Molecular nanomagnets: a viable path toward quantum information processing?

The Cr₇Ni ring [145], with its $S = 1/2$ ground state, has been one of the most studied molecular qubits (see table 1(i)). The seven Cr³⁺ ($s = 3/2$) and the Ni²⁺ (s = 1) ions are at the vertices of a regular octagon and their spins are antiferromagnetically coupled, yielding a $S = 1/2$ ground state with very little mixing with higher S states (at low magnetic fields). The ground doublet of Cr₇Ni was thus shown to be suitable for the encoding of the $| 0 \rangle$ and $| 1 \rangle$ logical states of a qubit [122]. In addition, the energy gap to the first excited multiplet ($S = 3/2$) is around 13 K, ensuring a small leakage to non-computational levels during single-qubit rotations. In addition, the properties of antiferromagnetic rings can be tailored for the realization of different QC architectures, from changing the ligands in order to facilitate their linking into multi-qubit structures [67] (see section 4.2), to tuning their anisotropy [57] and hence matrix elements of specific transitions. Moreover, Cr₇Ni molecules can be successfully deposited on surfaces preserving their magnetic properties [35, 146–150]. This is crucial for the realization of molecular quantum processors, since single molecules organized on a surface in a chip-like architecture would provide notable advantages in the individual addressing of qubits and eventual wiring into other nanodevices.

Table 1. Most important classes of molecular spin qubits and related features/achievements. (first row, second column) Reprinted with permission from [121]. Copyright (2015) American Chemical Society. Further permission related to the material excerpted should be directed to the ACS. (first row, fourth column) Reproduced with permission from [126]. CC BY-NC 3.0. (second row, first column) Reprinted (figure) with permission from [161], Copyright (2019) by the American Physical Society. (second row, third column) Reproduced from [163]. CC BY 4.0. (second row, fourth column) Reproduced from [165]. CC BY 4.0. (third row, first column) Reproduced from [16]. CC BY 4.0. (third row, fourth column) Reproduced with permission from [166]. CC BY-NC 3.0. (fourth row, first column) Reproduced from [155]. CC BY 4.0. (fourth row, third column) Reproduced with permission from [167]. CC BY-NC 3.0. (fifth row, first column) Reprinted (figure) with permission from [133], Copyright (2008) by the American Physical Society. (fifth row, third column) Reproduced from [160]. CC BY 4.0. (sixth row, first column) Reproduced from [156]. CC BY 4.0. (sixth row, second column) Reproduced with permission from [169]. CC BY-NC 3.0. (sixth row, third column) Reproduced with permission from [133]. CC BY-NC 3.0. (sixth row, fourth column) Reprinted (adapted) with permission from [125]. Copyright (2018) American Chemical Society.

	(i) Multi-spin clusters	(ii) Monomers (3d)	(iii) Ln qubits	(iv) Nuclear spins
Very long T2
		[121, 158, 159]		[126]
Electric control
	[160–162]		[129, 163]	[43, 164, 165]
Auxiliary states
	[14, 16, 122, 156]			[166]
Engineering
	[12, 155]	[132, 167]	[133]
Noise-protected encoding
	[128, 157, 160]		[163, 168]
Embedded quantum-error correction
	[156]	[169]	[133]	[125, 126, 170]

Antiferromagnetic rings also satisfy another crucial prerequisite for the deployment of MNMs in QC. Indeed, coherence times T₂ of the order of a few µs were already measured in 2007 on Cr₇Ni [118]. To further improve T₂ when measured in crystals, Cr₇Ni qubits can be diluted in a matrix of non-magnetic analogues (i.e. isostructural molecules where the magnetic ions are replaced by equivalent diamagnetic ones, like Ga₇Zn for Cr₇Ni) [151], thus reducing harmful inter-molecular interactions. In this regime, decoherence at low temperature mostly originates from the hyperfine interaction of each qubit with the surrounding nuclear spins [119]. A further improvement of the Cr₇Ni coherence time (up to T₂ $\gt$ 15 µs) was obtained by chemical engineering of the nuclear-spin environment of the magnetic ions [152, 153]. By combining these synthetic techniques significant values of $\eta\gt 10^3$ (i.e. the number of elementary operations which can be implemented within the coherence time) can be reached. All in all, Cr₇Ni and its fellow rings are still important building blocks for molecule-based QIP [154, 155]. A final comment on multi-spin molecules is in order. By a proper design of the molecular spin Hamiltonian and of the hierarchy and structure of the spin–spin couplings, it is possible to identify a regime in which decoherence does not increase with the number of levels, as discussed in sections 4.4 and 5 below [156, 157].

A decisive step forward to increase T₂ was achieved in [159] by shifting the attention from multi-spin clusters to much simpler complexes containing a single metal ion (see table 1(ii)). This strategy allowed reaching T₂ values as large as $68 \,\mu$s at 7 K in a (PPh₄)₂[Cu(mnt)₂] molecule, at the price of renouncing to the additional degrees of freedom typical of multi-spin clusters. Moreover, by shifting non-computational excited states from ${\sim} 1$ meV (as in Cr₇Ni) to the eV range, phonon-induced relaxation is strongly suppressed, resulting in coherence times as large as 0.6 µs at room temperature [159]. Record T₂ values, outperforming those observed in solid-state qubits, were observed in a V-based system, ((d₂₀-Ph₄P)₂[V(C₈S₈)₃]), reaching ~ 1 ms at 10 K after a proper chemical tuning of the nuclear spin content in both ligands and solvents [121]. By combining this T₂ with the capabilities of the highest-power available EPR spectrometer [136], $\eta \sim 10^5$ can be reached.

Performances at room temperature, however, are strongly limited in this system by the rapid decrease of the spin- lattice relaxation time T₁, on increasing temperature. Replacing the V-ion with a $S = 1/2$ vanadyl moiety (VO²⁺) has been decisive in this sense, with VOPc (Pc = phthalocyanine) displaying very long T₁ = 2.4 s, thanks to its stable coordination geometry [139]. In addition, pulsed EPR spectroscopy experiments on VOPc revealed quantum coherence up to room temperature with an almost constant $T_M \sim$ 1 µs [158]. These results were also accompanied by the observation of Rabi oscillations, demonstrating the possibility to coherently manipulate the VOPc molecular spin qubit up to room temperature. This and all the results that followed [171, 172], have established VO-based complexes as the new paradigm for temperature-resistant molecular spin qubits. Among these, VO(TPP) (TPP = tetraphenylporphyrinate) is one of the most promising molecular qubit [173]. It can be arranged in dimeric species where the two electronic spins are distinguishable and exchange-coupled to implement quantum gates [167]. The incoherent spin dynamics of VO-based systems was also extensively studied from both an experimental and theoretical point of view [113, 114, 174–176], shedding new light on the role of phonons and on spin-phonon couplings in molecular spin relaxation and decoherence. Last but not least, square- pyramidal vanadyl complexes can also be deposited on surfaces, obtaining monolayers of ordered arrays of intact molecules, retaining their paramagnetic nature [177] and can also be embedded into superconducting coplanar resonators [18, 178, 179], paving the way for the integration of molecular spin ensembles into microwave quantum architectures.

Interesting systems are also monomers containing a single spin S transition-metal ion like Fe³⁺, as they provide $(2S+1)$-levels qudits which can be coherently manipulated by microwave drives [180]. For instance, coherence times of a few µs at 5 K were shown for [Cr³⁺C₂O₄)₃]³⁻ spin 3/2 and [Fe³⁺C₂O₄)₃]³⁻ $S = 5/2$ complexes [137, 180].

Concurrently with these progresses obtained with 3d-based complexes, MNMs containing single lanthanide (Ln) ions have been also largely studied (table 1(iii)). The original idea behind the investigation of these molecules was to exploit the large magnetic anisotropy of Ln ions to increase the energy barrier of SMMs. Indeed, after the discovery of a SMM behavior of double-decker Pc complexes containing a single Tb or Dy ion [181] in the early 2000s, a lot of efforts were devoted to the theoretical design and the synthesis of Ln SMMs with increasingly high blocking temperatures [74, 86, 87, 182–192]. At the same time, Ln-based mononuclear complexes have also offered new alternatives for realizing molecular qubits and qudits [11]. For instance, rare-earth polyoxometalates (POM) have been considered as promising qubit candidates with competitive coherence times [13, 168, 193–196]. The simplest situation is given by Kramers ions, with a ground doublet that naturally encodes a qubit whose frequency can be tuned by external magnetic fields. The case of non-Kramers ions, with an integer angular momentum $\mathcal{J}$, exemplifies a more subtle crystal-field engineering. Transverse magnetic anisotropy terms induce avoided level crossings between states with opposite angular momentum projections $\pm m_{\mathcal{J}}$ at specific magnetic fields . The gap at these spin clock transitions can be maximized by adequately choosing the lanthanide ion and its local coordination (e.g. a Ho³⁺ with a $m_{\mathcal{J}} = \pm 4$ ground state in a fourth fold symmetric coordination, as it is the case with HoW₁₀ [168, 194, 197]). The qubit states are then encoded into symmetric and anti-symmetric superpositions of $\pm m_{\mathcal{J}}$ states resulting in a qubit which is practically insensitive to magnetic noise, as it was shown for HoW₁₀ [168], where a maximum in T₂ was observed at the avoided-crossing magnetic field. Other examples confirming that coherence times increase sharply near each clock transition have been found in 3d-metal-based compounds [198, 199], where the anticrossing gap can be tuned via the chemical design of the molecular structure [163, 200].

Nuclear spins of magnetic ions are also a valuable addition to their electronic counterparts in encoding qubits or qudits (see table 1(iv)). As discussed below, nuclear spins in transition metal or Ln ions can provide a significant number of states for qudit-based algorithms. They are in general more isolated from the environment and this yields both a higher protection from decoherence and longer manipulation times. Nevertheless, in contrast to organic radicals usually employed in NMR quantum computation [201, 202], nuclear spins exploited for QIP in MNMs are coupled to the electronic spins by a large hyperfine interaction (A). This is especially true for Ln-complexes, but also for transition metals. Such a sizable $A\sim 500$–800 MHz on the one hand increases the splitting between the ground and excited states (thus enabling initialization by cooling below ${\sim} 10$ mK). On the other hand, it makes nuclear spin manipulations by radio-frequency pulses much faster than in organic radicals and it distinguishes different energy gaps even in nuclei with small intrinsic quadrupole. In particular, it is usually possible to identify a regime at intermediate magnetic fields (0.2–0.3 T) where the electronic and nuclear spin wavefunctions are practically factorized, but (i) the Rabi frequency of nuclear spin oscillations is enhanced by several orders of magnitude compared to the case with A = 0 [125]; (ii) an effective quadrupole interaction allows one to distinguish all nuclear transitions and address each of them separately by resonant drives [126]. In this context, Yb(trensal) provided the first example of coupled electronic qubit-nuclear qudit system for the implementation of QEC algorithms [125, 203] (details in section 5). Another important example is given by ⁵¹V in VO(TPP), which embeds a nuclear spin 7/2 coupled to the electronic spin 1/2. This yields electronuclear spin states for the encoding of a qudit (see section 4.4 below), whose coherent manipulations was demonstrated via pulsed EPR and broadband NMR experiments [126, 166].

