Designed Polynuclear Lanthanide Complexes for Quantum Information Processing

The design of dissymmetric organic ligands featuring combinations of 1,3-diketone and 2,6-diacetylpyridine coordination pockets has been exploited to produce dinuclear and trinuclear lanthanide-based coordination compounds. These molecules exhibit two or more non-equivalent Ln ions, most remarkably enabling the access to well-defined heterolanthanide compositions. The site-selective disposition of each metal ion within the molecular entities allows to study each centre individually as a spin-based quantum bit, affording unparalleled versatility for quantum gate design. The inherent weak interaction between the Ln ions permits the performance of multi-qubit quantum logical operations realized through their derived magnetic states, or implementing quantum-error correction protocols. The different studies performed to date on these systems are revised, showing their vast potential within spin-based quantum information processing.


Introduction
Lanthanide compounds have been widely exploited due to their exceptional physical properties, 1-10 specially in their most common oxidation state (+3), when they exhibit the electronic configuration [Xe]4f n (n = 0 to 14). The coupling between the spin and orbital magnetic moments of the 4f electrons, particularly protected from external perturbations by full 5s and 5p electronic shells, generates a ladder of states that furnishes a rich variety of 4f-4f transitions spanning the near infrared (NIR) to the UV and visible region. [11][12][13] These transitions are formally forbidden, but the excites states can be accessed by energy transfer processes that lead to the emission phenomenon called sensitized metal luminescence. 14,15 Such peculiarities have inspired researchers to study these materials in a wide range of applications, as in agents for optical and magnetic resonance imaging 14,16 or devices for light-emitting diodes. 17,18 In addition, the shielding of these electrons makes the states of these ladders to be only perturbed by the crystal fields, which split its states into the so-called Stark levels by energies ranging from the microwave region to 100 cm -1 at the most. These levels respond differently to an external magnetic field, thus generating an energy barrier to the inversion of the magnetization of each individual ion. 19,20 These properties have allowed to consider lanthanide complexes for single-molecule magnetic information storage. [20][21][22][23][24] In the last years, lanthanide molecules have also been exploited for spin-based quantum information processing (QIP), considering the two low-lying magnetic states of Ln(III) ions as the realization of a quantum bit (the basic information unit of the quantum computer). 25,26 In contrast to their classical counterpart, the quantum bit (or qubit) is defined not only in its two quantum states (i.e., spin up and spin down, or |0⟩ and |1⟩) but also in any arbitrary quantum superposition of them (| ⟩ = |0⟩ + β|1⟩). This and other resources inherent to quantum mechanics allows QIP to perform tasks unachievable by current computation technologies. Thus, several quantum systems are being investigated to provide the hardware for this new type of computers. Among them, we can find the quantized levels of superconducting circuits, [27][28][29] the electronic states of trapped atoms or ions, 30,31 or the use of polarization states of photons. 32,33 Exploiting molecules to embed spin-based qubits offers important advantages compared to other technologies, since they can be produced in large scale and can be easily tuned by chemical methods. 34 One type of molecules studied extensively as spin-based qubits are the series of clusters with formula (R 4N)[Cr7NiF8(O2CR')16], which consist of a ring with the eight metals bridged pairwise by one F − and two carboxylate ligands. The antiferromagnetic coupling between adjacent metals leads to an S = ½ spin ground state, which encodes the qubit. 35 The quantum coherence 36 and ability of these rings to associate into complex supramolecular structures have been studied intensely. 