Abstract
Enhanced coherence in HoW_{10} molecular spin qubits has been demonstrated by use of clocktransitions (CTs). More recently it was shown that, while operating at the CTs, it was possible to use an electrical field to selectively address HoW_{10} molecules pointing in a given direction, within a crystal that contains two kinds of identical but inversionrelated molecules. Herein we theoretically explore the possibility of employing the electric field to effect entangling twoqubit quantum gates within a 2qubit Hilbert space resulting from dipolar coupling of two CTprotected HoW_{10} molecules in a diluted crystal. We estimate the thermal evolution of T_{1}, T_{2}, find that CTs are also optimal operating points from the point of view of phonons, and lay out how to combine a sequence of microwave and electric field pulses to achieve coherent control within a switchable twoqubit operating space between symmetric and asymmetric qubit states that are protected both from spinbath and from phononbath decoherence. This twoqubit gate approach presents an elegant correspondence between physical stimuli and logical operations, meanwhile avoiding any spontaneous unitary evolution of the qubit states. Finally, we found a highly protected 1qubit subspace resulting from the interaction between two clock molecules.
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Introduction
Electrical control of spins at the nanoscale offers significant architectural advantages in spintronics, because electric fields (Efields) can be confined over shorter length scales than magnetic fields (Bfields)^{1,2,3,4,5}. In the context of qubits, the use of Efields has been already suggested as a strategy to generate entangling twoqubit gates, either passing current pulses through molecules in singlemolecule spintronics, or by applying Efields in a P@Si crystal^{6,7,8,9}.
Magnetic molecules are considered as an ideal platform in this line since their spin Hamiltonian can be tailored by chemical design^{10}. Constructing twoqubit gates has been already explored theoretically based on heterodimer systems, where the qubit states are defined as identical to the spin states of the individual molecular component^{11,12,13,14,15,16}. These schemes are generally affected by a weak “alwayson” interqubit interaction that cannot be completely switched off due to residue dipolar coupling, which makes them hard to scale as it induces an unwanted twoqubit evolution of the wavefunction. Other strategies employ a single highspin metal center that encodes multiple qubits, where most of the logical operation, both singlequbit and twoqubit, require a series of physical multistep climbing on the energy levels of the molecule^{17}. Implementing twoqubit logical gates as independent physical operations is challenging in practice.
Another challenge for molecular spin qubits is that they exhibit fragile quantum coherence owing to the inevitable interaction with the environment (spin and phonon baths). A significant enhancement has been recently achieved via clocktransitions (CTs), which protect from magnetic noise^{18,19,20}. Nevertheless, at the CT field, the twoqubit states would present a vanishing firstorder Zeeman effect and a vanishing magnetic moment, due to perfect mixing within a tunnelingsplitting electronic spin doublet M_{J} = ± 4. We will explore herein how these limitations may be overcome by slightly moving away from the CT but preserving some coherent protection and, at the same time, the use of a directional Efield to electronically tune the transition frequencies of the two interacting qubit molecules.
The polyoxometalate molecular anion \({[{{{\rm{Ho}}}}{({{{{\rm{W}}}}}_{5}{{{{\rm{O}}}}}_{18})}_{2}]}^{9}\) (abbreviated to HoW_{10}) with crystal structure \({{{\rm{N{a}}}_{9}[{Y}_{1x}H{o}_{x}{({W}_{5}{O}_{18})}_{2}]\; \cdot \; 35{H}_{2}O}}\) (x = 1%) is a prime example of CT spin qubit^{18}. The crystal unit cell contains two inversionsymmetry related HoW_{10} anions, slightly distorted (D_{4d} symmetry) along their C_{4} axes (Fig. 1a). The magnetic levels of a single HoW_{10} can be described by a Hamiltonian, including crystal field, hyperfine, and Zeeman interactions:
where, \({B}_{k}^{q}\) corresponds to the crystalfield parameters (CFPs) in the Extended Steven Operator notation (\({\hat{O}}_{k}^{q}\)). J and I are the total electronic and nuclear angular momentum, respectively, A denotes the isotropic hyperfine interaction, g_{e} (g_{N}) and μ_{B} (μ_{N}) correspond to the electronic (nuclear) gyromagnetic ratio and Bohr magneton, respectively. The lowenergy region of interest comprises 16 electronuclear spin levels corresponding to the spin doublet M_{J} = ± 4 and a nuclear spin I = 7/2 (Supplementary Fig. 1). Each CT corresponds to an anticrossing between the 8th and the 9th levels (Supplementary Note 1).
A recent experimental study showed that one can achieve coherent control over the spin of HoW_{10} molecules by using an Efield pulse to manipulate the CT frequency^{21}. This is realized in practice due to a strong spin–electric coupling (SEC), which arises from intrinsic symmetry breaking, a soft and electrically polarizable environment of the spin carriers, and a spin spectrum that is highly sensitive to distortions. This allows to selectively address the spins of identical HoW_{10} molecules pointing at different directions.