The most traditional way to control the state of MNMs employs oscillating external magnetic fields, but their local application at the single-molecule level within short timescales, as required for QC, still represents a major challenge. For this reason, the electric control of molecular spin qubits has also been envisaged. Contrary to magnetic fields, electric fields involve no flowing currents (thus no dissipation) and can be confined by nanoscopic gates, which allows for enhancing the field intensity while achieving a high degree of integration. Yet, their coupling to electron spins is typically rather weak. To overcome this intrinsic hurdle, antiferromagnetically-coupled triangular molecules were proposed. Indeed, the exchanged-coupled magnetic ions interact with external electric fields through chirality of their spin structure and thus the spin-electric coupling is possible even in the absence of spin-orbit interaction [128, 160]. Moreover, the peculiar encoding of the qubits into the chirality degree of freedom makes them intrinsically protected from decoherence driven by the interaction with neighboring nuclear spins [128]. We mention here, in turn, a different proposal of a decoherence-protected encoding of logical qubit states, based on toroidal magnetic states [97], as those found in Dy₃ [204, 205] and Dy₆ systems [98].

The first experimental demonstrations of molecular spin control by means of electric fields were achieved some years later, along different proposals. A crucial milestone is represented by the electric-field modulation of the hyperfine interaction between a nuclear spin 3/2 and an Tb qubit (the so-called hyperfine Stark effect). Experiments were performed on a single TbPc₂ [43, 164] molecule arranged in a single-molecule transistor setup, presented in section 8. The Authors found a modification of the nuclear Rabi frequency induced by application of an electric field which periodically modulates A when the system is subject to microwave pulses, thus demonstrating an electrically-driven magnetic resonance. A remarkable dephasing time $T_2^* = 64 \, \mu$s was also reported, measured by Ramsey interferometry on the individual nuclear qudit.

Another approach was pursued in [162] to study MnPhOMe helices, by applying an oscillating electric field during a continuous-wave EPR experiment. A magnetoelectric effect was observed and attributed to a modulation of the exchange interaction induced by the electric field. Conversely, electric field pulses were embedded in EPR Hahn-echo sequences to investigate spin-electric properties of AF rings and of the triangular molecule Cu₃ [161]. In particular, an electric field pulse is applied just after the $\pi/2$ pulse. As a consequence, during free precession the spin accumulates an additional phase which depends linearly on the electric field via the spin-electric coupling constant (SEC). Hence, the method allows one to investigate the magnitude of the SEC, finding ${\sim} 1.9$ rad (V m⁻¹)⁻¹ in both Cr₇Mn and Cu₃. To increase this value, the attention was shifted to Ln complexes, in line with the aforementioned choice of TbPc₂ [43]. To this end, the method was applied to the POM HoW₁₀ [129], finding a significantly larger SEC of 11.4 Hz (V m⁻¹)⁻¹. This value is close to that obtained on Mn²⁺ impurities in ZnO [206], where the electric field was used to modulate the zero-field splitting anisotropy of spin 5/2 ions. The key ingredients to reach a sizable spin-electric coupling are identified to be an electrically polarizable and soft environment for the spin, together with spin energy levels strongly sensitive to distortions. The strategy applied to HoW₁₀ would allow one to reach Rabi frequencies larger than $1/T_2 \approx 1$ MHz by using an electric field of 1 mV nm⁻¹, as available in nanogaps. Further steps would be required to reach control capabilities comparable to magnetic drives. One possibility is also to design exchange-coupled multi spin molecules with sizable electric dipole, thus enabling to tune the exchange coupling by electric fields [207].

Chemistry can also be exploited to expand the computational space, moving from single to multiple qubits encoded in single molecules. A quite intuitive option of this 'scaling up by molecular design' path is to create molecular structures hosting several weakly coupled and addressable magnetic centers [208]. Examples include molecular dimers and trimers of lanthanide ions [133, 209, 210], as well as supramolecular structures able to bind several well-known molecular qubits, such as Cr₇Ni, and, in some cases, combine them with other complexes having an effective $S = 1/2$ [12, 16, 134, 211, 212].

In 2009, it was shown that AF rings like Cr₇Ni can be chemically and magnetically linked to each other through an interposed paramagnetic ion [12]. This work paved the way for the synthesis of different weakly-coupled supramolecular dimers, where the coherence of the individual units is also preserved [16, 154, 213]. In [12] the Authors also demonstrated that it is possible to generate maximally entangled states in these 'trimers' with simple microwave pulse sequences. Starting from this result, many supramolecular compounds containing linked antiferromagnetic rings were synthesized and investigated. Being entanglement an essential resource for QIP, dimers of weakly-coupled MNMs were also exploited as a playground for experimentally investigating entangled states. Initially, entanglement between molecular subunits was experimentally demonstrated by exploiting susceptibility as an entaglement witness or by fitting trial model Hamiltonians to electron paramagnetic resonance data [135]. More recently, a dimer of Cr₇Ni rings has been used as a benchmark to develop and test a method exploiting 4-dimensional inelastic neutron scattering to detect and quantify entanglement [110] in these supramolecular compounds.

A relevant question is how to implement in molecular systems the single and two-qubit gates required to perform universal operations. The possibility of introducing switchable spin–spin interactions will be discussed in the following sub-section. Here, we consider the situation in which the spin centers are permanently coupled. If, as it is often the case, the inter-qubit couplings are stronger than the Rabi frequencies $\Omega_{\mathrm{R}}$ of each qubit, the operations rely on the ability of addressing resonant transitions between different spin states by their resonant frequencies. It can be shown that this implies a certain asymmetry between each of the spin centers in the molecule to make different energy gaps distinguishable, anticipating the more general case of spin qudits discussed further below. Let's consider the spin Hamiltonian of two interacting (pseudo-) $s = 1/2$ spins

$$\begin{equation} H_\mathrm{dim} = {\textbf{s}}_1 \cdot {\textbf{J}} \cdot {\textbf{s}}_2 + \mu_B {\textbf{B}} \cdot \left( {\textbf{g}}_1 \cdot {\textbf{s}}_1 + {\textbf{g}}_2 \cdot {\textbf{s}}_2 \right) \end{equation} \tag{ 3 }$$

coupled by an exchange interaction tensor J and with different gyromagnetic tensors ${\textbf{g}}_{1}$ and g₂. The latter property makes both spin qubits separately addressable through their different resonance frequencies under any finite magnetic field. Because of their mutual coupling, the energy associated with flipping each spin also depends on the state of the other, which provides the necessary ingredient to implement conditional operations. These two conditions result in a fully 'anharmonic' energy level scheme, as shown in figure 2(a). Then, two-qubit gates can be realized by single shot resonant electromagnetic pulses. For instance, a pulse tuned at the $| 10 \rangle \leftrightarrow | 11 \rangle$ transition of a molecular spin dimer implements the prototypical cNOT gate.

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This scheme was proposed on an asymmetric Tb₂ [209] and CeEr [210] dimers (see figure 2(b)), where a Hamiltonian of the type of equation (3) was used. The necessary magnetic assymmetry was achieved via the different coordinations of the two Tb³⁺ ions and by the heteronuclear composition, respectively. The transition involved in the cNOT gate was spectroscopically resolved by continuous-wave and the associated decoherence time T₂ was measured by pulsed EPR experiments (figure 2(c)). Resolving such a transition (i.e. the excitation of the target spin depending on the state of the control) demonstrates the existence of a sizable coupling between the two distinguishable qubits. An analogous time-resolved experiment was implemented on a two-qubit assembly consisting of two NO radicals in a TEMPO molecule [214], made inequivalent by the differently-oriented g tensors. A coherent oscillation of the target qubit state (corresponding to a cNOT gate) was observed on a time-scale of about 200 µs. However, the implementation of single qubit rotations becomes more involved with permanently coupled qubits.

Spin-correlated radical pairs obtained from photodriven electron transfer from a molecular donor to an acceptor were also studied as potential two-qubit units [215]. These can be faithfully initialized in an entangled singlet state $(| 01 \rangle-| 10 \rangle)/\sqrt{2}$ even at room temperature and then manipulated by microwave pulses to implement one and two-qubit gates. Donor–acceptor entanglement can also be transferred to a third spin [216, 217] and controlled by charge recombination, thus in fact implementing a quantum teleportation protocol [218]. Using a chiral bridge to link donor and acceptor could result in a local spin polarization which can be exploited for initialization and readout of single molecules, as explained in section 8.

Showing that the system affords universal operations in the Hilbert space spanned by the two qubits reduces to demonstrating that the set of allowed resonant transitions can generate a complete set of operations, i.e. that any state can be generated via a sequence of such transitions [219]. Figure 2(d) shows such a 'universality plot' of the rates of operations $W_{i,j}$, with $i,j \in \left\{\vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle, \vert 11 \rangle \right\}$, linking any two basis states. The fact that $W_{i,j} T_{2} \gt 1$ for any ${i,j}$ shows that the molecular system is universal and provides a quantitative method to benchmark the performances of different molecules [8, 166, 196, 220].

An alternative scheme applies when the intramolecular spin–spin couplings J are very weak, i.e. when $J/\hbar \ll \Omega_{\mathrm{R}}$. Along the lines of nuclear magnetic resonance (NMR) quantum computing, it is then possible to let the system evolve freely into an entagled state [124, 221]. This approach was demonstrated on a family of Cr₇Ni supramolecular dimers [124] by applying double electron-electron resonance techniques to detect the evolution of a proper initial state on a timescale $h/J$. Later on, a similar method, combining single qubit rotations and free evolution of the coupled system, was proposed for implementing two-qubit gates between Cr₇Mn dimers, encoding clock-transitions qubits [221].

A serious drawback of these methods is that the permanent qubit–qubit interaction strongly limits scalability of the architecture. As it has been mentioned above, rotations of each qubit might require a large number of frequencies, depending on the state of other qubits in the qubit register. Moreover, in the second scheme the inability of turning off the inter-qubit coupling leads to an unwanted spontaneous evolution of the system, which should be corrected for a reliable computation.

Molecular multi-qubit structures introduced in the previous section present permanent spin–spin interactions, which cannot be reversibly removed on the time scale of qubit gates. Hence, alternative strategies must be found to switch on and off such mutual qubit–qubit interactions in an effective way. We overview the most important proposals in this direction in figure 3, based on using suitable driving fields and on exploiting non-computational states which can be easily obtained in MNMs, thanks to the high degree of chemical control.