37,38 In contrast to clusters, monometallic complexes do not present the need to isolate an S = ½ spin ground state from excited states. Of the possible metals, Cu(II) and V(IV) have been the most investigated 39,40 and their quantum coherence optimized progressively. [41][42][43][44] In this context, Ln-based compounds have shown particular appealing characteristics. First, they typically exhibit large separations between the ± mJ ground state and the first excited state, allowing to use the former as an effective S = ½ two-level system and thus embody the qubit states. Since the g factor of these doublets is usually very large, the qubit computational basis is highly polarizable, therefore offering better initialization capacities. In addition, the specific case of Gd(III) allows the possibility to encode several qubits in the magnetic states of one single Ln ion, exempt of spin orbit coupling (SOC), exploiting the high spin ground state and multiple transitions between different m J states. 45 Finally, most Ln ions have isotopes with nuclear spin, which allows using the derived electronuclear transitions to define multiple qubits or n-qudits (a quantum system with nquantum levels that can be coherently manipulated). 46 Quantum transitions within a spin-based qubit can be driven coherently using electromagnetic pulses. The resulting magnetic states must conserve their quantum coherence for a given time, called phase-memory time, T M, during which the information can be manipulated. This capacity was shown, for example, in a Gd III -doped polyoxometalate material ([GdxY1- 12-). For this compound, a coherence time of 410 ns was estimated for the superposition of a targeted pair of quantum states. 47 Similarly, coherent manipulations were observed within the ground-state doublet of a mononuclear compound doped with Yb III ([YbxLu1-x(trensal)]). Here all the possible electron-spin rotations with conservation of the nuclear spin were detected at specific magnetic fields and their quantum coherence characterized using, respectively, continuous wave and pulsed X-band EPR measurements on single-crystals. 48 Those studies, based on magnetically diluted compounds, evidenced the effect of magnetic dipolar interactions as the main source of decoherence. However, isolating the spin qubit in a diamagnetic matrix precludes interaction between them, and thus its role in multi-qubit logical operations. Decoherence problems caused by magnetic field fluctuations can be diminished when operating with atomic clock transitions, particularly protected from local field variations. This approach was shown in a Ho III -molecular nanomagnet, where the holmium ion is encapsulated by two molecular tungsten oxide moieties ([Ho(W 5O18)2] 9-). 49 This strategy allowed observing coherence times up to 8.4 microseconds, even for a highly concentrated sample. The realization of algorithms for quantum computing requires a universal set of quantum gates (qugates). The latter are fundamental quantum logic operations brought about by one or several qubits. The quantum versions of the CNOT and SWAP operations constitute archetypes of two-qubit qugates. In fact, the former, together with individual qubits are a universal set. The computational basis of two-qubit gates span the four possible combinations of two qubits, |11⟩, |10⟩, |01⟩ and |00⟩. The CNOT flips the state of the second (target) qubit only when the first (control) qubit resides in a particular state, for example in the state |1⟩ or spin up (ie. |11⟩ ↔ |10⟩). On the other hand, the SWAP gate exchanges the state of both qubits when those are in a different state (ie. |01⟩ ↔ |10⟩). Chemistry offers the possibility to incorporate two qubits within one molecule and confer them the ability to realize qugates. These two qubits need to be addressable, thus exhibit a different response to the external stimulus used to manipulate them. In the case of spinbased qubits the ideal technology for manipulating them is EPR, thus, addressable qubits must have different g factors. In addition, conditional operations require an interaction between qubits. However, the interaction must be weak enough so that both, factorized and entangled states are possible. All these requirements can be satisfied through chemical design, for example through incorporation of two lanthanides within molecules in the appropriate conditions. 34 Molecules hosting several lanthanides could actually allow the realization of multiqubit quantum gates of two or even more qubits. 25,26 or could be used to simplify the implementation of other complicated operations, such as quantum error corrections (see below), which otherwise require artificially wiring up several qubits. Another approach to address these challenges proposes to use the eight spin-orbit free, internal spin levels of the Gd ion within molecules.