In this work, we theoretically demonstrate the possibility of employing Efield pulses to effect entangling twoqubit quantum gates within a 2qubit Hilbert space resulting from two neighboring CTprotected molecules of HoW_{10}. This involves finding a Bfield constituting a compromise between keeping some of the protection from magnetic noise and allowing unitary evolution of the qubit states. We start with a theoretical estimate of the effect of temperature and Bfield on the longitudinal (T_{1}) and transverse (T_{2}) relaxation times of a HoW_{10} single qubit. Later, with the whole Hamiltonian, we indicate the conditions and procedure to implement arbitrary 2qubit manipulations in this system.
Results
Singlequbit relaxation
Prior to addressing the twoqubit system (i.e., the two inversionsymmetry related HoW_{10} units of the crystal cell), let us analyze the relaxation dynamics of an individual HoW_{10} molecule. Note that, despite the CT protection, HoW_{10} presents relatively short coherence times T_{2}(5K) ≃ 8 μs. We are interested in the role of temperature because of the lack of such information both in experiment and in theory, since previous works have focused on the role of the spin bath for this compound^{22,23}.
The dynamics of the entire system (electronic spins and phonons) can be described by the time evolution of the density operator, \(\hat{\rho }(t)\).
where \(\hat{H}={\hat{H}}_{{{{\rm{S}}}}}+{\hat{H}}_{{{{\rm{ph}}}}}+{\hat{H}}_{{{{\rm{Sph}}}}}\) is the total Hamiltonian describing the electronic spins (\({\hat{H}}_{{{{\rm{S}}}}}\)), the phonon bath (\({\hat{H}}_{{{{\rm{ph}}}}}\)) and their spinbath interaction (\({\hat{H}}_{{{{\rm{Sph}}}}}\)), respectively. When phonon dynamics is much faster than the spin relaxation (as it is the case here), the BornMarkov approximation is safely invoked integrating out the phonon component from the density matrix and making the problem purely electronic in the presence of a phonon bath^{24}. In this regime, the dynamics of the electronic spin states can be described by the reduced spin density operator (\({\hat{\rho }}^{{{{\rm{S}}}}}\)) within the Redfield equation in the eigenvector representation of the spin Hamiltonian \({\hat{H}}_{{{{\rm{S}}}}}\)^{25,26,27,28},
where ω_{ab} = (E_{a} − E_{b})/ℏ, and E_{a} and E_{b} are the corresponding eigenvalues. R_{ab,cd} is the full Redfield tensor, which accounts for the system relaxation due to the interaction with the thermal bath. To evaluate the Redfield tensor, we need a spectral function for the bath, which is taken from ref. ^{29}, and the spin–phonon couplings, which are estimated from ab initio calculations (Supplementary Note 2).
We then solve the master equation (Eq. (3)) in time at different temperatures. T_{1} and T_{2} are thus extracted by fitting the exponential decays of magnetization at any temperature (Supplementary Note 3). The temperature evolution of T_{1} predicted for HoW_{10} (Fig. 2a) reveals an exponential T_{1} – T dependence over the temperature range of 3–11 K. An Arrhenius fit of this Orbach process, T_{1} ∝ exp (U_{eff}/k_{B}T), gives us an effective energy barrier U_{eff} of 34.5 cm^{−1}, which is virtually identical to the half of the energy of the lowestfrequency molecular vibration of HoW_{10} (68.4 cm^{−1})^{30}. This relation is in line with the interpretation of the underbarrier relaxation in molecular nanomagnets by Lunghi et al.^{29} and indicates that the longitudinal relaxation of HoW_{10} within the ground doublet is assisted by the lowestfrequency phonon mode at low temperatures. Our analysis also shows that T_{2}s are in the same order as for T_{1}s, both following the similar temperature trend (Fig. 2b). This behavior is in accordance with the observed drastic T_{2} decrease upon heating^{18}, demonstrating that T_{2} is limited by T_{1}, which is mainly governed by the spin–phonon coupling with the lowestfrequency phonon mode. We notice that our calculation underestimates T_{1} and T_{2} compared with the experimental estimates determined at the CTs. This is partly due to the overestimation of spin–phonon couplings computed within a single HoW_{10} model in gas phase. On the other hand, Raman relaxation processes (not considered here) may be important to estimate T_{1} more precisely at low temperatures. However, a similar temperature dependence behavior for Orbach and Raman relaxation mechanisms have been shown as long as anharmonicity effects are included as here through the spectral density (Supplementary Equation 16)^{31}.
