A two-qubit molecular architecture for electron-mediated nuclear quantum simulation

Relativistic Douglas-Kroll-Hess calculations of hyperfine interactions within first-principles multireference methods

A relativistic magnetic hyperfine interaction Hamiltonian based on the Douglas-Kroll-Hess (DKH) theory up to the second order is implemented within the ab initio multireference methods, including spin-orbit coupling in the Molcas/OpenMolcas package. This implementation is applied to calculate relativistic hyperfine coupling (HFC) parameters for atomic systems and diatomic radicals with valence s or d orbitals by systematically varying active space size in the restricted active space self-consistent field formalism with restricted active space state interaction for spin-orbit coupling. The DKH relativistic treatment of the hyperfine interaction reduces the Fermi contact contribution to the HFC due to the presence of kinetic factors that regularize the singularity of the Dirac delta function in the nonrelativistic Fermi contact operator. This effect is more prominent for heavier nuclei. As the active space size increases, the relativistic correction of the Fermi contact contribution converges well to the experimental data for light and moderately heavy nuclei. The relativistic correction, however, does not significantly affect the spin-dipole contribution to the hyperfine interaction. In addition to the atomic and molecular systems, the implementation is applied to calculate the relativistic HFC parameters for large trivalent and divalent Tb-based single-molecule magnets (SMMs), such as Tb(III)Pc2 and Tb(II)(CpiPr5)2 without ligand truncation using well-converged basis sets. In particular, for the divalent SMM, which has an unpaired valence 6s/5d hybrid orbital, the relativistic treatment of HFC is crucial for a proper description of the Fermi contact contribution. Even with the relativistic hyperfine Hamiltonian, the divalent SMM is shown to exhibit strong tunability of HFC via an external electric field (i.e., strong hyperfine Stark effect).

Molecular nanomagnets: a viable path toward quantum information processing?

Molecular nanomagnets (MNMs), molecules containing interacting spins, have been a playground for quantum mechanics. They are characterized by many accessible low-energy levels that can be exploited to store and process quantum information. This naturally opens the possibility of using them as qudits, thus enlarging the tools of quantum logic with respect to qubit-based architectures. These additional degrees of freedom recently prompted the proposal for encoding qubits with embedded quantum error correction (QEC) in single molecules. QEC is the holy grail of quantum computing and this qudit approach could circumvent the large overhead of physical qubits typical of standard multi-qubit codes. Another important strength of the molecular approach is the extremely high degree of control achieved in preparing complex supramolecular structures where individual qudits are linked preserving their individual properties and coherence. This is particularly relevant for building quantum simulators, controllable systems able to mimic the dynamics of other quantum objects. The use of MNMs for quantum information processing is a rapidly evolving field which still requires to be fully experimentally explored. The key issues to be settled are related to scaling up the number of qudits/qubits and their individual addressing. Several promising possibilities are being intensively explored, ranging from the use of single-molecule transistors or superconducting devices to optical readout techniques. Moreover, new tools from chemistry could be also at hand, like the chiral-induced spin selectivity. In this paper, we will review the present status of this interdisciplinary research field, discuss the open challenges and envisioned solution paths which could finally unleash the very large potential of molecular spins for quantum technologies.

Proof-of-Concept Quantum Simulator Based on Molecular Spin Qudits

The use of d-level qudits instead of two-level qubits can largely increase the power of quantum logic for many applications, ranging from quantum simulations to quantum error correction. Magnetic molecules are ideal spin systems to realize these large-dimensional qudits. Indeed, their Hamiltonian can be engineered to an unparalleled extent and can yield a spectrum with many low-energy states. In particular, in the past decade, intense theoretical, experimental, and synthesis efforts have been devoted to develop quantum simulators based on molecular qubits and qudits. However, this remarkable potential is practically unexpressed, because no quantum simulation has ever been experimentally demonstrated with these systems. Here, we show the first prototype quantum simulator based on an ensemble of molecular qudits and a radiofrequency broadband spectrometer. To demonstrate the operativity of the device, we have simulated quantum tunneling of the magnetization and the transverse-field Ising model, representative of two different classes of problems. These results represent an important step toward the actual use of molecular spin qudits in quantum technologies.