These spin states in the compound [Gd(H 2O)(P5W30O110] 12-, for example, were exploited to perform operations using three qubits. 45 Moreover, all transitions were found to be coherently controlled, with times ranging from 470 to 600 ns at 6 K. Some years ago, we decided to explore the synthesis of polynuclear lanthanide complexes as possible multi-qubit qugates, based on the design of organic ligands able to encapsulate more than one Ln ion in different environments. Originally, the design of the asymmetric ligand H 3LA (Scheme 1) allowed the production of a complete series of dinuclear homometallic [LnLn] species featuring non-equivalent metal ions, thus potentially able to perform two-qubit quantum gates. 50-52 Such molecular system was further exploited to embed two different Ln ions, thus producing heterometallic [LnLn'] compounds through one-pot reactions (Figure 1)

Non-symmetric dinuclear Ln compounds as information processing devices
Dinuclear complexes featuring 4f ions are very common in the literature (more than 8000 hits in the CSD, version 5.42). However, in order to perform 2qubit logical operations, their corresponding Ln III ions must be individually addressable, and thus exhibit a different magnetic response. In addition, the quantum entanglement between both qubits required to perform such operations demands that they are weakly interacting. Molecules fulfilling these criteria are scarce, since Nature favours the formation of symmetric species. In order to promote the coupling of two different Ln III qubits within a molecule, we designed an asymmetric ligand, H The structural characterization of all the compounds featured in Figure 2 reveal that they crystalize in the monoclinic P2 1/c space group. This analysis evidenced only structural trait, differentiating two groups; the coordination mode of the nitrate ligand, which transits from bidentate for the lightest ions (from La to Eu) to monodentate for the heaviest ones (from Gd to Lu). Therefore, both sites of the [LnLn] molecules exhibit coordination number 9, unless when the NO 3ligand is chelating, which imposes coordination number 10. The different coordination environment for Ln1 and Ln2 was analysed by means of continuous shape measures (CShMs). While non-ideal polyhedron describes properly the environment of Ln1, the geometry for Ln2 was found close to either a spherical capped antiprism or a spherical tricapped trigonal prism. Moreover, the structural analyses for all homometallic [LnLn] compounds evidenced systematic longer Ln2-O bonds compared the Ln1-O ones ( Figure 2). The parameter ΔO served to assess this property (ΔO difference between average distances of Ln2-O and Ln1-O, Table 1). This striking feature, which was exploited to promote access to heterometallic molecules (see below), contributed as well to the difference between both 4f ions in the molecular scaffold. The interaction between Ln1 and Ln2 was evaluated with molar magnetic susceptibility (χ) measurements on the paramagnetic analogues. The decline of the magnetic response when cooling the samples showed the expected depopulation of the excited Stark sublevels of the corresponding 2S+1 LJ multiplet, as well as a weak antiferromagnetic interaction between the metal ions. 52 The latter was assessed for both [GdGd] analogues (X = Cland NO 3 -), which allows modelling the corresponding χ versus T dependence using the simple spin-only Hamiltonian H = -JS 1S2 thanks to the lack of orbital angular momentum of Gd(III). The best fit J values (-0.02 and -0.04 cm -1 for chloride 56  Please do not adjust margins Please do not adjust margins analogues, respectively) confirmed the proposed weak antiferromagnetic interaction, suggesting these homometallic [LnLn] molecules as ideal platforms for molecular quantum processors.

Site-selective one-pot formation of heterometallic [LnLn'] molecules
The structural characterization of the homometallic [LnLn] series demonstrated, not only different coordination environments between Ln1 and Ln2, but also a systematic difference of cavity size. As observed in Figure 2, the space available to Ln2 is larger than to Ln1. Thus, the molecular scaffold suggests a preference for larger metal ions in site 2, compared to site 1. This feature was exploited to create siteselective [LnLn'] molecules by following the same reaction procedure but using two different lanthanide ions together with H 3LA. 53, 57-59 The crystalline products obtained from these reactions were characterized both in solid state and in solution.
The different combinations reported so far are listed in Table 1  For almost all binary combinations explored, the best refinement solution of the structural data converges to the expected distribution. The solutions obtained for other metallic distributions produce an increment to the refinement parameters R 1 and wR2 (Table 1). The only analogues where the nature of Ln1 and Ln2 could not be clearly evidenced crystallographically derive from reactions involving Ln ions with ionic radii differences (∆r) smaller than 0.05 Å. As expected (Figure 3) Figure 4, illustrating the purity of the compounds. However, signals from the corresponding [LnLn] and [Ln'Ln'] homometallic analogues may be observed for compounds featuring ions with very similar sizes. These results are a reflect of a relaxation of the selectivity of the system in solution, which causes the marginal detection of non-heterometallic signals for some pairs with moderately higher ∆r values (Table 1). Scrambling upon dissolution, as suggested by these observations study was corroborated by means of DFT calculations, which concluded a decrease on the selectivity derived from the loss of the terminal NO 3 − , H2O and py ligands. 57 Table 1. Structural parameters for all the reported (Hpy)[LnLn'(HL1)3X(py)(H2O)] analogues (X = Clor NO3 -).
