum simulation on near-term quantum computing platforms.In view of the limitations on the computing capability imposed by the noises and the topological connectivity, our simulation algorithm circumvents the obstacles by adaptively partitioning the effects of huge bath into adequate groups based on the performance of the quantum devices.
We demonstrate our approach by simulating the FID process of the electron spin of an NV − center coupled to a huge nuclear spin bath and perform the simulation on IBMQ.We design a prototypical quantum circuit implementing the total Hamiltonian of an NV − center coupled to a huge number of nuclei via the dipole-dipole hyperfine interaction.Additionally, to reflect the experimental conditions, we also design a family of polarization oracles implementing the nuclear spin engineering by the DNP technique.
To investigate the capability of the quantum devices simulating the electron spin dynamics, we also perform a series of preliminary examinations simulating the effects a few number of nuclei.Based on their performance, we can simulate the FID process either in an collaboration with authentic device and quantum simulator, or fully on quantum simulator of IBMQ.With this adaptive partition approach, we can reproduce the effects account-ing for 520 nuclei on the FID process.In particular, we have taken into account the various values of magnetic fields and the nuclear spin polarizations in an experimental condition.
Although the simulation results are subject to imperfectness caused by both the noise of the authentic quantum devices and the algorithmic errors of the simulators, our approach remains reliable and reproduces critical physical phenomena, including the dynamical behavior of the electron spin, the variation of the profile of the CHER, and the nonclassicality in terms of the negativity in the CHER.We stress that, our approach is flexible in the sense that we can distribute the computing loading not only to different devices, but also to different qubit groups on a same device in a single task for improving the efficiency.Namely, the distribution strategy is adjustable depending on the condition of the devices and the required accuracy or efficiency.FIG. 9.The AQS results for ten nuclei obtained from ibm qasm simulator.We demonstrate the results of three polarizations, p (k) = (0, 0, 0) (upper panels), p (k) = (0, 0, 1) (middle panels), and p (k) = (1, 0, 0) (lower panels), at various values of the magnetic field.We find that the classical simulator ibm qasm simulator can simulate at most ten nucleus-ancilla qubit pairs in a single task.Regardless of the limitation on the number of qubits, the results fit the theoretical calculations very well besides tiny errors caused by the classical simulation algorithm.cation to the Headquarters of University Advancement at NCKU, and partially by the National Center for Theoretical Sciences, Taiwan.
as to benchmark the capability of the classical simulators provided by IBMQ, we have also performed larger prototypical circuits on the ibm qasm simulator.
Although the ibm qasm simulator provides 32 qubits, we find that it has a limited computing capability simulating up to ten nucleus-ancilla qubit pairs (21 qubits used in a single task).Errors occur in the backend operation if more than 21 qubits are included in a single task.This limitation can be understood from the giant Hilbert space of size 2 21 , corresponding to the propagation of a density matrix of dimension 2 21 × 2 21 .In Fig. 9, we show the prototypical simulations for the effects of ten 13 C nuclei on ibm qasm simulator.It can be seen that the results given by the simulator fit the theoretical calculations very well for three polarizations.However, there are still tiny errors due to the approximations introduced by classical simulation algorithm.

FIG. 1 .
FIG.1.(a) An NV − center in diamond lattice is a point defect consisting of a substitutional nitrogen (N) and a vacancy (V) in an adjacent lattice site.The axis joining V and N defines an intrinsic z-axis, along which an external magnetic field B = Bz ez is applied.(b) For the electron spin triplet ground state, there is a zero-field splitting D/2π = 2.87 GHz between the sublevels mS = 0 and mS = ±1.In the presence of an external magnetic field, the degeneracy between mS = ±1 can be lifted due to the Zeeman splitting.Then the two different spin transitions |0 ↔ | ± 1 can be selectively addressed with MW pulses at an appropriate frequency, forming a logical qubit.(c) Schematic illustration of an NV − center in diamond lattice interacting with13 C nuclear spin bath (dark gray spheres).To guarantee the validity of the dipole-dipole hyperfine interaction, all13 C nuclei lie outside a radius of 0.5 nm.Furthermore, we also assume that only the nuclei within a polarization area (yellow shadow) of radius 1 nm can be identically polarized in a controllable manner via the DNP technique.
II. DYNAMICS OF NV− CENTER A. Hamiltonian of NV − center

TABLE I .
Polarization oracle and polarization vector.