Oriented Triplet Excitons as Long-Lived Electron Spin Qutrits in a Molecular Donor-Acceptor Single Cocrystal

The photogeneration of multiple unpaired electron spins within molecules is a promising route to applications in quantum information science because they can be initialized into well-defined, multilevel quantum states (S > 1/2) and reproducibly fabricated by chemical synthesis. However, coherent manipulation of these spin states is difficult to realize in typical molecular systems due to the lack of selective addressability and short coherence times of the spin transitions. Here, these challenges are addressed by using donor-acceptor single cocrystals composed of pyrene and naphthalene dianhydride to host spatially oriented triplet excitons, which exhibit promising photogenerated qutrit properties. Time-resolved electron paramagnetic resonance (TREPR) spectroscopy demonstrates that spatially orienting triplet excitons in a single crystal platform imparts narrow, well-resolved, tunable resonances in the triplet EPR spectrum, allowing selective addressability of the spin sublevel transitions. Pulse-EPR spectroscopy reveals that at temperatures above 30 K, spin decoherence of these triplet excitons is driven by exciton diffusion. However, coherence is limited by electronic spin dipolar coupling below 30 K, where T2 varies nonlinearly with the optical excitation density due to exciton annihilation. Overall, an optimized coherence time of T2 = 7.1 μs at 20 K is achieved. These results provide important insights into designing solid-state molecular excitonic materials with improved spin qutrit properties.

Spin Qubit in a 2D GdIII NaI -Based Oxamato Supramolecular Coordination Framework

Qubits are the basic unit of quantum information and computation. To realize quantum computing and information processing, the decoherence times of qubits must be long enough. Among the studies of molecule-based electron spin qubits, most of the work focused on the ions with the spin S=1/2, where only single-bit gates can be constructed. However, quantum operations require the qubits to interact with each other, so people gradually carry out relevant research in ions or systems with S>1/2 and multilevel states. In this work, a two-dimensional (2D) oxygen-coordinated GdIII NaI -based oxamato supramolecular coordination framework, Na[Gd(4-HOpa)4 (H2 O)] ⋅ 2H2 O (1, 4-HOpa=N-4-hydroxyphenyloxamate), was selected as a possible carrier of qubit. The field-induced slow magnetic relaxation shows this system has phonon bottleneck (PB) effect at low temperatures with a very weak magnetic anisotropy. The pulse electron paramagnetic resonance studies show the spin-lattice and spin-spin relaxation times are T1 =1.66 ms at 4 K and Tm =4.25 μs at 8 K for its diamagnetically diluted sample (1Gd0.12 %). It suggested that the relatively long decoherence time is mainly ascribed to its near isotropic and the PB effect from resonance phonon trapped for pure sample, while the dilution further improves its qubit performance.

Low-temperature motion of the scandium bimetal in endofullerene Sc2@C80(CH2Ph)

The spin decoherence of the scandium bimetal in Sc2@C80(CH2Ph) is studied at low temperatures (20-120 K) by the electron spin echo technique. The correlation between the magnetic quantum number m of the total spin state of the scandium nuclei j and the decay rates is established. For the total spin j = 5, a decrease of the phase relaxation time by a factor of two is observed by changing the transition from m = -1 to m = +1 and m = -3 at 120 K. The observed results are rationalized in the framework of the rotational diffusion of the endohedral fragment in the fullerene cage. It is found that the characteristic rotation time is of the order of a microsecond at 100 K and increases at lower temperatures.

Molecular One- and Two-Qubit Systems with Very Long Coherence Times

General-purpose quantum computation and quantum simulation require multi-qubit architectures with precisely defined, robust interqubit interactions, coupled with local addressability. This is an unsolved challenge, primarily due to scalability issues. These issues often derive from poor control over interqubit interactions. Molecular systems are promising materials for the realization of large-scale quantum architectures, due to their high degree of positionability and the possibility to precisely tailor interqubit interactions. The simplest quantum architecture is the two-qubit system, with which quantum gate operations can be implemented. To be viable, a two-qubit system must possess long coherence times, the interqubit interaction must be well defined and the two qubits must also be addressable individually within the same quantum manipulation sequence. Here results are presented on the investigation of the spin dynamics of chlorinated triphenylmethyl organic radicals, in particular the perchlorotriphenylmethyl (PTM) radical, a mono-functionalized PTM, and a biradical PTM dimer. Extraordinarily long ensemble coherence times up to 148 µs are found at all temperatures below 100 K. Two-qubit and, importantly, individual qubit addressability in the biradical system are demonstrated. These results underline the potential of molecular materials for the development of quantum architectures.