ΔwR2 (%) [c] MSScramb. [ 59 Interestingly, the emission of these compounds was satisfactorily ascribed to one unique Ln site, despite that they exhibit scrambling by mass spectrometry. These results evidenced the drastic effect of DMSO on the stability of the molecular scaffold, and thus in the process of scrambling in solution. The cumulative evidence thus indicates that the amount of pure [LnLn'] derivatives accessible by this chemical approach is noteworthy, thus providing numerous possible [LnLn'] analogues for embedding quantum bits and performing quantum logical operations. Notably, few other molecular architectures promoting one-pot heterometallic Ln-based compounds have been reported. As an example, an hexadentate ditopic ligand combining a benzimidazole-pyridine-carboxamide unit and a bis(benzimidazole)pyridine moiety was found to encapsulate two different Ln ions with selectivity up to 90 % based on ESI-MS and 1 H NMR studies, only for the most favourable cases. 65,66 There are also examples of heterotrinuclear systems obtained with a tris-tridentate ligand, although with much decreased selectivity. 66 Interestingly, a quinolate ligand was used to produce trinuclear molecules with different metallic environments, thus favouring nonstatistical distributions of three different Ln ions. The selectivity, however, was very low. [67][68][69] The alternative way to obtain heterometallic compounds of high site-selectivity is through consecutive synthetic steps, thus sequentially positioning each Ln ion. [70][71][72][73][74][75] This strategy allows involves also binding two or more Ln discrete compounds, thus obtaining multinuclear complexes featuring two 76,77 or three 78, 79 different 4f ions. The main drawback of these methods is that they are tedious and challenging synthetically, preventing the screening of many Ln combinations, the production of sufficient amounts for physical studies or implementations, the preparation of magnetically dilute materials or repeated attempts to grow large single crystals.

Realisation of quantum logical operations inside the [LnLn'] molecular architecture
The structural characterization of the different [LnLn'] analogues evidenced the potential of this unique molecular scaffold to embed two Ln-based quantum bits. The molecules provide great versatility for "g engineering" by allowing to choose in great measure the nature of both ions, ensuring magnetic dissymmetry even for the homometallic derivatives. In addition, they promote weak magnetic coupling among both metal ions. The controlled encapsulation of two different Ln ions, offers a versatility that is very useful for other purposes. For example, it allows studying full or partial diamagnetic analogues, and thus characterizing each qubit separately in the same environment exhibited in the corresponding two-qubit molecule. In the following sections, the potential shown by the

Dinuclear lanthanide compounds for spin-based CNOT and SWAP quantum gates
As previously discussed, the low-lying magnetic states of a lanthanide ion can provide a good realization of the spin quantum bit. In general, the crystal field imposes a well-isolated ground state doublet, so that its ±m J states can be used as the two-level (effective S = ½ spin) system to encode the qubit basis states defined by the spin up, |1⟩ or spin down, |0⟩ ( Figure 5). Electromagnetic pulses can be then used to prepare quantum superpositions of those qubits. If several qubits are embedded in none molecule and conditional rotations of any of these are possible using also pulse sequences, these molecules may be used to implement multi-qubit qugates (see above). 25,26 Considering the magnetic inequivalence of both lanthanide ions in the above described dinuclear complexes and their weak coupling, four non-degenerated Zeeman states originate under an external magnetic field, featuring unequal energy separations. This allows selecting the proper transitions in order to promote specifically either the CNOT or the SWAP operation, as shown in Figure 5. The homometallic [TbTb] analogue (X = Cl -, Figure 6) was investigated down to sub-Kelvin temperatures to confirm this strategy. 