To gain an insight into the relaxation behavior in the vicinity of CTs, relaxation times are further analyzed at different Bfields and different temperatures. Figure 2c, d illustrate T_{1}, T_{2} divergences with the longest relaxation times observed at the CT, i.e., \(\left({T}_{1,2,Bz\ne 0}{T}_{1,2,{{{\rm{CT}}}}}\right)/{T}_{1,2,{{{\rm{CT}}}}}\). This protection against vibrational decoherence at the CT coincides with the wellknown T_{2} protection from magnetic noise. The latter originates in the fact that the Ho spin possesses a vanishing magnetic moment resulted from mixing between \(\left\vert {M}_{J}=+4\right\rangle\) and \(\left\vert {M}_{J}=4\right\rangle\) (Fig. 1b). When moving away from the CT, such mixing is broken by the Zeeman effect and thereby the dipolar decoherence is activated. Since we did not include dipolar decoherence in our model, it is not surprising that the calculated T_{2} drop ( ~40%) at 10 mT away from the CT is less intense than the sharp T_{2} divergence determined in experiments.
Twoqubit gates
To explore the possibility of coherent control over a 2qubit Hilbert space, including generating entangled states, we considered two nearest HoW_{10} qubits as indicated in Fig. 1a. With a protection of quantum coherence against spinspin interactions, it has been demonstrated that T_{2} was already saturated at 8 μs at 5 K in a 1% diluted HoW_{10} crystal^{18}, in which 0.01% abundance of HoW_{10} dimers can be expected. This value falls into the dilution range of nonCT molecular spin qubits conventionally used for an optimum T_{2}^{32,33}. Although fully realistic, a technical difficulty of operating on molecular spin qubit pairs in a disordered, diluted system is that isolated molecules with similar transition energies will be far more abundant than molecule pairs. We will outline two independent ways of addressing this problem, based on initialization and on twoqubit gates.
Figure 3 a illustrates a schematic diagram for a general twoqubit system with identical (left) and inequivalent (right) qubits. In the symmetric scenario (at the CT and zero Efield in HoW_{10} dimer case), one would define all four states as linear combination of the singlemolecule states. Specifically, \({\left\vert 00\right\rangle }^{{{{\rm{s}}}}}\) and \({\left\vert 11\right\rangle }^{{{{\rm{s}}}}}\), which are energetically wellseparated, correspond to the double ground and double excited states, respectively. In addition, \({\left\vert 01\right\rangle }^{{{{\rm{s}}}}}\) and \({\left\vert 10\right\rangle }^{{{{\rm{s}}}}}\) are degenerate in energy (Fig. 3a, left). Since the two molecules are identical and uncoupled, the states of the two molecules in \({\left\vert 01\right\rangle }^{{{{\rm{s}}}}}\) and \({\left\vert 10\right\rangle }^{{{{\rm{s}}}}}\) are separable and independent. That electronic situation is better understood not actually as a twoqubit system, but rather as two singlequbit systems that happen to be close to each other and function as two copies of the same qubit.
The key requirement for generating highfidelity entangled states is to have distinguishable transitions and switchable (symmetric to asymmetric) operating qubit space. The former can be achieved through activating dipolar interaction. However, the symmetric scenario maintains under this circumstance, which resembles previous theoretical proposals^{11,12,13,14}. In order to have switchable qubit states, an Efield is introduced in our case, which alters the physical nature of the qubit space by breaking symmetry in \({\left\vert 01\right\rangle }^{{{{\rm{s}}}}}\) and \({\left\vert 10\right\rangle }^{{{{\rm{s}}}}}\) and gives rise to asymmetric states (Fig. 3a right).
This feature will be discussed in detail later. At this stage, let us first address the response of the HoW_{10} dimmer system to an applied Efield. In reality, with an applied Efield, two symmetryinversion HoW_{10} molecules intrinsically experience opposite directions of the field but with the same magnitude, that is, +E for one molecule and E for another. To simplify the description and calculations, we apply the net Efield effect on just one of the molecules, which is equivalent to the experimental scenario. The microscopic description in terms of an effective Hamiltonian of this system is as follows:
where \({\hat{H}}_{{{{\rm{s}}}}}^{a}\) and \({\hat{H}}_{{{{\rm{s}}}}}^{b}({{{\bf{E}}}})\) are the spin (crystal field + hyperfine + Zeeman) Hamiltonians for HoW_{10} sites a and electrically tuned b. The crystalfield parameters (CFPs) \({B}_{k}^{q}(E)\) are determined by establishing a relation between the given Efield and spinenergy levels (Methods and ref. ^{21}). \({\hat{H}}^{{{{\rm{ex}}}}}\) denotes the interacting Hamiltonian, which accounts for the dipolar interaction (\({j}_{a,b}^{{{{\rm{dip}}}}}\)) between the two sites a and b. This dipolar interaction is dependent of Bfield for a given twosite distance and orientations (Supplementary note 4), and vanishes at the CTs where the molecule is effectively diamagnetic (Fig. 3b).