Quantum Gate Operations on a Spectrally Addressable Photogenerated Molecular Electron Spin-Qubit Pair

Sub-nanosecond photodriven electron transfer from a molecular donor to an acceptor can be used to generate a radical pair (RP) having two entangled electron spins in a well-defined pure initial singlet quantum state to serve as a spin-qubit pair (SQP). Achieving good spin-qubit addressability is challenging because many organic radical ions have large hyperfine couplings (HFCs) in addition to significant g-anisotropy, which results in significant spectral overlap. Moreover, using radicals with g-factors that deviate significantly from that of the free electron results in difficulty generating microwave pulses with sufficiently large bandwidths to manipulate the two spins either simultaneously or selectively as is necessary to implement the controlled-NOT (CNOT) quantum gate essential for quantum algorithms. Here, we address these issues by using a covalently linked donor-acceptor(1)-acceptor(2) (D-A₁-A₂) molecule with significantly reduced HFCs that uses fully deuterated peri-xanthenoxanthene (PXX) as D, naphthalenemonoimide (NMI) as A₁, and a C₆₀ derivative as A₂. Selective photoexcitation of PXX within PXX-d₉-NMI-C₆₀ results in sub-nanosecond, two-step electron transfer to generate the long-lived PXX^•+-d₉-NMI-C₆₀^•- SQP. Alignment of PXX^•+-d₉-NMI-C₆₀^•- in the nematic liquid crystal 4-cyano-4'-(n-pentyl)biphenyl (5CB) at cryogenic temperatures results in well-resolved, narrow resonances for each electron spin. We demonstrate both single-qubit gate and two-qubit CNOT gate operations using both selective and nonselective Gaussian-shaped microwave pulses and broadband spectral detection of the spin states following the gate operations.

Modular Approach to Creating Functionalized Surface Arrays of Molecular Qubits

The quest for developing quantum technologies is driven by the promise of exponentially faster computations, ultrahigh performance sensing, and achieving thorough understanding of many-particle quantum systems. Molecular spins are excellent qubit candidates because they feature long coherence times, are widely tunable through chemical synthesis, and can be interfaced with other quantum platforms such as superconducting qubits. A present challenge for molecular spin qubits is their integration in quantum devices, which requires arranging them in thin films or monolayers on surfaces. However, clear proof of the survival of quantum properties of molecular qubits on surfaces has not been reported so far. Furthermore, little is known about the change in spin dynamics of molecular qubits going from the bulk to monolayers. Here, a versatile bottom-up method is reported to arrange molecular qubits as functional groups of self-assembled monolayers (SAMs) on surfaces, combining molecular self-organization and click chemistry. Coherence times of up to 13 µs demonstrate that qubit properties are maintained or even enhanced in the monolayer.

An exchange coupled meso-meso linked vanadyl porphyrin dimer for quantum information processing

We report here the synthesis of a new meso-meso (m-m) singly linked vanadyl-porphyrin dimer that crystallizes in two different pseudo-polymorphs. The single crystal continuous-wave electron paramagnetic resonance investigation evidences a small but crucial isotropic exchange interaction, J, between the two tilted, and thus distinguishable, spin centers of the order of 10^-2 cm^-1. The experimental and DFT studies evidence a correlation between J values and porphyrin plane tilting angle and distortion. Pulsed EPR analysis shows that the two vanadyl dimers maintain the coherence time of the monomer. With the obtained spin Hamiltonian parameters, we identify suitable transitions that could be used as computational basis states. Our results, coupled with the evaporability of porphyrin systems, establish this class of dimers as extremely promising for quantum information processing applications.

Five-Spin Supramolecule for Simulating Quantum Decoherence of Bell States

We report a supramolecule that contains five spins of two different types and with, crucially, two different and predictable interaction energies between the spins. The supramolecule is characterized, and the interaction energies are demonstrated by electron paramagnetic resonance (EPR) spectroscopy. Based on the measured parameters, we propose experiments that would allow this designed supramolecule to be used to simulate quantum decoherence in maximally entangled Bell states that could be used in quantum teleportation.

Three individually addressable spin qubits in a single molecule

An asymmetric bis-phenol-β-diketone (H₄L) has been designed as a ligand programmed to promote the assembly of a molecular arrangement composed of three magnetically exchanged [NiCu] pairs, each exhibiting an S = 1/2 spin. The latter are shown by EPR and magnetometry to be good qubit realizations and non-equivalent within the molecule in the solid state, as required for conditional quantum gates.

Trinuclear Cobalt(II) Triple Helicate with a Multidentate Bithiazolebis(oxamate) Ligand as a Supramolecular Nanomagnet

The cobalt(II)-mediated self-assembly of the potentially tris(chelating) N,N'-2,2'-(4,4'-bithiazole)bis(oxamate) (dabtzox) ligand gives a new metal-organic supramolecular nanomagnet of formula K₆Co₃(dabtzox)₃·8H₂O·MeOH (1) featuring a unique linear triple-stranded trinuclear structure of the helicate type.