51 The χT vs T plot showed a drop below 100 K that was ascribed to the depopulation of the excited Stark sublevels derived from the crystal field ( Figure 6). From this data, an approximate value for the uniaxial anisotropy D was determined, which allowed estimating a separation between the ground state and the first excited state of 180 K and thus ensuring a well-isolated doublet for each of the Tb ions ( Figure  5). The second drop observed below 3 K can be attributed to the antiferromagnetic coupling between the two Tb qubits, which was corroborated by magnetic heat capacity measurements. Interestingly, a finite magnetic moment is observed despite the presence of such antiparallel exchange interaction, which can be modelled considering that the easy axes of each qubit are not parallel but tilted. A value of this tilting angle δ of 66° was found to reproduce satisfactorily both magnetic and heat capacity data. Most importantly, this model explains the magnetic asymmetry of the dinuclear entity, and thus the nonequivalence of the two spin qubits. In order to demonstrate that the system allows promoting the transitions associated to the logical operations, the energy level spectrum was generated by simulation ( Figure 6). Continuous-wave EPR measurements were performed in frozen solutions in order to show that transitions between |1⟩ ⊗ |0⟩ and |0⟩ ⊗ |1⟩ (SWAP gate) or |1⟩ ⊗ |0⟩ and |1⟩ ⊗ |1⟩ (CNOT gate) can be carried out by selecting the required magnetic field where only those are resonant with X-band photons. The study performed on [TbTb] demonstrated that the designed molecular scaffold meets the requirements to implement selected logical operations. In order to exploit their physical realization, and taking advantage of the chemical versatility of these compounds, the study was extended to the heterometallic analogue [CeEr], which offers several advantages. 53 The ions Ce III and Er III exhibit very different magnetic behaviour (J = 5/2 and g J = 6/7 for the former and J = 15/2 and g J = 6/5 for the latter). Also, since both are Kramer's ions, the crystal field interaction is expected to lead to doubly degenerated ground states, neatly isolated from the excited states. Moreover, the presence of magnetic nuclear spin is drastically reduced, carried only by 23 % of the stable isotopes for Er(III), thus contributing to reduce decoherence. The adaptability of the coordination architecture allowed studying each isolated Ln qubit by selecting, respectively, the [CeY] and [LaEr] analogues. These compounds exhibit both Ce III and Er III ions at the same position as in [CeEr], together with a diamagnetic ion (Y III and La III , respectively, Figure 7). Magnetic susceptibility and EPR data from these analogues were used to obtain information on the magnetic energy level structure and thus the nature of the ground state doublet for each Ln qubit. As expected, the effective gyromagnetic g i tensors found for Ce III and Er III ions were very different, quantifying their magnetic inequivalence. Again, the first excited sublevel was found to be well above the ground state doublet, with a ∆Ln energy difference of 43 and 230 K for Er III and Ce III , respectively (Figure 7). The physical characterization of [CeEr] allowed determining a weak antiferromagnetic coupling between both qubits, and thus again an optimal energy level diagram to promote two-qubit logical operations. In this case, the coherence of the resonance corresponding to the CNOT operation was examined by time-domain EPR spectroscopy. The echo signal detected was found to decay exponentially, providing a decoherence time of T M ≈ 410 ns ( Figure 7) and demonstrating for the first time the feasibility of coherent manipulations on a two-qubit molecular spin quantum gate made using coordination chemistry.