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Schemes reported in panels (a)–(c) employ global microwave pulses acting on the whole qubit register as the only manipulation tool. This removes the challenge of addressing single-molecules (discussed in section 8), but also brings along some limitations if the target is a universal QC (see below). Conversely, it represents a promising scheme for quantum simulation of a large class of model Hamiltonians (see section 6). In scheme (a), qubits are encoded into multi-spin MNMs, exploiting the richness of their spectrum as a resource for computation. In a first proposal [122], a chain of Cr₇Ni qubits with properly tuned inter-molecular couplings was shown to behave as a non-interacting register as long as each ring is in the computational subspace. Conversely, an effective finite coupling between pairs of neighboring qubits is obtained by temporarily exciting one of them to a specific state. However, in order to perfectly switch off the interaction within the computational subspace, a fine tuning of the inter-qubit couplings at the synthetic level is required ⁶ . Such a limitation is overcome by encoding qubits into anti-ferromagnetic isosceles triangles, as reported in figure 3(a). In that case, each qubit is described by the spin Hamiltonian

$$\begin{equation} H_\mathrm{tri} = J {\textbf{s}}_1 \cdot {\textbf{s}}_2 + J^{\prime} {\textbf{s}}_3 \cdot \left( {\textbf{s}}_1 + {\textbf{s}}_2 \right) + \mu_B {\textbf{B}} \cdot \sum_i {\textbf{g}}_i \cdot {\textbf{s}}_i, \end{equation} \tag{ 4 }$$

with $J\gt J^{\prime}$ the isotropic exchange interactions between spins s_i and ${\textbf{g}}_i$ the respective gyromagnetic tensors. Hamiltonian $H_\mathrm{tri}$ is diagonal in the total spin basis $| S_{12} S M \rangle$ ($\textbf{S}_{12} = {\textbf{s}}_1 + {\textbf{s}}_2$, $\textbf{S} = \textbf{S}_{12} + {\textbf{s}}_3$) and the energy spectrum results in two low-energy doublets $| 0,1/2,M \rangle$ and $| 1,1/2,M \rangle$ split by $\delta = J-J^{\prime}$ and a higher energy $S = 3/2$. The two doublets characterized by $S_{12} = 0,1$ are used as the computational and auxiliary states, respectively. The former are exploited to encode logical states $| 0 \rangle \equiv | 0,1/2, -1/2 \rangle$ and $| 1 \rangle \equiv | 0,1/2, 1/2 \rangle$, the latter to implement two qubit gates in a switchable manner. In particular, let us consider (figure 3(a)) a pair of neighboring triangular units, made distinguishable by slightly different parameters in H_tri (e.g. ${\textbf{g}}_3$), and coupled by $H_{AB} = {\textbf{s}}_3^A \cdot \left( j_1 {\textbf{s}}_1^B + j_2 {\textbf{s}}_2^B \right)$, with $j_{1,2} \ll J, J^{\prime}$. As long as both A and B triangles are in the computational subspace ($S_{12} = 0$), the effective AB interaction is off and single-qubit gates can be implemented by microwave pulses resonant with the $| 0,1/2,-1/2 \rangle \leftrightarrow | 0,1/2,1/2 \rangle$ transition on either A or B qubits in the reigister. The interaction is activated by bringing B triangle to the auxiliary state with $S_{12} = 1$. This transition is allowed, but generally slower compared to single-qubit ones, because the corresponding matrix element $\propto (g_1-g_2)$, which for transition metals is typically ${\sim} g_i/10$. A (semi)resonant 2π pulse (corresponding to excitation and de-excitation) from the computational to the auxiliary subspaces conditioned by the state $| \alpha \beta \rangle$ ($\alpha, \beta = 0,1$) of the AB pair implements a cϕ gate, as detailed in appendix C. Indeed, the inter-molecular interaction makes the excitation of B qubit (to the auxiliary state) dependent on the state of A, thus implementing it only for $| \alpha \beta \rangle = | 11 \rangle$ (i.e. for a specific M of both qubits in the computational and auxiliary states). The simplest implementations of these triangular qubits are represented by spin 1/2 ions such as V⁴⁺ or Cu²⁺ [222]. A drawback of these systems are possible (dipolar) interactions between vertices of different triangles, which can be reduced by replacing the single vertex spin with a larger molecule with a spin 1/2 ground state [14].

An alternative scheme (illustrated by figure 3(b)) was put forward in [15], where the switch of the interaction between neighbouring logical qubits was a separated antiferromagnetic unit with S = 0 ground state. Both logical qubits and switches are arranged in an alternating ABAB pattern, i.e. A/B qubits as well as A-B/B-A switches are spectroscopically distinguishable and hence can be separately addressed by global magnetic pulses. If we consider, for instance, Cr₇Ni qubits, such a selectivity can be obtained by tilting their respective orientations and exploiting the anisotropy of their g tensors. The simplest implementation of the switch unit is represented by a dimer of spins 1/2 coupled by antiferromagnetic interaction and characterized by different (or differently oriented) g tensors. As in scheme (a), this latter ingredient allows the excitation of the switch unit from the S = 0 ground state to a S = 1 excited one, thus dynamically turning on the effective coupling between neighboring qubits and implementing a controlled-phase gate among them, as discussed above. This proposal could be realized for instance by linking two non-parallel Cr₇Ni rings through a Cu₂ dimer with significantly different g on the two ions and a not too large exchange coupling. An even simpler implementation is represented by using a single spin (such as an effective spin 1/2) as a switch of the effective qubit–qubit coupling (see figure 3(c)). To this end, several compounds were synthesized in the following years [132, 211, 223]. The idea to dynamically turn on and off the qubit–qubit coupling is the following. By working in a sizable magnetic field, such that the qubit-switch coupling is much smaller than the difference of Zeeman energies between qubits and switch, the state of the qubits and of the switch are factorized. In the idle configuration, the switch is frozen into its ground state and hence its coupling to the qubits only renormalizes the magnetic field on the qubits. However, the excitation energy of the switch depends on the states of both qubits to first order via the (longitudinal component of the) qubit-switch exchange interaction. Hence, an excitation of the switch by a semi-resonant pulse only for a specific state of the qubit pair (e.g. when the two qubits are in $| 11 \rangle$) can be exploited to implement a cϕ gate, as explained before. The same scheme can also be adapted to an electro-nuclear spin system [132], where electronic spin provide a fast effective switch, while logical qubits are encoded in hyperfine-coupled nuclear spins. Although simpler to be synthesized compared to scheme (b), the approach of figure 3(c) allows one to switch on and off the effective qubit–qubit coupling only to first order. Indeed, the transverse component of the qubit-switch exchange interaction induces a second-order effective coupling between the two qubits, arising from mixing with excited states of the switch. Such an unwanted time evolution can be partially suppressed by using a pair of inequivalent qubits [16]. However, it limits scalability of the platform [16] compared to proposals (a), (b), were the interaction was completely turned off in the idle phase.

Remarkably, schemes (a)–(c) on a chain of ABAB qubits allow one to implement gates in the absence of local qubit control [122], by exploiting non-computational qubit states to embed an auxiliary control-unit [224], which marks specific sites along the chain and can be moved by SWAP gates. This approach makes the proposed ABAB linear register universal, but it also implies a sizeable number of SWAP gates which could make the resulting computation very long.

Another proposal, differing both in the manipulation tool and in the nature of the switch, is illustrated by figure 3(d). It is based on using a scanning tunneling microscope (STM) tip to locally control the state of a redox-active unit interposed between pairs of molecular qubits. The latter can be brought back and forth from a reduced (s = 0) to an oxidized ($s = 1/2$) state by quickly adding/removing an electron via the STM tip, thus turning on/off the inter-qubit interaction. An implementation of this scheme on a pair of VO²⁺ qubits, connected to a central switch consisting of mixed valence [PMo₁₂O₄₀] Keggin unit, was proposed in [225]. In the on state, the three-spin Hamiltonian is of the form

$$\begin{align} H_{3} & = J_{qq} \; {\textbf{s}}_1 \cdot {\textbf{s}}_3 + J_{qs} \; {\textbf{s}}_2 \cdot \left( {\textbf{s}}_1 + {\textbf{s}}_3 \right) \nonumber\\ & = \left(J_{qq}-J_{qs}\right) \; {\textbf{s}}_1 \cdot {\textbf{s}}_3 + J_{qs} \; \textbf{S}^2/2 \nonumber\\ & \quad + \mu_B B \cdot \sum_i g_{iz} s_{iz}, \end{align} \tag{ 5 }$$

where $s_{1,3}/s_2$ are the qubits/switch spins, J_qq and J_qs are the qubit–qubit and qubit-switch exchange interactions and it is fixed B = 0. An entangling gate such as the $\sqrt{{\mathrm{SWAP}}}$ between s₁ and s₃ is here obtained by exploiting the free evolution induced by Hamiltonian (5). To achieve this, one then needs to cancel the evolution due to the term proportional to $\textbf{S}^2 = (\sum_i {\textbf{s}}_i)^2$, which can be achieved for particular sets of parameters of H₃ after specific time intervals. However, high fidelity for the $\sqrt{{\mathrm{SWAP}}}$ gate is guaranteed only for fixed $J_{qq}/J_{qs}$ ratios.

A more flexible scheme, implementing the equivalent iSWAP$^\alpha$ gate, was proposed in [16, 212], and applied to a three spin system with a redox-active unit (either a Co$^{3+}\rightarrow$ Co²⁺ or a Ru$_2^{3+,4+}$Co$^{2+}\rightarrow$ Ru$_2^{3+}$Co²⁺) linking two Cr₇Ni qubits. The scheme works in an applied magnetic field (along z in the following) and requires the two qubits to have the same Zeeman energy, but different from that of the oxidized switch. In particular, if $J_{qq}\ll (g_{2z}-g_{1z}) \mu_B B$, one can derive an effective qubit–qubit Hamiltonian of the form

$$\begin{equation} H_{qq} = \Gamma \left(s_{1x} s_{3x} + s_{1y} s_{3y}\right) + \lambda_1 s_{1z} + \lambda_3 s_{3z}, \end{equation} \tag{ 6 }$$

by restricting to the subspace in which the switch is practically frozen into its $m = -1/2$ state. Here $\Gamma = J_{qq}^2/2 \mu_B B (g_{1z}-g_{2z})$, $\lambda_1 = \lambda_3 = -J_{qq}/2 + \Gamma/2 + g_{iz} \mu_B B$. Apart from single qubit R_z rotations, free evolution induced by H_qq implements the iSWAP$^\alpha$ gate, as described in section 3.C. Simulations performed on Cr₇Ni–Co–Cr₇Ni and Cr₇Ni–Ru₂Co–Cr₇Ni complexes, including experimentally measured values of T₂, demonstrate that both single- and two-qubit gates can be implemented with high fidelity along these lines. Here we are neglecting small direct dipolar interactions between s₁ and s₃, which would worsen the performance of the scheme, by introducing an always on coupling which would require proper schemes to be corrected.

A iSWAP$^\alpha$-like gate can be implemented on any qubit pair with a mutual XY interaction (see appendix A), provided that states $| 01 \rangle$ and $| 10 \rangle$ are degenerate (i.e. $\lambda_1 = \lambda_3$ in (6)). Otherwise, if $\Gamma \ll |\lambda_1 - \lambda_3|$, the qubit–qubit coupling is ineffective. Nevertheless, there exist several possibilities to tune single-qubit energies and hence activate an interaction which implements the iSWAP$^\alpha$ gate, as sketched for instance in figure 3(e). A first option is based on applying longitudinal microwave pulses to the qubit pair, with $\hbar \omega = \lambda_1-\lambda_3$, as suggested in [154]. This longitudinal pulse perturbatively compensates the difference between single-qubit transition energies, thus implementing an iSWAP$^\alpha$ evolution. However, this gate occurs on a rather long time-scale, because the effective inter-qubit interaction is re-scaled (for spins 1/2) by a factor $\approx B_1/B_0$, where B₁ is the amplitude of the oscillating field.

An alternative recent proposal exploits an electric field to tune single-qubit energies between pairs of HoW₁₀ CT qubits coupled by a weak dipole-dipole interaction [226]. By properly choosing the applied magnetic field, one can identify an experimental configuration (close to the clock transition) which guarantees protection from decoherence, while also activating a dipolar coupling (which would instead vanish exactly at the CT). As sketched in figure 3(e), in presence of an electric field E states $| 01 \rangle$ and $| 10 \rangle$ are split and each of the two qubits can be selectively addressed by microwave pulses; by turning E temporarily off $| 01 \rangle$ and $| 10 \rangle$ become degenerate and therefore a iSWAP$^\alpha$ gate is implemented via the free evolution induced by the dipolar coupling. The proposal requires a diluted crystal of qubits and a precise dipolar interaction between two neighboring HoW₁₀, thus making scalability challenging. However, pulse sequences can be designed to only affect the desired kind of molecular pairs, leaving isolated molecules (and, to a lesser extent, also other pairs) unaffected [226]. The implementation of two qubit gates with both these methods based on varying the static Hamiltonian is rather slow. In addition, in the absence of a switch, the always on spin–spin coupling will induce a non-trivial evolution of the register, making, e.g. rotation of each qubit dependent on the state of those coupled to it.