Stoichiometrically Controlled Assembly of Lanthanide Molecular Complexes of the Heteroditopic Divergent Ligand 4'-(4-Pyridyl)-2,2':6',2″-terpyridine N-Oxide in Hypodentate or Bridging Coordination Modes. Structural, Magnetic, and Photoluminescence Studies

Mononuclear rare-earth tris-β-diketonato complexes RE(tta)₃dme [RE = Y (1), La (2), Dy (3), or Eu (4); Htta = 2-thenoylacetone; dme = 1,2-dimethoxyethane] react cleanly at room temperature in a 1:1 molar ratio with the heteroditopic divergent ligand 4'-(4-pyridyl)-2,2':6',2″-terpyridine N-oxide (pyterpyNO) to yield RE₂(tta)₆(pyterpyNO)_n, where n = 2 for RE = Y (5), Dy (6), or Eu (7) and n = 3 for RE = La (8). The crystal structure of 5 revealed a dinuclear compound with two pyterpyNO's bridging through the oxygen atom in a hypodentate mode leaving the terpyridine moieties uncoordinated. Using a metal:pyterpyNO molar ratio of 2 for RE = Y (9), Dy (10), or Eu (11), it was possible to isolate the molecular complexes RE₄(tta)₁₂(pyterpyNO)₂, while using a 5:3 molar ratio, the product La₅(tta)₁₂(pyterpyNO)₃ (12) can be obtained. ⁸⁹Y nuclear magnetic resonance spectroscopy revealed two different yttrium centers at room temperature for 9. An X-ray diffraction study of 10 showed a symmetrical tetranuclear structure resulting from the coordination of two Dy(tta)₃ fragments to the two hypodentate terpyridines of the dinuclear unit and presenting two different coordination sites for metals with coordination numbers of 8 and 9. Magnetic studies of 6 and 10 revealed the presence of an antiferromagnetic interaction between the two Dy(III) atoms bound by the NO bridges. These compounds displayed a slow relaxing magnetization through Orbach (6) and Raman (10) processes in the absence of an applied magnetic field; the rate increased upon application of a 1 kOe field. 7 and 11 showed a bright red emission typical of Eu³⁺. The two complexes have similar emission properties mainly determined by the employed β-diketonato ligands.

Exchange-Bias Quantum Tunneling of the Magnetization in a Dysprosium Dimer

Single-molecule magnets (SMMs) have been shown to possess bewildering phenomena leading to their proposal in several futuristic applications ranging from data storage devices to the basic unit of quantum computers. The main characteristic for the proposal of SMMs in such schemes is their inherent and intriguing quantum mechanical properties, which in turn, could be exploited in novel devices with larger capacities, such as for data storage or enhanced properties, such as quantum computers. In the quest of SMMs displaying such intriguing quantum effects, herein, we explore the synthesis, structural, and magnetic characterization of a dimeric dysprosium-based SMM composed of a tetradentate Schiff-base ligand with formula [Dy₂(HL)₂(benz)₂(NO₃)₂]. Magnetic studies show that the complex is an SMM, while sub-Kelvin μ-SQUID studies revealed the exchange-bias characteristics of the system attributed to the presence of exchange interaction between the Dy³⁺ pair.

Controlled coherent dynamics of [VO(TPP)], a prototype molecular nuclear qudit with an electronic ancilla

We show that [VO(TPP)] (vanadyl tetraphenylporphyrinate) is a promising system to implement quantum computation algorithms based on encoding information in multi-level (qudit) units. Indeed, it embeds a nuclear spin 7/2 coupled to an electronic spin 1/2 by hyperfine interaction. This qubit-qudit unit can be exploited to implement quantum error correction and quantum simulation algorithms. Through a combined theoretical and broadband nuclear magnetic resonance study, we demonstrate that the elementary operations of such algorithms can be efficiently implemented on the nuclear spin qudit. Manipulation of the nuclear qudit can be achieved by resonant radio-frequency pulses, thanks to the remarkably long coherence times and the effective quadrupolar coupling induced by the strong hyperfine interaction. This approach may open new perspectives for developing new molecular qubit-qudit systems.

Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers

Magnetic molecules are prototypical systems to investigate peculiar quantum mechanical phenomena. As such, simulating their static and dynamical behavior is intrinsically difficult for a classical computer, due to the exponential increase of required resources with the system size. Quantum computers solve this issue by providing an inherently quantum platform, suited to describe these magnetic systems. Here, we show that both the ground state properties and the spin dynamics of magnetic molecules can be simulated on prototype quantum computers, based on superconducting qubits. In particular, we study small-size anti-ferromagnetic spin chains and rings, which are ideal test-beds for these pioneering devices. We use the variational quantum eigensolver algorithm to determine the ground state wave-function with targeted ansatzes fulfilling the spin symmetries of the investigated models. The coherent spin dynamics are simulated by computing dynamical correlation functions, an essential ingredient to extract many experimentally accessible properties, such as the inelastic neutron cross-section.

Increasing the Hilbert space dimension using a single coupled molecular spin

Quantum technologies are expected to introduce revolutionary changes in information processing in the near future. Nowadays, one of the main challenges is to be able to handle a large number of quantum bits (qubits), while preserving their quantum properties. Beyond the usual two-level encoding capacity of qubits, multi-level quantum systems are a promising way to extend and increase the amount of information that can be stored in the same number of quantum objects. Recent work (Kues et al. 2017), has shown the possibility to use devices based on photonic integrated circuits to entangle two qudits (with "d" being the number of available states). In the race to develop a mature quantum technology with real-world applications, many possible platforms are being investigated, including those that use photons, trapped ions, superconducting and silicon circuits and molecular magnets. In this work, we present the electronic read-out of a coupled molecular multi-level quantum systems, carried by a single Tb₂Pc₃ molecular magnet. Owning two magnetic centres, this molecular magnet architecture permits a 16 dimensions Hilbert space, opening the possibility of performing more complex quantum algorithms.

Targeting molecular quantum memory with embedded error correction

The implementation of a quantum computer requires both to protect information from environmental noise and to implement quantum operations efficiently. Achieving this by a fully fault-tolerant platform, in which quantum gates are implemented within quantum-error corrected units, poses stringent requirements on the coherence and control of such hardware. A more feasible architecture could consist of connected memories, that support error-correction by enhancing coherence, and processing units, that ensure fast manipulations. We present here a supramolecular {Cr7Ni}-Cu system which could form the elementary unit of this platform, where the electronic spin 1/2 of {Cr7Ni} provides the processor and the naturally isolated nuclear spin 3/2 of the Cu ion is used to encode a logical unit with embedded quantum error-correction. We demonstrate by realistic simulations that microwave pulses allow us to rapidly implement gates on the processor and to swap information between the processor and the quantum memory. By combining the storage into the Cu nuclear spin with quantum error correction, information can be protected for times much longer than the processor coherence.

Design of pure heterodinuclear lanthanoid cryptate complexes

Heterolanthanide complexes are difficult to synthesize owing to the similar chemistry of the lanthanide ions. Consequently, very few purely heterolanthanide complexes have been synthesized. This is despite the fact that such complexes hold interesting optical and magnetic properties. To fine-tune these properties, it is important that one can choose complexes with any given combination of lanthanides. Herein we report a synthetic procedure which yields pure heterodinuclear lanthanide cryptates LnLn*LX3 (X = NO3 - or OTf-) based on the cryptand H3L = N[(CH2)2N[double bond, length as m-dash]CH-R-CH[double bond, length as m-dash]N-(CH2)2]3N (R = m-C6H2OH-2-Me-5). In the synthesis the choice of counter ion and solvent proves crucial in controlling the Ln-Ln* composition. Choosing the optimal solvent and counter ion afford pure heterodinuclear complexes with any given combination of Gd(iii)-Lu(iii) including Y(iii). To demonstrate the versatility of the synthesis all dinuclear combinations of Y(iii), Gd(iii), Yb(iii) and Lu(iii) were synthesized resulting in 10 novel complexes of the form LnLn*L(OTf)3 with LnLn* = YbGd 1, YbY 2, YbLu 3, YbYb 4, LuGd 5, LuY 6, LuLu 7, YGd 8, YY 9 and GdGd 10. Through the use of 1H, 13C NMR and mass spectrometry the heterodinuclear nature of YbGd, YbY, YbLu, LuGd, LuY and YGd was confirmed. Crystal structures of LnLn*L(NO3)3 reveal short Ln-Ln distances of ∼3.5 Å. Using SQUID magnetometry the exchange coupling between the lanthanide ions was found to be anti-ferromagnetic for GdGd and YbYb while ferromagnetic for YbGd.