Within the two dipolarly coupled molecules, the 16 levels of the individual molecules are combined into a 256 manifold (Fig. 3c). Without loss of generality, and since our Hamiltonian does not include terms that are extradiagonal in the nuclear spin, we consider only states involved in the first CT at \({B}_{\min }\) = 24 mT. Our operating twoqubit space is thus defined as the 4 levels resulting from the weak dipolar coupling between \({\left\vert {M}_{J} = \pm 4,{M}_{I} = 1/2\right\rangle }^{(a)}\) and \({\left\vert {M}_{J} = \pm 4,{M}_{I} = 1/2\right\rangle }^{(b)}\) states for sites a to b, respectively. We label these four electronuclear spin states as \(\left\vert 00\right\rangle\), \(\left\vert 01\right\rangle\), \(\left\vert 10\right\rangle\), and \(\left\vert 11\right\rangle\), independently of their physical nature, which will depend on the applied B or Efield.
As discussed above, dipolar coupling is firstly needed to make two molecules inequivalent (\(\delta f=E(\left\vert 10\right\rangle )E(\left\vert 01\right\rangle )\, \ne \,0\)), which is realized by moving Bfield away from the CT region. Figure 3d illustrates δf as a function of the deviation from the CT. A high Bfield is not actually required to achieve this inequivalence, indeed around B = 12 mT the exchange becomes nonnegligible (δf = 0.1 MHz).
In the next step, an Efield is applied to generate asymmetric twoqubit states. In our working conditions (Bfield = 12 mT, Efield = 300 V per 2 mm), the spins of the two molecules are coupled whereas the two qubits are uncoupled. This is evidenced by the fact that each of the states \({\left\vert 00\right\rangle }^{{{{\rm{as}}}}}\), \({\left\vert 01\right\rangle }^{{{{\rm{as}}}}}\), \({\left\vert 10\right\rangle }^{{{{\rm{as}}}}}\), \({\left\vert 11\right\rangle }^{{{{\rm{as}}}}}\) are eigenstates of the static Hamiltonian (eq. (4)), as can be seen from the wavefunction composition of the HoW_{10} pair (Supplementary Tables 5–8). Besides, the two singlequbit frequencies ℏω_{1} and ℏω_{2} no longer correspond to the two singlemolecule excitation energies, i.e., the eigenstates of the pair as our working space are not the eigenstates of the individual molecules due to dipolar interaction. Despite that, singlequbit addressability is not disturbed since transition frequencies are still distinct.
This framework is able to effect twoqubit gates, but first we briefly address the issue of initialization since our starting state would be the \({\left\vert 4,1/2\right\rangle }^{(a)}\otimes {\left\vert 4,1/2\right\rangle }^{(b)}\) (Fig. 3c, e, black). One can apply a sequence of pulses to transfer the population from the ground state of the bimolecular Hamiltonian to the ground state of our operating space. We estimated a possible initialization sequence by analyzing the wavefunction within a hyperfine basis (\(\left\vert {M}_{I}^{a},{M}_{I}^{b}\right\rangle\), Supplementary Note 5). Note that the dipolar interactions are strong enough that this initialization step is able to distinguish between isolated molecules and pairs of neighboring HoW_{10} qubits within the diluted crystal.