All in all, the list of proposals presented above are promising, but have either met difficulties at the synthetic stage (a), (b) or present serious issues to scalability (d,e). Proposal (c) lies in between, since it can be applied to existing compounds, but can only be scaled up to a limited extent (up to 5–10 qubits) [16]. As discussed in detail in section 7, the real way to scale up a molecular spin quantum processor is by placing one or a few molecules (such as a short register of kind (c)) within superconducting resonators [227] and achieving the strong coupling between an individual molecule and a single photon. Recent progresses in this direction, combining top-down and bottom-up approaches in the design of both molecules and circuits [228], make this ambitious target reachable in the near future. Along these lines, schemes to implement two-qubit gates between a pair of molecules, whose interaction is mediated by the virtual or real exchange of photons (figure 3(f)), have been already put forward and are described in section 7.

Besides the design of peculiar QIP schemes, the capability to combine magnetic ions into complex structures brings along the most important tool of MNMs for quantum computation: a multi-level spectrum where both coherence and matrix elements can be engineered to a large extent. This allows one to move from a binary to a qudit logic, thus increasing the computational power of the architecture. The qudit approach expands the tools of quantum logic compared to qubit-based platforms by introducing a larger range of quantum states and computational opportunities. Qudits represent quantum information using systems with d > 2 levels. While qubits can exist in a superposition of just two states, a qudit can exist in a superposition of up to d states. This larger state space allows for the representation of more information in a single quantum unit and can provide more efficient execution of some specific applications, such as quantum simulation of field theories [229], nuclear [230], fermionic and bosonic models [231] and especially QEC (discussed in sections 5 and 6). Qudits also allow for the encoding of multiple qubits within a single unit. This can lead to more efficient quantum simulation algorithms, where part of the interaction terms, usually implemented by two-qubit gates, is represented as simpler single-qudit operations. For other quantum algorithms, the use of qudits yields a logarithmic advantage in the number of logical units along with more efficient decompositions [232, 233], which are not impressive for a perspective fault-tolerant machine but can be important in the NISQ era. There are then approaches to quantum computing besides the quantum circuit model, such as the one-way (measurement-based) and adiabatic paradigms, which might also benefit from qudit encoding [232]. Qudits can also be advantageous for quantum communication protocols, such as quantum key distribution [234] and for quantum sensing applications [235]. Working with qudits also presents challenges, including a possible increase of vulnerability to noise and decoherence, due to the larger computational space. Qubit-based architectures have been more extensively developed and researched, and transitioning to qudit-based ones requires addressing these challenges and developing new tailored techniques and algorithms. We show below that the specific features of MNMs can be exploited to address these issues.

The first proposal for exploiting a MNM as a qudit dates back to 2001 [9], when Grover's search algorithm was adapted to the 21 energy levels of the ground S = 10 multiplet of Fe₈ or Mn₁₂ SMM (figure 1(a)), split by a nearly axial anisotropy and Zeeman interaction. A scheme to implement the algorithm exploiting multi-frequency EPR pulses was designed, based on complex multi-photon processes occurring at high-order perturbation theory. Later on, a simpler sequence of radiofrequency pulses with proper amplitudes, phases and frequencies was proposed using a generalized rotating frame representation for smaller nuclear spin qudits [236] with nuclear spin I ranging from $3/2$ to $9/2$. This scheme was then experimentally applied to three of the four levels of a TbPc₂ nuclear $I = 3/2$ qudit [44] in a single-molecule transistor setup (figure 4(a)). A first pulse sequence was employed to prepare the system in a superposition of the qudit states with equal weights, implementing in fact a qudit-Hadamard gate H_d . A second pulse sequence is subsequently used to enhance the amplitude of the searched state, by setting the duration of the pulses to obtain the proper time-evolution in the generalized rotating frame, as shown in figure 4(a).

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Nuclear spins $I\gt1/2$ represent indeed the simplest realization of a qudit [125, 126, 166, 237]. These can be controlled by resonant radiofrequency pulses in a broadband NMR setup [125, 126] either at intermediate fields, where the electronic and nuclear spin wavefunctions are practically factorized, or at low field, where electronic and nuclear spins are entangled and behave as a unique $(2S+1)(2I+1)$-level electro-nuclear qudit [166]. The capability to implement universal qudit unitaries in times significantly shorter than T₂ on an entangled electro-nuclear qudit was demonstrated in [166] on a ⁵¹VO(TPP) molecule, consisting of a nuclear spin 7/2 coupled to an electronic spin 1/2 and thus providing a 16-level qudit (equivalent to 4 qubits). The proof uses the same method introduced in section 4.2, by showing that any pair of logical states could be connected by an arbitrary gate before losing coherence [219, 238]. An example of Rabi oscillations between a pair of $\Delta m_I = \pm 1$ levels is shown in figure 4(b) for ¹⁷³Yb(trensal) $I = 5/2$ qudit, coupled by a strong hyperfine interaction to an effective electronic qubit [125]. Such a strong coupling enables nuclear spin manipulations (dependent on the electronic spin state) 3 orders of magnitude faster than for a bare nuclear spin with A = 0, even in presence of a moderate magnetic field ($B\sim 0.3$ T) which yields a small electro-nuclear mixing [125].

Similarly, single $S\gt1/2$ compounds such as transition metal $S = 3/2$ or $S = 5/2$ complexes [180] or giant-spin systems originating from the interaction of multiple spins can be used to encode a qudit. The simplest existing examples are transition metal single-ion complexes [137] or Gd³⁺ compounds [196] (figure 4(c)), for which Rabi oscillations between neighboring $\Delta m_S = \pm 1$ levels were already achieved experimentally by resonant microwave pulses. The 8 levels of the $S = 7/2$ Gd qudit were exploited in [196] to encode the state of three qubits. Within such an encoding, a single resonant pulse addressing the $| 110 \rangle\leftrightarrow | 111 \rangle$ transition can be used to implement the important three qubit Toffoli gate.

In order to reach even larger Hilbert spaces, the idea of exploiting internal spin states can be combined with the ability of coupling several magnetic centers within the same molecular or supramolecular entity that was described in section 4.2. Examples are an asymmetric Gd₂ dimer, with up to 64 electronic spin levels [220] and Tb₂Pc₃ molecules, hosting two mutually coupled nuclear spin qudits [239]. Yet, these experiments also illustrate some inherent limitations to scaling up quantum resources within each molecule. Increasing the qudit dimension eventually introduces a spin level crowding and unwanted degeneracies between spin transitions that limit manipulation and readout to a subset of all spin states.

A further drawback of using a giant-spin as a qudit is the increase of the effect of decoherence with the system size. Indeed, as detailed in section 5, coherence between pairs of $| m \rangle, | m^{\prime} \rangle$ states decays with a rate $(m-m^{\prime})^2/T_2$, thus making superpositions of several $| m \rangle$ states particularly error-prone. To overcome this limitation, multi-spin clusters with anti-ferromagnetic competing interactions have been recently proposed [157]. Remarkably, these systems were shown to be protected from decoherence and in particular from its increase with the number of levels used for the qudit encoding (see section 5). A hypothetical molecule belonging to this class of systems is shown in figure 4(d), with a bi-pyramidal structure analogous to that of the existing Ni₇ cluster [93]. Besides protection from decoherence, the system Hamiltonian was engineered to increase the connectivity between the energy levels (by slight symmetry distortions and/or anisotropic terms in the spin Hamiltonian), thus enabling efficient decomposition of single-qudit unitaries such as the generalized Hadamard (H_d ) gate. For instance, it is possible to allow transitions between each of the energy levels belonging to the computational subspace (black lines in figure 4(d)) and an auxiliary state (red). Both suppression of decoherence and increase of inter-level connectivity can be obtained by satisfying general requirements on the hierarchy of the spin–spin couplings in the Hamiltonian, thus making the synthesis of these optimal systems within the capabilities of coordination Chemistry. The error reduction compared to a giant spin molecule with the same number of levels in the implementation of the paradigmatic H_d gate (corresponding to the quantum Fourier transform on a qudit) is shown in the right part of figure 4(d). The final gain (ratio between the two errors) increases with the system size and reaches values of ${\sim} 50$ for an 8-level qudit. We note that here leakage effects are neglected. Nevertheless, these are strongly system specific and can be largely limited by using quantum optimal control techniques, as demonstrated in [127] on a Gd 8-level qudit.

The most straightforward approach is to translate the block-encoding onto a molecule containing several weakly-interacting spins 1/2 (each representing a physical qubit), in the spirit of pioneering works on NMR QC [249–252]. A first proposal to implement the three-qubit code was put forward in [253], based on using ¹⁵⁹Tb³⁺ or ⁶³Cu²⁺ trimers. However, a crucial point is that a QEC code must target the leading errors affecting the physical hardware and this must be reflected by a proper correspondence between the errors which occur on the hardware and those managed by the code (this is obviously important also for the qudit approach). For instance, the simplest (three-qubit) block-code is designed to correct independent errors on any of the physical qubits, while correlated errors are not managed. Therefore, its implementation requires a platform consisting of weakly interacting spins 1/2, such that two-qubit errors on the real hardware are very unlikely.

Moreover, in the specific case of MNMs, the leading error at low temperatures is represented by pure dephasing, induced by the interaction between the central spins and the surrounding nuclear spin bath. Hence, on our molecular hardware we must focus on encodings designed to fight pure dephasing, such as the phase-flip version of the three qubit code. Although it is not a full quantum code (it only corrects a specific biased noise), it is tailored to protect from the most important error in this class of systems. Protection is achieved by introducing the code words $| {+}{+}{+} \rangle$ and $| {-}{-}{-} \rangle$ [with $| \pm \rangle = (| 0 \rangle\pm | 1 \rangle)/\sqrt{2}$] for the logical states $| 0_L \rangle$ and $| 1_L \rangle$, respectively. A phase flip error on any of the three qubits (i.e. a Kraus operator $\propto \sigma_{k,z}$) induces the transformation $| + \rangle \leftrightarrow | - \rangle$, thus bringing $| \psi_L \rangle$ to an orthogonal (erred) state which can be detected by the code and corrected by a final Toffoli (ccNOT) gate on the target spin.

Realization of this code on a properly synthesized rare-earth trimer of weakly-interacting effective spins 1/2 (Er-Ce-Er) was proposed in [133]. The molecule was shown to be promising for such an implementation (see figure 6) thanks to the small coupling between different spins combined with largely different and anisotropic g tensors. As a consequence, the system eigenstates are practically factorized and represent a suitable basis for the code.