Broad-band spectroscopy of a vanadyl porphyrin: a model electronuclear spin qudit

We explore how to encode more than a qubit in vanadyl porphyrin molecules hosting a S = 1/2 electronic spin coupled to a I = 7/2 nuclear spin. The spin Hamiltonian and its parameters, as well as the spin dynamics, have been determined via a combination of electron paramagnetic resonance, heat capacity, magnetization and on-chip magnetic spectroscopy experiments performed on single crystals. We find low temperature spin coherence times of micro-seconds and spin relaxation times longer than a second. For sufficiently strong magnetic fields (B > 0.1 T, corresponding to resonance frequencies of 9-10 GHz) these properties make vanadyl porphyrin molecules suitable qubit realizations. The presence of multiple equispaced nuclear spin levels then merely provides 8 alternatives to define the '1' and '0' basis states. For lower magnetic fields (B < 0.1 T), and lower frequencies (<2 GHz), we find spectroscopic signatures of a sizeable electronuclear entanglement. This effect generates a larger set of allowed transitions between different electronuclear spin states and removes their degeneracies. Under these conditions, we show that each molecule fulfills the conditions to act as a universal 4-qubit processor or, equivalently, as a d = 16 qudit. These findings widen the catalogue of chemically designed systems able to implement non-trivial quantum functionalities, such as quantum simulations and, especially, quantum error correction at the molecular level.

Controlling the Dynamics of Three Electron Spin Qubits in a Donor-Acceptor-Radical Molecule Using Dielectric Environment Changes

Photogenerated entangled electron spin pairs provide a versatile source of molecular qubits. Here, we examine the spin-dependent dynamics of a covalent donor-acceptor-radical molecule, D-A-R^•, where the donor chromophore (D) is peri-xanthenoxanthene (PXX), the acceptor (A) is pyromellitimide (PI), and the radical (R^•) is α,γ-bisdiphenylene-β-phenylallyl (BDPA). Selective photoexcitation of D within D-A-R^• in butyronitrile/propionitrile at 140 K and butyronitrile at 105 K results in the spin-selective reactions D-A-R^• → D^•+-¹(A^•--R^•) and D^•+-³(A^•--R^•). Subsequently, at 140 K, D^•+-¹(A^•--R^•) → D^•+-A-R^-, whereas D^•+-³(A^•--R^•) → D-A-R^•. In contrast, at 105 K, D^•+-³(A^•--R^•) → ^3*D-A-R^• and D-A-R^•. Time-resolved EPR spectroscopy shows that ^3*D-A-R^• is highly spin-polarized, where the m_s = ±1/2 spin sublevels of the doublet-quartet manifolds are selectively populated. These results demonstrate dielectric environment control over different spin states in the same molecule.

Chemical tuning of spin clock transitions in molecular monomers based on nuclear spin-free Ni(ii)

We report the existence of a sizeable quantum tunnelling splitting between the two lowest electronic spin levels of mononuclear Ni complexes. The level anti-crossing, or magnetic "clock transition", associated with this gap has been directly monitored by heat capacity experiments. The comparison of these results with those obtained for a Co derivative, for which tunnelling is forbidden by symmetry, shows that the clock transition leads to an effective suppression of intermolecular spin-spin interactions. In addition, we show that the quantum tunnelling splitting admits a chemical tuning via the modification of the ligand shell that determines the crystal field and the magnetic anisotropy. These properties are crucial to realize model spin qubits that combine the necessary resilience against decoherence, a proper interfacing with other qubits and with the control circuitry and the ability to initialize them by cooling.

Probing Vibrational Symmetry Effects and Nuclear Spin Economy Principles in Molecular Spin Qubits

The selection of molecular spin qubits with a long coherence time, Tm, is a central task for implementing molecule-based quantum technologies. Even if a sufficiently long Tm can be achieved through an efficient synthetic strategy and ad hoc experimental measurement procedures, many factors contributing to the loss of coherence still need to be thoroughly investigated and understood. Vibrational properties and nuclear spins of hydrogens are two of them. The former plays a paramount role, but a detailed theoretical investigation aimed at studying their effects on the spin dynamics of molecular complexes such as the benchmark phthalocyanine (Pc) is still missing, whereas the effect of the latter deserves to be examined in detail for such a class of compounds. In this work, we adopted a combined theoretical and experimental approach to investigate the relaxation properties of classical [Cu(Pc)] and a CuII complex based on the ligand tetrakis(thiadiazole)porphyrazine (H2TTDPz), characterized by a hydrogen-free molecular structure. Systematic calculations of molecular vibrations exemplify the effect of normal modes on the spin-lattice relaxation process, unveiling a different contribution to T1 depending on the symmetry of normal modes. Moreover, we observed that an appreciable Tm enhancement could be achieved by removing hydrogens from the ligand.