Here, we employ a specific case for illustration of single and doublequbit operations, namely the generation of a Bell state involving \({\left\vert 00\right\rangle }^{{{{\rm{as}}}}}\) and \({\left\vert 11\right\rangle }^{{{{\rm{as}}}}}\). Once the initial state \({\left\vert 00\right\rangle }^{{{{\rm{as}}}}}\) is prepared, the Efield is turned on and a microwave πpulse is applied to promote the \({\left\vert 00\right\rangle }^{{{{\rm{as}}}}}\to {\left\vert 10\right\rangle }^{{{{\rm{as}}}}}\) transition (Fig. 3e, f). The eigenstates after switching off the Efield are \({\left\vert 10\right\rangle }^{{{{\rm{s}}}}}\) and \({\left\vert 01\right\rangle }^{{{{\rm{s}}}}}\), thus Rabilike oscillations start between \({\left\vert 10\right\rangle }^{{{{\rm{as}}}}}\) and \({\left\vert 01\right\rangle }^{{{{\rm{as}}}}}\). These coherent oscillations constitute a twoqubit SWAP gate by switching \({\left\vert 10\right\rangle }^{{{{\rm{as}}}}}\to {\left\vert 01\right\rangle }^{{{{\rm{as}}}}}\). More exactly, any desired rotation between these two states can be achieved by choosing the time of the operation, including the notable \(\sqrt{SWAP}\) that together with singlequbit rotations forms a universal gate set. The \(\sqrt{SWAP}\) gate operation would result in entangled states that can be readout in Bell states as \({\Psi }_{23}^{\pm }=\frac{1}{\sqrt{2}}({\left\vert 01\right\rangle }^{{{{\rm{as}}}}}\pm {\left\vert 10\right\rangle }^{{{{\rm{as}}}}})\). To generate the desired Bell state, applying a πpulse to \({\Psi }_{23}^{\pm }\) enables population transfer from \({\left\vert 10\right\rangle }^{{{{\rm{as}}}}}\to {\left\vert 00\right\rangle }^{{{{\rm{as}}}}}\) and \({\left\vert 01\right\rangle }^{{{{\rm{as}}}}}\to {\left\vert 11\right\rangle }^{{{{\rm{as}}}}}\) simultaneously because of their indistinguishable energy gaps, i.e., \({\Phi }_{14}^{\pm }=\frac{1}{\sqrt{2}}({\left\vert 00\right\rangle }^{{{{\rm{as}}}}}\pm {\left\vert 11\right\rangle }^{{{{\rm{as}}}}})\) (Fig. 3g). In other words, the transition frequency corresponding to each qubit is independent on the state of the twoqubit system. This unambiguous correspondence between logical operation and physical operation in a switchable twoqubit space constitutes a key difference between our proposal and previous approaches^{11,12,13,14,17,34,35}.
Physical implementation and pulse duration time
Let us give some estimates on practical details. The typical times for twoqubit gate operations will be given by the inverse of the interaction energy between the two molecular spins. With δf = 0.1 MHz at 12 mT as discussed above, a half rotation (\(\sqrt{SWAP}\)) would take in the order of 5 μs. The duration of microwave πpulses (800 ns) was used in experiment to selectively excite qubits with narrow frequency around 3 MHz in the presence of the Efield^{21}. Thus, the overall time needed (≃7 μs) for the pulse sequence illustrated in Fig. 3f is comparable to the T_{2} value at 5 K. However, our calculations above show that the phononmediated decoherence below 3 K would be so weak that T_{2} would rise to the order of 1 ms, indicating that the decoherence caused by the spin bath would dominate. The nuclear spin bath had been previously estimated to produce a T_{2} ≃ 300 μs^{22}, and the electron spin bath can be conveniently lowered by dilution if needed, meaning a conservative estimate of T_{2} ≃ 30 μs should be easily achievable, and at the same time sufficiently longer than our estimated operation time.
The pair of states \({\left\vert 10\right\rangle }^{{{{\rm{as}}}}}\) and \({\left\vert 01\right\rangle }^{{{{\rm{as}}}}}\) merit a separate discussion. We have employed them here as two of the four states in a twoqubit system, but they also could be employed as an exceptionally protected single qubit (3rdCT, Supplementary Fig. 11), with the other states being auxiliary. The idea that interqubit dipolar interactions and spinphonon interactions need to be suppressed, and the proposal to do that by having antiferromagnetically ordered spin qubits via chemical design has been around for some time now^{36}. A similar strategy has been achieved experimentally in socalled flipflop qubits^{4}. One can appreciate the unusual protection against magnetic noise in the energy differences in Fig. 3d, and in the energy levels themselves in Supplementary Figs. 8 and 9.
As mentioned above, a key difficulty from working with pairs of magnetic entities within a diluted crystal is getting past the signal from the monomers, which will be statistically much more abundant. To supress singlequbit signal one needs to design pulse sequences, as in the example above, where all singlequbit operations add up to full 2π singlequbit rotations, so all the single qubits will be back to the ground state and not contribute to the detected signal. For qubit pairs, which experience twoqubit rotations, the same pulse sequence results in nontrivial operations. The extension of the same strategy to other quantum circuits is discussed in Supplementary Note 6, where other challenges for this scheme that may arise from the low symmetry of the crystal structure are discussed, together with possible strategies to address them.
Discussion
Our work presents a general methodology to investigate the possibility of generating entanglement in a 2qubit Hilbert space constructed from two molecular spin qubits. To do so, one must determine the qubit relaxation times, and the details of the pulse sequence, including transition frequencies and pulse duration times. Of course, to obtain a reliable entangled state, relaxation times place an upper limit to total pulse duration times. Here, we investigated a CT qubit in HoW_{10}. The inversionsymmetric HoW_{10} pair in a diluted crystal offers several features including the robust coherence close to the CTs, the strongest spinelectric response and the resulting switchable operating space between symmetric and asymmetric twoqubit states. Our results offers a promising strategy towards the use of HoW_{10} molecular spin qubits in constructing a fully addressable twoqubit quantum processor, which not only avoids the “alwayson” interqubit interaction by local Efield control, but also realizes onetoone correspondence between logical and physical operations.