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The sequence of operations is outlined in figure 6(a), with the central (Ce) ion representing the logical qubit in which information is initially stored and the two lateral Er ions acting as ancillae. Gates were applied to the well-characterized Er-Ce-Er system (reported in panel (b)), by simulating the time evolution of the system under the sequence of resonant pulses implementing them. An example of these microwave pulses is shown in figure 6(c). Despite the code is designed to correct only errors occurring during the memory time τ (in which the system evolves freely), simulations were performed by including pure dephasing during the whole sequence. This yields an uncorrectable error which limits the performance of the code and that makes it rather strongly dependent on T₂. Nevertheless, the break-even point compared to an uncorrected spin 1/2 with the same T₂ was already crossed for $T_2 = 0.5 \, \mu$s.

Hereafter, we consider the alternative qudit encoding. This formal approach (in which a single qudit encodes a qubit protected from a given error set) was first proposed on abstract [254, 255] or bosonic [256, 257] multi-level systems. It is particularly appropriate for MNMs, because it allows one to combine the design of targeted codes with that of the hardware Hamiltonian, in order to maximize protection of the system from unwanted errors.

A first simple application of these ideas was put forward in [125], where the ¹⁷³Yb(trensal) complex (figure 4(b)) was considered, consisting of a nuclear spin $I = 5/2$ coupled to an electronic (effective) spin 1/2. The system was characterized by broad-band NMR measurements and the nuclear spin coherence was probed by Hahn-echo and Carr–Purcell–Meibum–Gill rf-pulse sequences. In a static field of $0.2-0.3$ T, the eigenstates are practically factorized $| m_S,m_I \rangle \equiv | m_S \rangle \otimes | m_I \rangle$ states with well defined electronic and nuclear spin projections along the magnetic field $m_S = \pm 1/2$, $m_I = -5/2, \dots, 5/2$. All the electronic and nuclear spin transitions are well resolved thanks to the sizeable quadrupole and hyperfine interactions. Hence, the d = 6 (nuclear) qudit space with $m_S = -1/2$ was exploited to encode a qubit protected from a generic amplitude shift error $E_{\pm} = \sum_{m_I} | m_I \pm 1 \rangle \langle m_I |$. This brings the logical state $| \psi_L \rangle = | m_S = -1/2 \rangle \otimes \left( \alpha | m_I = -3/2 \rangle + \beta | m_I = 3/2 \rangle\right)$ into one of the orthogonal error words $| \psi_+ \rangle = | m_S = -1/2 \rangle \otimes \left( \alpha | m_I = -1/2 \rangle + \beta | m_I = 5/2 \rangle\right)$ or $| \psi_- \rangle = | m_S = -1/2 \rangle \otimes \left( \alpha | m_I = -5/2 \rangle + \beta | m_I = 1/2 \rangle\right)$, corresponding to a $\Delta m_I = \pm 1$ error. Note that these error words preserve the original superposition without overlap with $| \psi_L \rangle$, thus enabling error detection according to KLc. This is achieved by two selective microwave pulses inducing a conditional excitation of the electronic spin only for two specific states of the nuclear qudit. For instance, two pulses resonant with $| m_S = -1/2, m_I = -1/2 \rangle \rightarrow | m_S = 1/2,m_I = -1/2 \rangle$ and $| m_S = -1/2,m_I = 5/2 \rangle \rightarrow | m_S = 1/2,m_I = 5/2 \rangle$ transitions bring $| \psi_+ \rangle$ to $| m_S = 1/2 \rangle \otimes \left( \alpha | m_I = -1/2 \rangle + \beta | m_I = 5/2 \rangle\right)$. Hence, the electronic spin is flipped if $E_+$ error has happened . Therefore, the occurrence of $E_+$ can be detected by a measurements of the electronic spin only (without projecting the nuclear spin component). In a superconducting resonator architecture (see below), this could be achieved by a two-tone dispersive measurement setup [120, 258], in which a two-frequency photon drive is used to excite the resonator and the measurement is done by a broadband high-efficiency photon detector that is only sensitive to the amplitude of the output field. Finally, depending on the measurement outcome, a simple recovery procedure is applied, i.e. if the electronic spin is measured to be in $m_S = 1/2$, it is de-excited and then two radio-frequency pulses are employed to transform back $| \psi_+ \rangle$ into $| \psi_L \rangle$.

This error model constitutes a useful playground to clarify the basic principles of qudit-based QEC. However, it does not model the main errors occurring in real molecular spin systems. To this aim, specific encodings to fight pure dephasing [120, 156, 170] were designed. Two of them are shown in figure 7, targeted to different molecular systems. On the left panels (a,b), we consider a spin S (either a single ion or the total spin ground state of a molecule with ferromagnetic exchange interactions), while on the right (panels c,d) a set of spins coupled by competing anti-ferromagnetic interactions. In both cases, dephasing of the spin system at low temperature originates from the coupling between the system spins and the surrounding nuclear spin bath. Conversely, phonon-mediated relaxation and dephasing processes can be neglected at low temperature, as long as the energy gaps are sufficiently smaller than the Debye energy (${\gtrsim} 5$ meV) [159, 171, 172]. Then, the incoherent dynamics of the spin systems can be modeled by a Lindbald master equation (see appendix D). For a spin S ion subject to an axial spin Hamiltonian and characterised by $| m \rangle$ eigenstates ($S_z | m \rangle = m | m \rangle$), this takes the form

$$\begin{equation} \dot{\rho}\left(t\right) = \frac{1}{T_2} \left( 2 S_z \rho\left(t\right) S_z - \left\{S_z^2,\rho\left(t\right)\right\} \right), \end{equation} \tag{ 10 }$$

where $\rho(t)$ is the density operator of the qudit. By performing a perturbative expansion for small $t/T_2$ of the solution of equation (10) $\rho(t) = \sum_k E_k \rho(0) E_k^\dagger$, one can write an analytical expression for the error operators [170]:

$$\begin{equation} E_k = \sqrt{\frac{\left(2t/T_2\right)^k}{k!}} e^{-S_z^2 t/T_2} S_z^k. \end{equation} \tag{ 11 }$$

Starting from this expansion for the error-operators, it is then possible to find code words satisfying KLc

$$\begin{align} | \mu_L \rangle = \frac{1}{\sqrt{2^{2S-1}}} \sum_{k = 0}^{S-1/2} \sqrt{\binom{2S}{2k+1-\mu} } | 2k+1-\mu -S \rangle, \end{align} \tag{ 12 }$$

with $\mu = 0,1$ and half-integer S. We note that $d = 2S+1$ levels are enough to correct $d/2$ errors E_k , i.e. dephasing up to order $(t/T_2)^{d/2}$ and the corresponding powers of S_z . The structure of these spin-binomial code words is reported in figure 7(a) as a bar plot. The smallest qudit dimension d enabling a minimal correction is d = 4, corresponding to a $S = 3/2$ system. In that case the code words are given by $| 0_L \rangle = (\sqrt{3} | -1/2 \rangle + | 3/2 \rangle)/2$, $| 1_L \rangle = (\sqrt{3} | 1/2 \rangle + | -3/2 \rangle)/2$ and the lowest order S_z error brings them to the orthogonal error words $| e_0 \rangle = (- | -1/2 \rangle + \sqrt{3} | 3/2 \rangle)/2$, $| e_1 \rangle = ( | 1/2 \rangle - \sqrt{3} | -3/2 \rangle)/2$, which are therefore clearly distinguishable and make the S_z error detectable by the code. In general, the correcting power can be quantified by considering the error $\mathcal{E}(| \psi_L \rangle,\rho)$ on an initial state $| \psi_L \rangle = (| 0_L \rangle+| 1_L \rangle)/\sqrt{2}$ subject only to pure dephasing for a memory time t and then to a cycle of error correction, ending up in the mixed state ρ. Results are plotted in figure 7(b) for spins up to 9/2 (d = 10) and evidence an increased correcting power by increasing S and hence the number of levels in the encoding.

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Many different ways can be identified to increase the number of levels in a single molecular spin qudit by combining spins via synthetic Chemistry methods. As mentioned in section 4.4, multi-spin clusters with a proper hierarchy of spin–spin interactions can yield to particularly advantageous choices. We illustrate this below by deriving the key ingredients driving decoherence in a molecule containing several interacting spins s_j . Within the same approximations used for (10), the Lindblad equation becomes:

$$\begin{equation} \dot{\rho}\left(t\right) = \sum_{\mu \nu} \gamma_{\mu \nu} \langle \mu | \rho\left(t\right) | \nu \rangle | \mu \rangle\langle \nu |, \end{equation} \tag{ 13 }$$

with $| \mu \rangle, | \nu \rangle$ the eigenstates of the system. This leads to a decay of the coherences $\rho_{\mu\nu}$ with a rate

$$\begin{align} \gamma_{\mu\nu} & = \sum_{j j^{\prime}} \zeta_{jj^{\prime}}^{zz} \left[ - 2 \langle \mu | s_j^z | \mu \rangle \langle \nu | s_{j^{\prime}}^z | \nu \rangle \right.\nonumber\\ &\quad\left.+ \langle \mu | s_j^z | \mu \rangle \langle \mu | s_{j^{\prime}}^z | \mu \rangle + \langle \nu | s_j^z | \nu \rangle \langle \nu | s_{j^{\prime}}^z | \nu \rangle \right]. \end{align} \tag{ 14 }$$

where $s^z_j$ are local spin operators, we are assuming for simplicity axial symmetry, and coefficients $\zeta_{jj^{\prime}}^{zz}$ contain the dipolar couplings between the spins of the system and nuclear spin bath operators [156]. Starting from this expression, we can now design strategies to improve the capabilities of the code. In the case of a spin S, $| \mu \rangle\equiv | m \rangle$, $s_j^z$ can be summed to give S_z and equation (13) reduces to (10), with $\gamma_{m,m^{\prime}} = (m-m^{\prime})^2/T_2$. Hence, decoherence is much faster for superpositions of states with large $|m-m^{\prime}|$, enhancing its impact as S increases. This limits the performance of the spin-binomial code words for large S, as shown in figure 7(b).

In order to suppress decoherence and its increase with the qudit size, we focus on the expression for $\gamma_{\mu\nu}$. We note that, apart from the system specific coefficients $\zeta_{jj^{\prime}}^{zz}$ depending on the nuclear spin bath and on its coupling to the system, decoherence is ruled by differences of expectation values of $s^z_j$ on different eigenstates. To enhance the capabilities of the code, one needs $\langle \mu | s_j^z | \mu \rangle$ and its variation between different eigenstates $| \mu \rangle$ to be as small as possible. As already stated in section 4.4, a general class of molecules showing this property is represented by half-integer spins coupled by antiferromagnetic competing interactions, giving rise to several total spin $1/2$ multiplets in the lowest part of the spectrum. As demonstrated in [156] for the illustrative hypothetical system sketched in the inset of figure 7(c), these low energy multiplets can be used to numerically find optimized code words involving an increasing number of levels. Compared to an analogous system with ferromagnetic couplings (characterized by a ground $S = 9/2$ multiplet) this molecular qudit shows an exceedingly better performance, with a 5-order of magnitude reduction of the final error, as shown in figure 7(d) and in the inset of panel (b). From that it also emerges that the error suppression is almost exponential in the qudit size d and rather small values of d already bring errors below 10⁻¹⁰. This result certifies the impact that the qudit approach to QEC, combined with molecular design, could have on quantum computing.