When Molecular Magnetism Meets Supramolecular Chemistry: Multifunctional and Multiresponsive Dicopper(II) Metallacyclophanes as Proof-of-Concept for Single-Molecule Spintronics and Quantum Computing Technologies?

Molecular magnetism has made a long journey, from the fundamental studies on through-ligand electron exchange magnetic interactions in dinuclear metal complexes with extended organic bridges to the more recent exploration of their electron spin transport and quantum coherence properties. Such a field has witnessed a renaissance of dinuclear metallacyclic systems as new experimental and theoretical models for single-molecule spintronics and quantum computing, due to the intercrossing between molecular magnetism and metallosupramolecular chemistry. The present review reports a state-of-the-art overview as well as future perspectives on the use of oxamato-based dicopper(II) metallacyclophanes as promising candidates to make multifunctional and multiresponsive, single-molecule magnetic (nano)devices for the physical implementation of quantum information processing (QIP). They incorporate molecular magnetic couplers, transformers, and wires, controlling and facilitating the spin communication, as well as molecular magnetic rectifiers, transistors, and switches, exhibiting a bistable (ON/OFF) spin behavior under external stimuli (chemical, electronic, or photonic). Special focus is placed on the extensive research work done by Professor Francesc Lloret, an outstanding chemist, excellent teacher, best friend, and colleague, in recognition of his invaluable contributions to molecular magnetism on the occasion of his 65th birthday.

Molecular Nanomagnets as Qubits with Embedded Quantum-Error Correction

We show that molecular nanomagnets have a potential advantage in the crucial rush toward quantum computers. Indeed, the sizable number of accessible low-energy states of these systems can be exploited to define qubits with embedded quantum error correction. We derive the scheme to achieve this crucial objective and the corresponding sequence of microwave/radiofrequency pulses needed for the error correction procedure. The effectiveness of our approach is shown already with a minimal S = 3/2 unit corresponding to an existing molecule, and the scaling to larger spin systems is quantitatively analyzed.

Molecular Nanomagnets as Qubits with Embedded Quantum-Error Correction

We show that molecular nanomagnets have a potential advantage in the crucial rush toward quantum computers. Indeed, the sizable number of accessible low-energy states of these systems can be exploited to define qubits with embedded quantum error correction. We derive the scheme to achieve this crucial objective and the corresponding sequence of microwave/radiofrequency pulses needed for the error correction procedure. The effectiveness of our approach is shown already with a minimal S = 3/2 unit corresponding to an existing molecule, and the scaling to larger spin systems is quantitatively analyzed.

Electrically tuned hyperfine spectrum in neutral Tb(II)(CpiPr5)2 single-molecule magnet

Molecular spin qubits with long spin coherence time as well as non-invasive operation methods on such qubits are in high demand. It was shown that both molecular electronic and nuclear spin levels can be used as qubits. In solid state systems with dopants, an electric field was shown to effectively change the spacing between the nuclear spin qubit levels when the electron spin density is high at the nucleus of the dopant. Inspired by such solid-state systems, we propose that divalent lanthanide (Ln) complexes with an unusual electronic configuration of Ln2+ have a strong interaction between the Ln nuclear spin and the electronic degrees of freedom, which renders electrical tuning of the interaction. As an example, we study electronic structure and hyperfine interaction of the 159Tb nucleus in a neutral Tb(ii)(CpiPr5)2 single-molecule magnet (SMM), which exhibits unusually long magnetization relaxation time, using the complete active space self-consistent field (CASSCF) method with spin-orbit interaction included within the restricted active space state interaction (RASSI). Our calculations show that the low-energy states arise from 4f8(6s,5dz2)1, 4f8(5dx2-y2)1, and 4f8(5dxy)1 configurations. We compute the hyperfine interaction parameters and the electronic-nuclear spectrum within our multiconfigurational approach. We find that the hyperfine interaction is about one order of magnitude greater than that for Tb(iii)Pc2 SMMs. This stems from the strong Fermi contact interaction between the Tb nuclear spin and the electron spin density at the nucleus that originates from the occupation of the (6s,5d) orbitals. We also uncover that the response of the Fermi contact term to electric field results in electrical tuning of the electronic-nuclear level separations. This hyperfine Stark effect may be useful for applications of molecular nuclear spins for quantum computing.