The approach presented in the present study can be employed to CT and nonCT systems. Further, our scheme can be directly employed to study the recently discovered Lu (II) CT system and its viability for entanglement generation^{19}. Finally, note that the existence of a highly protected subspace might be exploited for further schemes in a way that this twoqubit system can be considered as two physical qubits able to temporarily store a single logical qubit, as an approach to builtin quantum error protection.
Methods
Ab initio calculations
The timeindependent electronic structure of HoW_{10} was computed using the multiconfigurational Complete Active SelfConsistent Field SpinOrbit (CASSCFSO) method as implemented in the OpenMOLCAS program package (version 18.09)^{37}. The molecular geometry was extracted from the singlecrystal Xray structure. Scalar relativistic effects were taken into account with the DouglasKrollHess transformation using the relativistically contracted atomic natural orbital ANORCC basis set with VDZP quality for all atoms. The active space consisted of 10 electrons on the 7 forbitals of Ho^{3+} ion. The molecular orbitals were optimized at the CASSCF level in a stateaverage (SA) over 35 quintets of the ground state term (L = 6 for Ho^{3+}). The wave functions obtained at CASSCF were then mixed by spin–orbit coupling by means of the RASSI approach. The crystalfield parameters (CFPs) used for the system Hamiltonian (Eq. (1)) were calculated using SINGLE_ANISO module implemented in OpenMOLCAS^{38}. .
DFT calculations
The structural optimization of the crystallographic coordinates (in vacuum) and the vibrational modes calculations were carried out at DFT level using the Gaussian16 package in its revision A.03^{39}. Vibrational frequency calculations were performed using both the fully optimized structure and the Xray crystal structure with no optimization. The PBE0 hybrid exchangecorrelation functional was used for both optimization and frequency calculations in combination with Stuttgart RSC ANO basis set with effective core potential (ECP) for the Ho^{3+} cation. CRENBL basis set have been used for W with corresponding ECP potential and 631G(d,p) basis set had been used for oxygen. An “ultrafine” integration grid and “very tight” SCF convergence criterion were applied. Dispersion effects were taken into account using the empirical GD3BJ dispersion correction.
Spin–electric couplings
To quantify the spin–electric couplings (SEC), we established a relation between the change in the dipole moment and spinenergy levels. As the dipole moment depends on the electronic cloud distribution, it will be directly affected by an induced Efield. To quantify the change of dipole moment, we used vibrational normal mode basis, as they are orthogonal, and we know for fact that each molecular perturbation can be decomposed of linear combination of normal basis. A central assumption in our methodology is that the rise in potential energy (\({U}_{\alpha }=\frac{1}{2}\kappa {q}_{\alpha }^{2}\)) due to the displacement of the atomic positions in the form of a harmonic oscillator is exactly matched by the stabilization of the potential energy (U_{E} = − p ⋅ E) due to the change in the molecular electric dipole in presence of an external electric field. In absence of Efield, spinenergy levels in lanthanidebased complexes can be characterized by timeindependent crystalfield Hamiltonian \({\hat{H}}_{{{{\rm{CF}}}}}(J)\),
where \({B}_{k}^{q}\) and \({\hat{O}}_{k}^{q}(J)\) correspond to the crystalfield parameter and the Stevens operator of rank ‘k’ respectively, and ‘J’ is total angular momentum. From eq. (6), it is evident that external Efield will modify the crystalfield parameters; i.e., “\({B}_{k}^{q}\to {B}_{k}^{q}({{{\bf{Q}}}}(E))\)” and Hamiltonian takes the form \({\hat{H}}_{{{{\rm{CF}}}}}(J)\to {\hat{H}}_{{{{\rm{CF}}}}}(J,{{{\bf{Q}}}}(E))\), where Q(E) accounts for the perturbative displacement caused by the change in the electronic cloud as a consequence of the Efield.
To numerically simulate this effect, we firstly distort the molecular geometry along the displacement vector for each normal mode α. The distortions are quantified by zeropoint displacement for each normal mode, i.e., \({q}_{\alpha }=\pm \sqrt{\hslash \omega /{\kappa }_{\alpha }}\), here κ_{i} is the spring mass constant, which can be easily extract from DFT calculations. At each distorted geometry, we compute the dipole moment ‘p_{α}’ by singlepoint calculations in the presence of the environment described as a pointcharge approximation. The obtained ‘p_{α}’ is then used in E(q_{α}) = − U_{α}/∣Δp_{α}∣ to quantify how much distortion q_{α} is produced in each normal mode “α” by an induced Efield. The effective total distortion will be a linear combination of orthogonal basis defined in the normal mode basis:
where q_{α}(E) is the distortion induced at a given electrical field for the normal mode α, which can be further expressed in terms of normal vector as:
n_{x,y,z,α} are the normal vector coordinates in x, y, and z direction. Lastly, the effective total distortion (Q(E)) for given Efield is added to the equilibrium molecular geometry to recompute the spinenergy levels at the CASSCF level.