The Lindbald description of the bath dynamics provided so far is useful to highlight the essential ingredients of the system spin Hamiltonian which rule decoherence, but it also involves several approximations. It neglects, in particular, any memory effect in the bath. A more realistic (although much more computationally demanding) description of the many-body bath dynamics can be obtained by means of the cluster-correlation expansion, as done in [120] to describe decoherence of a single spin S ion in a nuclear spin bath. With a real molecule at hand (i.e. knowing the position of the nuclear spins), this technique allows for a precise quantitative prediction of the coherence decay of the spin system (usually not exponential as given by the Lindblad approximation) and then for the derivation of code words tailored for that specific noise. Remarkably, apart from some quantitative changes, conclusions for QEC on a spin S ion are not significantly modified by this more accurate description of the bath [120].

Several simple existing molecular systems can already embed QEC within the scheme outlined in this section. They consist of a nuclear spin I > 1 coupled by hyperfine interaction to an electronic spin 1/2 used as ancilla, along the lines described above. Besides Yb(trensal) [125], it is worth mentioning V-based compounds (VO(TPP) and VCp₂Cl₂) hosting a nuclear spin 7/2 together with an electronic spin with remarkable coherence. These systems have been characterized by combined NMR [126, 259] and EPR studies [259] both in terms of spin Hamiltonian parameters and of electronic and nuclear coherence times. As already noted in section 4.1, the sizeable hyperfine interaction (yielding well distinguished gaps and fast manipulations of the qudit), combined with the remarkable coherence times, make VO(TPP) a promising nuclear qudit [126], where transitions between specific pairs of levels can be driven by resonant radio-frequency pulses (see figure 8).

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Examples of electronic spin systems promising for implementing this code are given by Gd³⁺ (electronic spin $S = 7/2$) monomers [196] or dimers [220] or by transition metal dimers, as proposed, e.g. in [169]. Another possibility, offered by the flexibility of chemistry, is to improve the performance of a processing molecular qubit by adding a magnetic ion with a nuclear spin qudit acting as a quantum memory with QEC. This was proposed in [154] on a Cr₇Ni-Cu compound, where ⁶³Cu ion provides a nuclear spin $3/2$ coupled to an electronic ancilla, used both for error detection and for swapping information to the Cr₇Ni processor by virtual excitations.

The scheme proposed above is for protecting a quantum memory from its most important noise source. This represents the first step in the QEC framework, but it does not take into account errors occurring during the detection steps or during logical operations between encoded states.

In addition, thus far we have considered a code correcting only a biased noise acting on the system (pure dephasing) when this is left idle. However, this error can be transformed into other not correctable ones by unitary operations such as the detection/correction step or logical gates between encoded states. For instance, the result of decoherence acting during a correction procedure is a plateau in the error $\mathcal{E}$ that would appear at short $t/T_2$ in figures 7(b) and (d), thus limiting the corrective power of the code [156, 170]. This effect can be reduced by engineering the pulses in order to reduce their duration without increasing leakage to neighboring levels (an effect usually induced by fast, spectroscopically broad, pulses). For instance, one could employ derivative removal by adiabatic gate (DRAG) methods as in [169, 260, 261] or optimal quantum control techniques [127], which, in turn, also suppress possible leakage during manipulations. However, a residual relevant not-correctable error will always be present unless one designs bias-preserving schemes for error correction and implementation of logical gates. This was proposed in other qudit-based architectures [262, 263] and recently also on a molecule-based setup [264, 265]. In [265] a general Fault-Tolerant scheme against pure dephasing errors is introduced, which allows for the implementation of error correction and of a universal set of quantum gates without error propagation. As a result, the QEC advantage increases almost exponentially with the qudit size (as in figure 7(d)), even during the implementation of stabilizers for error-correction and of a universal set of logical gates.

A more demanding task which still requires work on the theoretical side is the design of a fully quantum QEC code, able to correct a wide class of errors on the qudit and not only a specific one. Often, proposals of fully quantum codes [255] are more general and mathematically elegant, but less targeted to the investigated physical system. Hence, they could fail in providing an effective protection in case a single source of noise dominates. The challenge in this direction is to design a code which is both able to correct a generic error but also tailored to the typical noise occurring on a molecular hardware.

As already mentioned in section 4.3, simplified versions of the architecture described above were proposed in the following years. In particular, two families of Cr₇Ni dimers with an interposed spin 1/2 switch (see figure 3(c)), and Cr₇Ni rings arranged in either a parallel or slightly tilted configuration, were investigated in [211]. The former restricts the classes of target Hamiltonians which can be simulated to those invariant by permutation of the two-qubits, while the latter (see figure 9(a)) would allow one to distinguish them and hence would perfectly fit the AB scheme described above. Both systems were used to numerically simulate the transverse field Ising model [$\mathcal{H}_{TIM} = \lambda s^z_1 s^z_2 + b (s_1^x + s_2^x)$] with 10 Trotter steps, corresponding to the subsequent implementation of 10 two-qubit and 20 single-qubit operators. This example constitutes a feasible proof-of-principle experiment, where the final output could be obtained by measuring the magnetization of the molecular crystal after the pulse sequence implementing the simulation. This lasts about 350 ns, a time which is ~ one order of magnitude shorter than typical coherence times of deuterated Cr₇Ni rings [118]. This makes decoherence practically ineffective on the simulated magnetization, which reproduces very well the expected behavior, as shown in figure 9(b).

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An alternative two-qubit elementary unit of a molecular quantum simulator is shown in figure 9(c). As in the scheme of figure 3(d), the switch between a pair of Cr₇Ni rings is here provided by a redox active unit (a Ru₂Co triangle) which in the oxidized state is diamagnetic and can be reversibly reduced into an effective $s = 1/2$ [212]. As illustrated in section 4.2, this scheme allows one to implement an iSWAP$^\alpha$ gate, corresponding to the time evolution induced by an interaction term $\mathcal{H}_{XY} = \lambda (s^x_1s^x_2 +s_1^y s_2^y)$ between the two qubits. This interaction can be easily transformed into analogous $\mathcal{H}_{XZ}$ or $\mathcal{H}_{YZ}$ by combining the time evolution $U_{XY}(t) = e^{-i \mathcal{H}_{XY} t}$ with suitable $R_{x,y}(\pm \pi/2)$ gates on both qubits. In particular, one can easily show that $U_{XZ} = R_{x}(\pi/2) U_{XY} R_{x}(-\pi/2)$ and $U_{Z Y} = R_{y}(\pi/2) U_{XY} R_{y}(-\pi/2)$. Then, since $[\mathcal{H}_{XZ},\mathcal{H}_{XY}] = [\mathcal{H}_{XZ},\mathcal{H}_{YZ}] = [\mathcal{H}_{YZ},\mathcal{H}_{XY}] = 0$, the simulation of the Heisenberg coupling can be obtained by the product $U_{XY}(\lambda t/2)U_{XZ}(\lambda t/2)U_{YZ}(\lambda t/2)$. Numerical simulations reported in figure 9(e) with real system parameters (including also the measured values of T₂) show a very good agreement with the exact time evolution, thus making also the system of figure 9(d) promising for proof-of-principle experiments on a molecular crystal.

Nuclear spin-based quantum simulators can also be envisaged, where hyperfine-coupled electronic spins mediate the effective interaction between a pair of nuclear spins. To this end, a promising compound was synthesized [132], consisting of a symmetric VO dimer where the two electronic spins are coupled by a dipole-dipole interaction (see figure 9(e)). Qubits are encoded into the lowest energy states ($m_I = 7/2, 5/2$) of the two $I = 7/2$ nuclei. Along the same lines of the scheme described in figure 3(c), here the switch of the interaction between the nuclei is provided by the electronic spin dimer. Indeed, the effective qubit–qubit coupling can be turned on by a fast 2π microwave pulse dependent on the state of the two nuclear spins via the hyperfine interaction, resonant with the electronic spin transition $| S = 1,M = -1 \rangle \leftrightarrow | S = 1,M = 0 \rangle$. The use of nuclear instead of electronic spins for the encoding protects information from decoherence. In parallel, two-qubit gates are implemented much faster than in standard NMR quantum computing approaches, because the effective coupling is mediated by electronic spin excitations. The VO dimer [132] was proposed as a quantum simulator of the tunneling of the magnetization in a spin 1 system, induced in an axial spin Hamiltonian by a rhombic term $E S_{x}^2$. Results are reported in figure 9(e), in very good agreement with the exact behavior (continuous line), with the inclusion of the measured $T_2 = 1 \,\mu$s.

Another interesting application connected with quantum simulations is the adiabatic quantum algorithm proposed in [268]. The scheme was applied to a phthalocyanine molecule containing three electron spins on NO radicals and to a trans-glutaconic acid radical molecule [269] embedding one electron spin bus qubit and two nuclear client qubits [201]. The algorithm solves the factorization problem using fewer qubits compared to Shor's and is based on defining an adiabatic path from an initial hardware Hamiltonian to a target one whose ground state encodes the solution of the optimization problem [268]. The unitary evolution bringing the system to such a ground state is obtained in [269] by a Suzuki–Trotter decomposition of the target Hamiltonian evolution into a sequence of required microwave/radiofrequency pulses.

The larger Hilbert space of qudits can provide additional resources for quantum simulations, enabling to address problems involving quantum objects with many degrees of freedom, like nuclear or bosonic Hamiltonians [230, 231]. Indeed, despite its utmost importance for applications ranging from solid-state to high-energy Physics, the quantum simulation of models including fermionic or bosonic fields is usually inefficient on a qubit register. Indeed, on the one hand JW mapping leads in general to non-local many-body interactions in the resulting spin Hamiltonian. On the other hand, standard techniques to encode a bosonic mode into qubits either employ an exponentially large Hilbert space [270] (with a number of qubits equal to the number of boson states) or require non-local interactions between distant qubits in the array [271, 272].

Recently, a working proof-of-concept quantum simulator, based on an ensemble of nuclear ¹⁷³Yb(trensal) qudits manipulated with multi-frequency radiofrequency pulses, has been realized [17]. The operativity of the simulator is demonstrated by implementing the quantum simulation of two representative problems, reproducing the correct physical behavior in both cases: an integer spin S subject to quantum tunneling of the magnetization and the transverse-field Ising model.

In general, molecular spin qudits can be used to greatly simplify the QS of multi-level problems. For instance, their multi-level structure can encode a boson mode, with the qudit levels representing the number of boson states (truncated to the number of qudit levels), as done in [231]. This scheme greatly reduces both the complexity of operations and the number of physical computational units, compared to the aforementioned qubit implementations. In particular, the Authors considered the so-called Rabi model, describing the atom-photon interaction via the Hamiltonian $\mathcal{H}_{Rabi} = \omega_a \sigma_z + \Omega a^\dagger a + 2G \sigma_x (a + a^\dagger)$, with $a^\dagger$(a) bosonic creation (annihilation) operators, satisfying $[a,a^\dagger] = 1$. It was shown how to simulate $\mathcal{H}_{Rabi}$, even in the ultra-strong coupling regime ($G/\Omega \gtrsim 1/4$), by employing a spin S qudit modeling the photon, coupled to a spin 1/2, describing the two-level atom.

An example assuming a simple spin $S = 3/2$ qudit is reported in figure 10. Green (black) arrows indicate transitions (induced by suitable microwave pulses) between qubit (qudit) states. The coupling between the two spins makes the excitation of each of the two units dependent on the state of the other, thus enabling full control of the system. This was exploited to compute first the ground state energy (by the variational quantum eigensolver algorithm [273], as shown in figures 10(b) and (c)) and then the time evolution of observables on the target system such as average photon number or atom excitation in a wide range of $G/\Omega$. Simulation of the hardware dynamics under the sequence of microwave pulses implementing the proposed procedures and using parameters of existing molecular systems (also including decoherence) demonstrate the effectiveness of the proposed approach. It is worth noting that this qudit-based approach could be exploited to simulate a wide range of models, including gauge fields, spin-phonon couplings and fermionic systems.