Exploiting the [LnLn'] coordination architecture to embed six addressable qubits
One of the main advantages of using lanthanides for quantum computation is the possibility to increase the number of spin qubits within the metal ion. As observed before for the [TbTb] or [CeEr] analogues, the isolated ground state ± mJ doublet of each Ln may be exploited to realize one quantum bit, thus allowing two-qubit quantum operations in these molecules. However, several quantum bits can be embedded in a single Ln ion if multiple spin levels are accessible. This would enable to increase the density of quantum information. In particular, the eight different m J states produced by the S =7/2 spin state of Gd III (which features no orbital angular momentum, i.e. L = 0) offer a unique energy level diagram to realize three quantum bits. Since the intrinsic anisotropy of this ion is negligible, its zero field splitting arises from the crystal field and thus from small distortions of the coordination environment. The transitions between the corresponding magnetic levels are, consequently, all accessible via conventional magnetic spectroscopic techniques. Our molecular platform can undoubtedly provide the required compounds to explore such strategy. 63 The heterometallic [LaGd] and [GdLu] analogues, for example, allowed characterizing the energy levels of Gd III in the two available coordination positions (site 1 and site 2, Figure 8). Being accompanied by diamagnetic Ln partners, the properties of the isolated Gd III ions were assessed by magnetic and spectroscopic measurements. Using continuous-wave EPR and heat capacity, the zero-field splitting parameters D and E were deduced, and found to be different from one metal to the other as expected from their different coordination environments (Figure 8). From these data, the energy level structure for each Gd III ion was obtained. The coherence of various of the transitions within these manifolds was assessed by electron spin-echo decay studies in frozen solutions at different magnetic fields. The values of T M obtained (from 1.05 to 0.73 µs for [LaGd] and from 1.31 to 0.85 µs for [GdLu], respectively) confirmed both compounds as potential three-qubit molecular platforms. In order to explore the system with both sites incorporating Gd III , the homometallic [GdGd] analogue was assessed ( Figure  9). Considering the presence of magnetic coupling between both ions, 64 unequally spaced energy levels can be generated, thus codifying six qubits (all combinations from |000000⟩ to |111111⟩). The physical characterization of [GdGd] evidenced an antiferromagnetic interaction in the compound, and thus the confirmation of such energy diagram (Figure 9). In this case, the coherence for transitions between some of those levels was found to be slightly reduced (T M = 0.73 µs, virtually constant for the fields explores), suggesting the interaction between both ions as a source of decoherence. Nevertheless, the experiments performed on the three systems confirmed that hyperfine interactions with solvent molecules are the main influence on T M. This study underscores the possibility to increase the number of accessible quantum bits in the molecular platform, and thus the potential of this chemical approach for quantum technology with higher performance. However, one major drawback to overcome is the fact that the energy diagram hosting the computational basis becomes excessively crowded for individual addressing (as is evident in Figure 9). The alternative approach to attain higher complexity is the increase of the number of metals acting as individual addressable qubits within the molecule.

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Increasing the complexity: Ligand-based strategy to encapsulate three Ln qubits
The discovery of the highly performant [LnLn'] system unveiled a synthetic avenue for the preparation of novel families of coordination chemistry molecules featuring site-selective hetero-lanthanide composition. In H 3LA, the two different chelating units, β-diketone (O,O) and dipicolinate-like (O,N,O), directly influence the preference for Ln ions with smaller or larger radii, respectively. Following on this principle, we decided to explore the coordination capabilities of the ligand H 2LB, featuring an additional β-diketone unit (Scheme 1). 80 By analogy with H 3LA, this organic molecule could potentially incorporate three Ln ions while keeping the two-metal selectivity. In that sense, the central dipicolinate-like (O,N,O) group was to allocate the largest Ln ion, while Ln featuring smaller ionic radii would be chelated in both diketonate units. The realization of these trinuclear [LnLn'Ln] heterometallic compounds would allow studying a molecular system with three Ln qubits and thus realizing more complex quantum operations, such as quantum error correction protocols (see below). As previously seen for [LnLn'], mixing H 2LB with two different Ln(NO 3)3 salts in pyridine through one-pot reactions produced the expected [LnLn'Ln] trinuclear entities. The different Ln/Ln' combinations reported so far are shown in Table 2. As observed from their crystal structure, in all cases the molecule exhibits three lanthanide ions chelated in this case by four ligands (Figure 10). Interestingly, two of the ligands are the result of partial in-situ hydrolysis, one β-diketone group being converted to a carboxylic acid (H 2LC, Scheme 1). Together with this new ligand, the structural characterization revealed the exact nature of these heterometallic and heteroleptic complexes with formula [Ln 2Ln'(LB)2(LC)2(py)(H2O)2](NO3) ([LnLn'Ln]). The stoichiometric synthesis of the compounds from their components was carried out by producing the ligand H 2LC and conducting the corresponding reactions. Interestingly, in this case, the difference in coordination environment is even more severe than in the [LnLn'] system. The central Ln2 ion is undecacoordinated, with two dipicolinate-like (O,N,O) units, two diketonate (O,O) groups and one pyridine ligand. The peripheral Ln1 ions are, in contrast, octa-coordinated, featuring two diketonate groups, one dipicolinate-like unit and one molecule of water. The positive charge of the resuting cationic species is compensated with a nitrate anion in the crystal lattice. The identity of each lanthanide ion was clearly evidenced by the refinement of the corresponding crystal structures, since other metallic distributions produced significant worst parameters (  Figure 11). These results demonstrated the success of the ligand-based strategy to control the disposition of three lanthanide ions in a molecule through one-pot reactions, and thus the possibility to explore its potential for new molecular quantum processors.  reference Δr (Å) [a] av.Ln1-O (Å) [b] av.Ln2-O (Å) [b] ΔO (Å) ΔR1 (%) [c] ΔwR2 (%) [c] MSScramb. [ [b] Only oxygen atoms from chelating LB 2and LC 2ligands were considered.