Spin vibrational couplings
The interaction of the electronic spins with the nuclear degrees of freedom (phonons) also known as spin–phonon couplings (SPC) is a source of decoherence in molecular qubits. Spin, phonons, and spinphonon Hamiltonians are defined below. (Note that the spin Hamiltonian is equivalent to the CF Hamiltonian previously described.)
ω_{α} denotes the frequency for mode α whereas n_{α} corresponds to the phonon level. \({\hat{q}}_{\alpha }\) denotes the a dimensional phonon coordinate and the term \({\left(\frac{\partial {B}_{k}^{q}}{\partial {q}_{\alpha }}\right)}_{0}\) is the SPC constant for a given phononα. To evaluate this coupling term, we start by computing the zeropoint energy displacements; i.e., \({q}_{\alpha }=\sqrt{\hslash {\omega }_{\alpha }/{k}_{\alpha }}\), where k_{α} is the spring mass constant for a given phononα. We then distort the equilibrium molecular geometry q_{eq} within a limit of −q_{α} → + q_{α} along the displacement vectors n_{x,y,z} for each modeα using: q_{dist,α} = q_{eq} + q_{j}n_{x,y,z}. For each distorted geometry q_{dist,α}, we performed ab initio electronic structure calculations (CASSCFSO) and extracted the CFPs (\({B}_{q}^{k}\)) in Steven’s operator definition. The obtained CFPs are fitted with second order polynomials to evaluate the firstorder SPC for a given phonon, i.e., \({\left(\frac{\partial {B}_{k}^{q}}{\partial {q}_{\alpha }}\right)}_{0}\).
Data availability
The data reported in this work and in the Supplementary Information are available from the corresponding author upon reasonable request.
Code availability
The code used in this work and in the Supplementary Information are available from the corresponding author upon reasonable request.
References
Kane, B. E. A siliconbased nuclear spin quantum computer. Nature 393, 133–137 (1998).
Trif, M., Troiani, F., Stepanenko, D. & Loss, D. Spinelectric coupling in molecular magnets. Phys. Rev. Lett. 101, 217201 (2008).
Laucht, A. et al. Electrically controlling singlespin qubits in a continuous microwave field. Sci. Adv. 1, e1500022 (2015).
Tosi, G. et al. Silicon quantum processor with robust longdistance qubit couplings. Nat. Commun. 8, 1–11 (2017).
Asaad, S. et al. Coherent electrical control of a single highspin nucleus in silicon. Nature 579, 205–209 (2020).
Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Phys. Rev. A 57, 120 (1998).
Godfrin, C. et al. Operating quantum states in single magnetic molecules: implementation of grover’s quantum algorithm. Phys. Rev. Lett. 119, 187702 (2017).
Madzik, M. T. et al. Conditional quantum operation of two exchangecoupled singledonor spin qubits in a moscompatible silicon device. Nat. Commun. 12, 1–8 (2021).
Savytskyy, R. et al. An electricallydriven singleatom ‘flipflop’ qubit. Preprint at https://arxiv.org/abs/2202.04438 (2022).
GaitaAriño, A., Luis, F., Hill, S. & Coronado, E. Molecular spins for quantum computation. Nat. Chem. 11, 301–309 (2019).
Timco, G. A. et al. Engineering the coupling between molecular spin qubits by coordination chemistry. Nat. Nanotechnol. 4, 173–178 (2009).
Aguilá, D. et al. Heterodimetallic \(lnln^{\prime}\) lanthanide complexes: toward a chemical design of twoqubit molecular spin quantum gates. J. Am. Chem. Soc. 136, 14215–14222 (2014).
Collett, C. A., Santini, P., Carretta, S. & Friedman, J. R. Constructing clocktransitionbased twoqubit gates from dimers of molecular nanomagnets. Phys. Rev. Res. 2, 032037 (2020).
FerrandoSoria, J. et al. A modular design of molecular qubits to implement universal quantum gates. Nat. Commun. 7, 1–10 (2016).
Lehmann, J., GaitaArino, A., Coronado, E. & Loss, D. Spin qubits with electrically gated polyoxometalate molecules. Nat. Nanotechnol. 2, 312–317 (2007).
FerrandoSoria, J. et al. Switchable interaction in molecular double qubits. Chem 1, 727–752 (2016).
Jenkins, M. et al. Coherent manipulation of threequbit states in a molecular singleion magnet. Phys. Rev. B 95, 064423 (2017).