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Although already envisioned by Lloyd [274], the quantum simulation of open systems has so far been much less explored. Nevertheless, it represents a very important objective. Indeed, it would allow one to understand complex behaviors related to relaxation or decoherence, which are at the basis of fundamental phenomena such as photosynthetic processes or transport and which must be necessarily kept under control in the design of novel quantum technologies. Unfortunately, these incoherent processes arise from the interaction of a relatively small subset of system qubits ($\mathcal{S}$) with a very large number of environment ($\mathbb{E}$) degrees of freedom. Although the overall $\mathcal{S}+\mathbb{E}$ dynamics is still governed by a Schrödinger equation and hence could in principle be simulated as described in the previous section, the need to include a huge collection of qubits to model the bath makes this approach unfeasible.

Therefore, one needs to focus on the dissipative dynamics of the reduced system density matrix, described by a completely-positive trace-preserving map $\varepsilon (\rho) = \sum_k^D E_k \rho E_k^\dagger$ with suitable Kraus operators E_k , and develop proper techniques to simulate it on a quantum hardware. A common approach [275] is based on implementing a unitary dynamics on an extended register (including system qubits and log₂ D additional environment ones) which generates $\mathcal{S}$-$\mathbb{E}$ entanglement, and then re-initializing $\mathbb{E}$ qubits through local measurements [276, 277]. This last irreversible step makes the resulting dynamics on $\mathcal{S}$ non-unitary. This general approach allows one to perform a quantum simulation of both coherent and incoherent dynamics by implementing a series of dynamical maps [278] or (equivalently) by performing a digital Suzuki–Trotter decomposition of the Liouvillian operator [277]. However, due to the need to reset a subset of qubits during the simulation of each incoherent step (usually done via projective measurements), it would require to address individual-molecules (see section 8), at difference from the proposals discussed above, which could instead also be implemented on molecular crystals. A possibility in this direction is offered by a novel three-qubit supra-molecule [279], where a Cr₇Ni is interposed between a Cu²⁺ porphyrin and a nitroxide. In this case, the very different relaxation times T₁ between the constituents molecular spins (24 µs for Cr₇Ni, tens to a hundred ms for the external spins) enabled to propose the system as a simulator of the incoherent dynamics of the two external qubits, with the central Cr₇Ni mimicking the environment ($\mathbb{E}$). After generating an entangled $\mathcal{S}$-$\mathbb{E}$ state by selective resonant pulses, the $\mathbb{E}$ qubit is re-initialized by waiting for its relatively fast relaxation, thus obtaining the reduced incoherent dynamics of $\mathcal{S}$. To be more specific, relaxation of the $\mathbb{E}$ qubit must be faster then the $\mathcal{S}$ qubits coherence but slower than $\mathcal{S}$-$\mathbb{E}$ manipulations.

Alternatively, one can extract the final reduced $\mathcal{S}$ density matrix after simulation of the incoherent dynamics by full state tomography of only $\mathcal{S}$ qubits, thus in fact summing over all possible $\mathbb{E}$ states, as proposed in the context of NMR quantum computing [275]. This latter method can be implemented on an ensemble of molecules and hence it is well suited to the molecular crystals described above, without requiring single-molecule addressing. A scheme for the quantum simulation of decoherence following this line was conceived, based on the five-qubit supra-molecule synthesized in [155] and shown in figure 11(a). The molecule consists of a chain of alternating Cu²⁺ and Cr₇Ni qubits with tailored weak and strong interactions. In particular, the larger coupling identifies a core of two $\mathcal{S}$ (Cr₇Ni) qubits (S₁ and S₂ in figure 11(a)), with a switchable interaction provided by the central Cu²⁺ (M). The smaller one is exploited to simulate the weak coupling of the system with the environment, represented by the two external Cu²⁺ (E₁, E₂). As a prototypical test case, the Authors considered the simulation of decoherence acting on an entangled Bell pair, a typical resource for quantum teleportation experiments. Preparation of the Bell pair is achieved by combining a two-qubit controlled-Z gate (achieved by an excitation of the central M switch dependent on the state of both S₁ and S₂) with symmetric rotations of S₁ and S₂. Then, pure dephasing is simulated by implementing an evolution of E$_{1,2}$ depending on the state of S$_{1,2}$, consisting of a rotation of the environment qubits followed by a cNOT gate between each S_i -E_i pair. The simulation time (or the $t/T_2$ ratio) is controlled by the rotation angle. Results of the simulated error (arising from breaking entanglement via decoherence) in a quantum teleportation experiment, along with the sequence of required pulses, are reported in figures 11(b) and (c), showing a very good agreement with the expected behavior, even with the inclusion of experimentally measured coherence times of the individual sub-units.

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This technology takes the classic light–matter interaction problem to a chip, in which artificial 'atoms', typically solid state qubits, interact with radiation modes of superconducting resonant circuits [3, 280, 281]. The underlying Physics can be described with the Jaynes–Cummings Hamiltonian

$$\begin{equation} H_\mathrm{JC} = \hbar \omega_{\mathrm{r}} \, a^{\dagger} a + \frac{\hbar \Omega} {2} \sigma_{z} + \hbar G \left( a^{\dagger} \sigma^{-} + \sigma^{+} a \right) \end{equation} \tag{ 16 }$$

that includes a term for the isolated resonator, with $a^\dagger$ and a bosonic creation and annihilation operators and $\omega_{\mathrm{r}}$ the frequency of the mode, a second one describing the isolated the qubit, where Ω is the transition frequency, plus a mutual interaction term. Here, G is the single qubit to single photon coupling strength. This scheme allows achieving coupling regimes that are beyond those attainable with 'real' atoms in three dimensional cavities. When G exceeds the decoherence rates of the qubit γ and of the superconducting cavity κ and the two subsystems are tuned to each other ($\Omega \simeq \omega_{\mathrm{r}}$), excitations over the ground state become hybridized light-matter bonding and antibonding states, whose energies differ by 2G. Besides, this strong coupling provides a tool to interface with the qubits. In particular, when the qubit and the resonator are detuned, i.e. $\vert \Omega - \omega_{\mathrm{r}} \vert \gg G$, the resonance frequency depends on the state of the qubit, which allows performing quantum non-demolition measurements of the latter. In the same dispersive regime, the coupling of two qubits to the same cavity mode introduces an effective interaction between them, which can be exploited to generate two qubit gates.

The ability to dispersively read out and coherently communicate qubits by means of such a circuit QED scheme was first shown by experiments performed on superconducting qubits [282–284]. More recently, it has also been implemented successfully with semiconducting quantum dots [285, 286]. In both cases, the qubit-photon coupling is driven by the photon electric field, either directly as in the case of superconducting charge qubits or mediated via an effective spin-orbit interaction in the case of spin qubits in semiconductors. This interaction affords reaching strong coupling to individual qubits.

The situation is far more challenging in the case of 'real' spins, such as those present in magnetic molecules, which interact with the cavity magnetic field. The generalization of Jaynes–Cumming model (16) to this situation gives [287, 288]

$$\begin{equation} H_{JC} = \hbar \omega_{\mathrm{r}} \, a^{\dagger} a + H_{S} + 2\left( a^{\dagger} + a \right) \mu_{\mathrm{B}} \textbf{b} \cdot {\textbf{g}} \cdot \textbf{S} \end{equation} \tag{ 17 }$$

where the second term is the Hamiltonian of the spin qudit, the third accounts for its interaction via the spectroscopic tensor g with the resonator and b is the root mean squared (rms) value of the magnetic field generated by one of its photon modes. The latter depends on the rms value of the superconducting current in the resonator, which equals $i_{\mathrm{rms}} = \left( \hbar \omega_{\mathrm{r}}/ 2 L \right)^{1/2}$, where L is the effective resonator inductance. The role of G is now played by the transition matrix element [287]

$$\begin{equation} \hbar G = g \mu_{\mathrm{B}} \vert \langle 0 \vert \textbf{b} \cdot \textbf{S} \vert 1 \rangle \vert \end{equation} \tag{ 18 }$$

where $\vert 0 \rangle$ and $\vert 1 \rangle$ are the two logical qubit states (or the two lowest energy states of a qudit). Typical values of G for standard coplanar resonators are of the order of a few Hz, thus much smaller than the decoherence rates of molecular spin qubits $\gamma = 1/T_2 \unicode{x2A7E} 1$ kHz and even smaller than the photon loss rate $\kappa = \omega_r/2\pi Q$. The latter can reach values as low as 5–10 kHz for $\omega_r/2\pi = 5$–10 GHz and resonator quality factors of 10⁶, achievable with state-of-the-art technology.

A way to reach stronger couplings is to work with macroscopic ensembles of N identical spins. Their collective coupling to each photon is then enhanced with respect to that of a single spin $G_{N} = \sqrt{N} G$ [289]. This trick has allowed observing strong coupling to diverse electronic spin systems, both inorganic [290–294] and molecular [18, 295, 296]. Also, high cooperativity coupling ($G^2 \gt \kappa/T_2$) to the $I = 5/2$ nuclear spins of ¹⁷³Yb-trensal molecules has been achieved (figure 12) [19]. These results lay the basis for reading out the states in crystals of molecular spins. Combined with proper methods to coherently control them, e.g. by combining resonators with broad band transmission lines, and with suitable method to refocus the spin wave function, in order to compensate for the effect of the inhomogeneous broadening, they could afford proof-of-concept implementations of quantum simulations or even of quantum computing algorithms in these molecules. Yet, these are limited to the Hilbert space of one of them (the large number in the crystal would here serve only to amplifying the readout signal). Scaling up beyond this requires coherently coupling photons to individual molecular spins.

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Equation (18) provides a hint for the design of strategies to approach this goal, either by acting on the circuit, which determines the amplitude of b, or on the spin states. The very small coupling between spins and photons reflects the large mismatch that exists between the dimensions of the molecules and that of the resonant circuits. This suggests looking for methods which fully exploit the nanoscopic dimension of molecules by confining the magnetic component of the photon modes in mode volumes closer to the molecular size, thus contributing to locally enhance b. A direct 'brute force' approach, illustrated by figure 13, is to locally reduce the width w of the resonator transmission line [287, 297]. Near such a constriction, b scales as $1/w$. Experiments performed on model free-radical spins deposited onto coplanar resonators [228] confirm that it is possible to reach G values close to 1 kHz for relatively modest photon frequencies (and energies) ${\simeq} 1.3$ GHz. They also point to the crucial role that the molecule-circuit interface plays [179], as it limits how close individual molecules can be from the superconducting currents that generate b. Coplanar resonators impose stringent limitations on the design of the resonator geometry, which determines the effective mode volume of the resonator. A promising alternative is to use lumped- element LC resonators [19, 298–300] coupled to a readout transmission line. Since, near resonance, the magnetic energy stored in these circuits is equally shared between electric and magnetic components, working with low inductance resonators, e.g. with L made of a single micro-wire, leads to an enhancement of the current generated by each photon, thus also of G [301]. This approach, combined with state-of-the-art parametric amplifiers for the readout of the transmitted signal, has led to the implementation of magnetic resonance at the level of a few tens of spins [298–300]. Needless to say, the two methods can eventually be combined, which should allow taking G close to 100 kHz for $S = 1/2$ spins located on 20 nm constrictions fabricated in minimum inductance 5 GHz resonators [21].