[c] Increase of percentage obtained by comparing with the values for the metal distribution producing the second best refinement parameters.

Complexes [LnLn'Ln] as three-qubit molecular processors: Implementing quantum error correction
One of the main concerns regarding QIP is the fragility of the quantum information conveyed by qubits, since their superposition states can be easily altered due to interactions with the environment. Protocols for quantum error correction (QEC) must be therefore developed for ensuring the proper realization of quantum operations. 81 These protocols may become extremely simplified if they can be performed by conditional operations on three addressable qubits connected to each other. In this protocol, two auxiliary qubits are used to protect an encoded qubit from phase flips (the most significant source of decoherence in molecular magnets). 82 This is carried out following consecutive operations on these three quantum bits, known as three-qubit phase-flip repetition code, in three different stages. The first step (encoding) consists of two CNOT operations between the qubit carrying the logical information (central qubit) and each of the two auxiliary qubits initialized into the |0⟩ state. Together with a π/2 rotation on the three qubits, these operations bring the encoded qubit into a state protected from phase flips, and thus robust against dephasing.
The three qubits are left undisturbed during a memory time τ, where their evolution depends only on pure dephasing. Then, the same gates from the encoding step are carried out in reverse order (π/2 rotations and CNOT operations) thus producing the decoding step. The final step is the correction one, consisting of a controlled-controlled-NOT operation (CCNOT) where the (central) qubit is the target and the ancillae are the control qubits. If there has been an error, this step corrects it. In order to promote these operations, the molecular architecture must consequently exhibit three addressable weakly coupled effective S = ½ ions. The trinuclear [LnLn'Ln] compound perfectly fulfils these requirements, in particular those analogues featuring anisotropic lanthanide ions with wellisolated ground states. The inequality between the distal qubits is ensured by their different respond to a magnetic field, since they have their anisotropy axes oriented differently. In order to study and implement the correction code, [ErCeEr] was chosen, Ce III being the encoded qubit and the two Er III ions acting as auxiliary qubits. 83 The complex was thus magnetically and spectroscopically characterized, together with the analogues [ErLaEr] and [LuCeLu], which allow studying each type of Ln qubit individually. The experimental data obtained from the latter allowed determining the corresponding g tensors, obtaining best values of g Ce = (1.7, 1.7, 2.2) and gEr = (1, 2, 11.5). Measurements performed on [ErCeEr] were then used to assess the Er-Ce interaction, providing dipole-dipole interaction tensors of J ErCe = (0.01, 0.07, -0.29) cm −1 and JErEr = (0.00, 0.02, -0,22) cm −1 . The results clearly demonstrated the ability of [ErCeEr] to promote the correction code, since it showed the existence of spin-spin interactions, and indicated a large and highly anisotropic g tensor for erbium, significantly different from that of cerium, while the three qubits being addressable.