Shiddiq, M. et al. Enhancing coherence in molecular spin qubits via atomic clock transitions. Nature 531, 348–351 (2016).
Kundu, K. et al. A 9.2ghz clock transition in a Lu (ii) molecular spin qubit arising from a 3,467mhz hyperfine interaction. Nat. Chem. 14, 392–397 (2022).
RubínOsanz, M. et al. Chemical tuning of spin clock transitions in molecular monomers based on nuclear spinfree ni (ii). Chem. Sci. 12, 5123–5133 (2021).
Liu, J. et al. Quantum coherent spin–electric control in a molecular nanomagnet at clock transitions. Nat. Phys. 17, 1205–1209 (2021).
EscaleraMoreno, L., GaitaAriño, A. & Coronado, E. Decoherence from dipolar interspin interactions in molecular spin qubits. Phys. Rev. B 100, 064405 (2019).
Chen, J. et al. Electron spin echo envelope modulation at clock transitions in molecular spin qubits. Preprint at https://arxiv.org/abs/2106.05185 (2021).
Breuer, H.P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford University Press, 2002).
Lunghi, A. & Sanvito, S. The limit of spin lifetime in solidstate electronic spins. J. Phys. Chem. Lett. 11, 6273–6278 (2020).
Briganti, M. et al. A complete ab initio view of orbach and raman spin–lattice relaxation in a dysprosium coordination compound. J. Am. Chem. Soc. 143, 13633–13645 (2021).
Lunghi, A. Toward exact predictions of spinphonon relaxation times: An ab initio implementation of open quantum systems theory. Sci. Adv. 8, eabn7880 (2022).
Gu, L. & Wu, R. Origins of slow magnetic relaxation in singlemolecule magnets. Phys. Rev. Lett. 125, 117203 (2020).
Lunghi, A., Totti, F., Sessoli, R. & Sanvito, S. The role of anharmonic phonons in underbarrier spin relaxation of single molecule magnets. Nat. Commun. 8, 1–7 (2017).
Blockmon, A. L. et al. Spectroscopic analysis of vibronic relaxation pathways in molecular spin qubit [ho (w5o18) 2] 9–: sparse spectra are key. Inorg. Chem. 60, 14096–14104 (2021).
Lunghi, A. & Sanvito, S. Multiple spin–phonon relaxation pathways in a kramer singleion magnet. J. Chem. Phys. 153, 174113 (2020).
Bader, K. et al. Room temperature quantum coherence in a potential molecular qubit. Nat. Commun. 5, 1–5 (2014).
Atzori, M. et al. Roomtemperature quantum coherence and rabi oscillations in vanadyl phthalocyanine: toward multifunctional molecular spin qubits. J. Am. Chem. Soc. 138, 2154–2157 (2016).
Luis, F. et al. Molecular prototypes for spinbased cnot and swap quantum gates. Phys. Rev. Lett. 107, 117203 (2011).
Luis, F. et al. A dissymmetric [gd2] coordination molecular dimer hosting six addressable spin qubits. Commun. Chem. 3, 1–11 (2020).
Stamp, P. C. & GaitaArino, A. Spinbased quantum computers made by chemistry: hows and whys. J. Mater. Chem. 19, 1718–1730 (2009).
Fdez. Galvan, I. et al. Openmolcas: From source code to insight. J. Chem. Theory Comput. 15, 5925–5964 (2019).
Ungur, L. & Chibotaru, L. F. Ab initio crystal field for lanthanides. Eur. J. Chem. 23, 3708–3718 (2017).
Frisch, M. et al. Gaussian 16 revision a. 03. (Gaussian Inc., Wallingford, CT, 2016).
Acknowledgements
This work is supported by the European Commission (FETOPEN project FATMOLS (No. 862893)); the Spanish MICINN (grant CTQ201789993 and PGC2018099568BI00 cofinanced by FEDER, grant MAT201789528 and the Unit of excellence “María de Maeztu” CEX2019000919M); and the Generalitat Valenciana (CIDEGENT/2021/018 and PROMETEO/2019/066). J.A. is indebted to the MICINN for his “Ramón y Cajal" fellowship (RyC201723500). We thank J. Liu for his insightful comments.
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A.G.A. and J.A. conceived the project. A.U. conducted singlequbit relaxation with the assistance of Z.H. and J.C. guided by J.A. A.U. and Z.H. conducted twoqubit entanglement generation guided by A.G.A. A.G.A., J.A., A.U., Z.H. wrote the paper. All authors contributed to the manuscript.
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Ullah, A., Hu, Z., Cerdá, J. et al. Electrical twoqubit gates within a pair of clockqubit magnetic molecules. npj Quantum Inf 8, 133 (2022). https://doi.org/10.1038/s41534022006478
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DOI: https://doi.org/10.1038/s41534022006478
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