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One of the trademarks of the molecular approach is the ability to tune the spin properties via a careful choice of the molecular composition and structure. Therefore, besides enhancing G by designing the circuit properties, one can also attempt to improve it by molecular design of adequate spin states. From equation (18), it follows that the spin-photon matrix element can be enhanced in molecules having a spin $S \gt 1/2$. In these systems, the level structure and the wavefunctions are determined, to a large extent, by the magnetic anisotropy. For simplicity, we consider a uniaxial spin system defined by a spin Hamiltonian $H_S = D S_{z}^{2} + g \mu_{\mathrm{B}} {\textbf{B}} \cdot \textbf{S}$. The microwave photon field then couples states with spin projections m and $m \pm 1$ along the anisotropy axis z. Interestingly, the coupling strength between the lowest energy states depends on the sign of D [287]. For systems with easy axis anisotropy (D < 0, ground states $m = \pm S$) $G \propto \sqrt{2S}/2$. An even larger enhancement is expected for systems with easy- plane anisotropy (D > 0), for which $G \propto \sqrt{S(S+1)}/2$ in the case of integer S (ground state m = 0) and $G \propto \sqrt{S(S+1) + 1/4}/2$ in the case of half-integer S (ground state $m = \pm 1/2$).

Increasing S should not lead to a stronger decoherence, especially if we work on single molecules (where inhomogeneous broadening is not an issue) and we focus on transitions between $\Delta m = \pm 1$ states. To further limit decoherence, one could encode the qudit in clock transition states arising at level anticrossings [168], see section 4.1. The spin-photon coupling is then dominated by the matrix element of S_z between states $(\vert S \rangle \pm \vert -S \rangle)/\sqrt{2}$ [227, 287], which leads to $G \propto 2S$, with a further gain of a factor of two compared to an easy-plane molecular spin S. Being already protected from magnetic noise, such spin clock transitions simultaneously optimize spin coherence and the coupling to circuits [302], the two parameters that determine the strong coupling condition.

Once a strong spin-photon coupling is reached, the basic idea to scale-up a hybrid quantum processor with molecular spins integrated in a solid state device (see also [8, 21, 227]) is illustrated in figure 14. It consists of a superconducting circuit hosting multiple magnetic molecules [21]. The nodes of the quantum processor are defined by nanoscopic constrictions fabricated in the inductor line of a lumped element resonator. Spins located at these nanobridges are coherently controlled by suitably shaped local microwave pulses generated by open transmission lines. Their coupling to the resonator allows for the dispersive or resonant read out of the final states (see next section). Moreover, the resonator frequency is tuned by changing the magnetic flux through a (series of) SQUID connected in series with the inductor line, which is exploited for resonant two-qudit gates (see below) and also to better address the multiple transitions that characterize the qudits.

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But, more importantly for scaling up, the circuit can help to 'wire up' two of these basic units. The coupling of each spin to the resonator introduces an effective interaction between the two qudits, which depends on the relative tuning between their respective frequencies and that of the resonator. This dependence allows switching on and off this coupling, thus creating a sort of 'artificial molecular dimer' in a circuit. In the following, we examine two scenarios for such interaction, which correspond to the dispersive and resonant interaction regimes. They lead to distinct operation modes. While the former has been successfully applied to superconducting qubits, the latter might be more suitable to cope with the weaker coupling expected for molecular spins.

7.2.1. Photon mediated spin–spin interactions in the dispersive regime.

The coupling of multiple spins to a single resonator mode can be treated by an extension of equation (17) [288]. For illustrative purposes, we consider here the case of two spins S with uniaxial anisotropy and a geometry in which the external magnetic field points along the molecular axes z, while the cavity field is perpendicular to it. The Hamiltonian of the hybrid system is then

$$\begin{align} H_{hyb} & = \hbar \omega_{\mathrm{r}} \left( a^{\dagger} a \right) \nonumber \\ &\quad + \sum _{j = 1,2} \left[ D_j S_{j,z}^{2} + g_{z} \mu_{\mathrm{B}} B_{j,z} S_{j,z} + g_{x} \mu_{\mathrm{B}} \left( a^{\dagger} + a \right) b_{j} S_{j,x} \right], \end{align} \tag{ 19 }$$

where $a^\dagger/a$ are bosonic creation/annihilation operators, $\omega_{\mathrm{r}}$ is the frequency of the resonator mode, D_j are the ZFS parameters describing the uniaxial anisotropy of the two spins S_j and $g_{z,(x)}$ are the z(x) components of their spectroscopic splitting tensor. When $S = 1/2$, the situation is analogous to that met with other qubits. If both spins are detuned from the resonator, i.e. the dispersive regime defined above applies to both, it is possible to derive from equation (19) a Hamiltonian for the two 'undressed' spins

$$\begin{equation} H_{disp} \simeq \sum _{j = 1,2} g_{z} \mu_{\mathrm{B}} B_{j,z} S_{j,z} + J \left( S_{1}^{+} S_{2}^{-} + S_{1}^{-} S_{2}^{+} \right) \end{equation} \tag{ 20 }$$

that includes an effective spin–spin interaction term with a constant

$$\begin{equation} J \simeq \hbar \, G_{1}G_{2} \left( \frac{\omega_{\mathrm{r}}}{\Omega_{1}^{2}-\omega_{\mathrm{r}}^{2}} + \frac{\omega_{\mathrm{r}}}{\Omega_{2}^{2}-\omega_{\mathrm{r}}^{2}} \right), \end{equation} \tag{ 21 }$$

where $G_{1,2}$ are the qubit-photon coupling strengths and $\Omega_{1,2}$ are the spin transition frequencies. This effective coupling connects transitions in the two spins (e.g. $m \longrightarrow m \pm 1$ in spin 1 with $m^{^{\prime}} \longrightarrow m^{^{\prime}} \mp 1$ in spin 2). It induces a net evolution between these states provided that their frequencies are degenerate, i.e. when $\Omega_{1} = - \Omega_{2}$. This process takes place therefore at constant energy and involves no excitation or de-excitation of photon modes in the resonator. Energy conservation provides a way to switch on and off the interaction and, in arrays of N > 2 qubits, to select which two are on 'speaking terms', e.g. by tuning and de-tuning the frequencies at different sites via local magnetic fields. While the interaction is 'on', state $\vert +1/2 \rangle_{1}, \vert-1/2 \rangle_{2}$ evolves into $\vert -1/2\rangle_1 \vert+1/2 \rangle_{2}$. Therefore, it allows for implementing a generalized iSWAP or $\sqrt{i {\mathrm{SWAP}}}$ gate between the two qubits at a rate determined by $J/\hbar$.

This result can be generalized to the case of $S\gt1/2$ spin qudits, provided that the dispersive condition applies to all their transition frequencies [288]. The effective spin Hamiltonian (20) becomes then more complex and includes a state dependent coupling tensor. Different transitions $m_{1} \longrightarrow m_{1}^{^{\prime}}$ and $m_{2} \longrightarrow m_{2}^{^{\prime}}$ of the two qudits can be connected by adequately tuning their respective frequencies, thus giving rise to a variety of conditional gates between them. The gate operation rate scales $\approx G_{1}G_{2}/\vert \omega_{\mathrm{r}} - \Omega^{m_j,m_j^{^{\prime}}}_j \vert$, with $\Omega^{m_j,m_j^{^{\prime}}}_j$ the $m_j\rightarrow m_j^{\prime}$ energy gap of spin j.

For both the qubit and qudit cases, it follows that making two-qudit operations faster than decoherence is more challenging, by the factor $G_{j} /\vert \omega_{\mathrm{r}} - \Omega^{m_j,m_j^{\prime}}_j \vert \ll 1$, than reaching strong coupling for each spin. This calls for exploring operation protocols beyond the dispersive limit.

7.2.2. Photon mediated spin–spin interactions in the resonant regime.

An alternative proposal has been recently put forward [21] to speedup photon-mediated two-qudit gates, thus overcoming the aforementioned weakness. In [21], photons are actually exchanged by bringing the field in resonance with specific spin transitions, as outlined in figure 3(f). The only manipulation tool here is represented by the tunability of the resonator frequency, obtained via SQUID elements coupled to the circuit [303, 304]. The scheme assumes to have two distinguishable spins in a magnetic field, with at least one additional auxiliary level ($| e \rangle$ in figure 3(f)) besides those used for encoding the states of the two qudits. The effect of photon loss, more important than in the dispersive regime, is strongly limited by setting $n_{ph} = 0$ in the idle configuration. A two-qubit (qudit) cϕ gate is then implemented by (i) bringing the cavity into resonance with the first qubit energy gap. If the qubit is in the excited state $| 1 \rangle$, a photon is resonantly emitted. Then (ii), the photon frequency is moved close to the $| 1 \rangle-| e \rangle$ gap of the second qubit (detuned by an amount δ), yielding a (semi-)resonant Rabi oscillation between the two states. The photon is finally re-emitted and (iii) re-absorbed by the first qubit, once its frequency has been properly tuned to match its $| 0 \rangle-| 1 \rangle$ gap. At the end of the sequence, a phase ϕ (set by δ) is added to $| 11 \rangle$ while all other two-qubit states are left unaffected, thus implementing a cϕ gate. A comparative plot of the performance of the resonant and dispersive schemes is shown in figure 15 for the quantum simulation of the Heisenberg Hamiltonian on a pair of qubits. This is a particularly good benchmark, since the simulation can be decomposed using an equal number of iSWAP$^\alpha$ or cϕ gates, as described in section 6. In this way, we focus on the capability of the two approaches, rather than on the specific algorithm decomposition. Simulations are performed by taking the best values for the spin-photon coupling reported in figure 13 and considering a SMM with S = 10 ground multiplet and easy-plane axial anisotropy, which provides a further enhancement of a factor ≈ 10 compared to a spin 1/2. In the upper panels the resulting local magnetization on the two spins is plotted, while in the lower ones the fidelity. Results obtained with the resonant approach are remarkably good, even with not too large $T_2 = 50 \, \mu$s. Then, although for short simulation angles $\lambda t$ the fidelity of the two methods is similar, for larger $\lambda t$ the resonant approach perform remarkably better. This is due to the practically constant gating times which are obtained by working in resonance (in the order of 3 µs), in contrast to their ~ linear increase up to ${\sim} 17 \, \mu$s when $G_j \approx 20|\omega_r-\Omega_{mm^{\prime}}|$, as used here. All in all, large but reachable values of T₂ enable the implementation of an accurate simulation of the model in the resonant regime.

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Just as with the dispersive coupling, the resonant scheme can be extended to an arbitrary number of qudit levels and to couple molecules placed into different resonators (by enabling photon hopping between them [303]) in a completely scalable setup. Indeed, we could consider an array of resonators each hosting one or a few molecules, and characterized by a difference in their bare frequencies significantly larger than their mutual interaction. In that configuration, photon hopping is suppressed and molecules placed in different resonators can be independently addressed. In order to implement a conditional dynamics between distant molecules, one can tune the resonator frequencies to bring them into mutual resonance, thus enabling resonant photon exchange.

To conclude this section, it is worth noting that the proposed architectures based on coupling MNMs to superconducting chips fully exploit the best features of both molecular spins (namely their long coherence, chemical tunability and multi-level structure) and photons (i.e. their high mobility). Such a mobility would enable, for instance, implementation of entangling gates between distant molecules, without needing very demanding SWAP gates.