From the whole set of data, the level diagram for the three interacting qubits was determined, showing that specific transitions between each of the magnetic energy levels can be driven by means of microwave resonant pulses ( Figure 12). Thus, each CNOT operation can be implemented a different pulse. The CNOT 2→3 (targeting the third qubit, Er III , controlled by the central Ce III ion), for example, is promoted by transitions between |110⟩ and |111⟩ states, and between |010⟩ and |011⟩. π/2 rotations require four pulses of slightly different frequencies, while CCNOT gate can be performed by a single pulse resonant with the |101⟩ ↔ |111⟩ transition. Pulsed EPR measurements allowed assessing the phase memory time T M at different fields where the transitions of interest take place, yielding values close to 0.5 µs. Time-dependent numerical simulations were then performed in order to demonstrate the viability of the protocol. From these, the difference between the initial stored logical state ψ〉 and the corrected or uncorrected state of the Ce III qubit were assessed. The simulations evidenced that the error is efficiently reduced by the correction code, allowing the implementation of 50-100 logical gates before its repetition becomes necessary. The results thus evidenced the possibility of exploiting such Ln-based molecules not only to encode qubits but also to embed quantum error correction.

Conclusions and outlook
This perspective has outlined a novel and very promising synthetic strategy to prepare site-selective heterometallic lanthanide molecules of different nuclearity, through ligand design, based on atomic radii differences. The different polynuclear Ln compounds revised here highlight the ability of specifically targeted systems for certain quantum logic operations. Thus, by combining two different chelating units such a diketonate and dipicolinate-like groups, different cavities can be promoted in a molecular scaffold, providing different environments (and physical properties) to the corresponding Ln qubits. Undoubtedly, this synthetic approach is very promising, since new original architectures can be designed exploiting and tuning organic ligands. As an example, the dinuclear [LnLn'] molecular system has been accessed from partially deprotonated HLA 2ligands, leaving the phenol group unaltered. However, as carried out for similar ligands, the latter could be further deprotonated under stronger basic conditions, thus promoting an additional "diketonate" pocket and potentially larger nuclearity. 84 On the other side, new asymmetric ligands featuring additional dipicolinate-like units could promote a third chemical environment in the molecular scaffold. As it has been observed, both [LnLn'] and [LnLn'Ln] compounds adopt larger Ln ions in cavities exhibiting higher number of dipicolinate-like groups. Through this approach one can imagine, for example, selectively locating three different types of 4f ions discriminated by their size, producing heterotrimetallic entities directly through one-pot reactions. On that respect, the use of similar chelating units, such as picolinate moieties, could also promote additional different coordination environments. 85 On the other hand, and despite the potential of molecular design, it should be pointed out that for molecules to perform quantum logical operations, a crucial step must be made: the integration of these spin-based quantum material into a technological device to allow writing, processing and readingout the information. In this regard, an appealing proposal is based on the design of new hybrid devices by coupling individual molecules with superconducting coplanar resonators and transmission lines. 86 The device is based on a coplanar superconducting resonator where the molecules are deposited near superconducting wave-guides. The former and the latter are responsible for creating general and local magnetic fields, which can be taken in and out-of resonance with a specific molecular entity to perform a quantum operation. In order to accomplish these hybrid devices, considered as the last step for controlling and communicating the information of the molecules, further specific considerations should be taken into account regarding the molecular design. Aside from embedding magnetically inequivalent and weakly coupled Ln ions, the molecular entities must be robust to permit their deposition. In this respect, both [LnLn'] and [LnLn'Ln] complexes have shown great stability in solution, opening the possibility to be exploited via deposition using the proposed DIP-Pen nanolithography. 86 Additionally, the interaction of the molecules with the surface must be sufficiently strong, and with a preferred orientation, to ensure homogeneous coupling. In that sense, additional grafting groups could be incorporated into the corresponding organic ligands of the Ln molecules. Considering the two addressable surfaces used for the hybrid quantum processor (metallic Nb or Si/SiO 2), several possibilities might be explored. Metallic surfaces could be, for example, exploited with molecules featuring thiol or aromatic groups, while the addition of carboxylate or phosphonate groups could be used to allow deposition of the molecule over oxide-based surfaces. The design of Ln systems thus drifts to a laborious but very attractive challenge for synthetic coordination chemists devoted to construct the future devices for QIP.