Exploiting clock transitions for the chemical design of resilient molecular spin qubits

Molecular spin qubits are chemical nanoobjects with promising applications that are so far hampered by the rapid loss of quantum information, a process known as decoherence. A strategy to improve this situation involves employing so-called Clock Transitions (CTs), which arise at anticrossings between spin energy levels. At CTs, the spin states are protected from magnetic noise and present an enhanced quantum coherence. Unfortunately, these optimal points are intrinsically hard to control since their transition energy cannot be tuned by an external magnetic field; moreover, their resilience towards geometric distortions has not yet been analyzed. Here we employ a python-based computational tool for the systematic theoretical analysis and chemical optimization of CTs. We compare three relevant case studies with increasingly complex ground states. First, we start with vanadium(iv)-based spin qubits, where the avoided crossings are controlled by hyperfine interaction and find that these S = 1/2 systems are very promising, in particular in the case of vanadyl complexes in an L-band pulsed EPR setup. Second, we proceed with a study of the effect of symmetry distortions in a holmium polyoxotungstate of formula [Ho(W₅O₁₈)₂]^9- where CTs had already been experimentally demonstrated. Here we determine the relative importance of the different structural distortions that causes the anticrossings. Third, we study the most complicated case, a polyoxopalladate cube [HoPd₁₂(AsPh)₈O₃₂]^5- which presents an unusually rich ground spin multiplet. This system allows us to find uniquely favorable CTs that could nevertheless be accessible with standard pulsed EPR equipment (X-band or Q-band) after a suitable chemical distortion to break the perfect cubic symmetry. Since anticrossings and CTs constitute a rich source of physical phenomena in very different kinds of quantum systems, the generalization of this study is expected to have impact not only in molecular spin science but also in other related fields such as molecular photophysics and photochemistry.

A bis-vanadyl coordination complex as a 2-qubit quantum gate

A new bis-hydroxyphenylpyrazolyl ligand, H₄L, allows isolating and structurally characterizing vanadyl and titanyl dinuclear complexes (Bu₄N)₂[(MO)₂(HL)₂] (M = V, Ti). The weak dipolar coupling and relatively short quantum coherence of the divanadyl anions are optimal for a 2-qubit molecular architecture proposed to implement electron-mediated nuclear quantum simulations.

Magnetization Dynamics and Coherent Spin Manipulation of a Propeller Gd(III) Complex with the Smallest Helicene Ligand

A homoleptic gadolinium(III) complex with the smallest helicene-type ligand, 1,10-phenanthroline-N,N'-dioxide (phendo) [Gd(phendo)₄](NO₃)₃·xMeOH (phendo = 1,10-phenanthroline-N,N'-dioxide, MeOH = methanol), shows slow relaxation of the magnetization characteristic for Single Ion Magnets (SIM), despite negligible magnetic anisotropy, confirmed by ab initio calculations. Solid state dilution magnetic and EPR studies reveal that the magnetization dynamics of the [Gd(phendo)₄]³⁺ cation is controlled mainly by a Raman process. Pulsed EPR experiments demonstrate long phase memory times (up to 2.7 μs at 5 K), enabling the detection of Rabi oscillations at 20 K, which confirms coherent control of its spin state.

A heterometallic [LnLn′Ln] lanthanide complex as a qubit with embedded quantum error correction

We show that a [Er–Ce–Er] molecular trinuclear coordination compound is a promising platform to implement the three-qubit quantum error correction code protecting against pure dephasing, the most important error in magnetic molecules. We characterize it by preparing the [Lu–Ce–Lu] and [Er–La–Er] analogues, which contain only one of the two types of qubit, and by combining magnetometry, low-temperature specific heat and electron paramagnetic resonance measurements on both the elementary constituents and the trimer. Using the resulting parameters, we demonstrate by numerical simulations that the proposed molecular device can efficiently suppress pure dephasing of the spin qubits.

We show that a [Er–Ce–Er] molecular trinuclear coordination compound is a promising platform to implement the three-qubit quantum error correction code protecting against pure dephasing, the most important error in magnetic molecules.

The dynamic ligand field of a molecular qubit: decoherence through spin–phonon coupling

A ligand field model highlights chemical design principles for the development of room temperature coherent materials for quantum information processing.

Mapping Magnetic Properties and Relaxation in Vanadium(IV) Complexes with Lanthanides by Electron Paramagnetic Resonance

Vanadium(IV) complexes are actively studied as potential candidates for molecular spin qubits operating at room temperatures. They have longer electron spin decoherence times than many other transition ions, being the key property for applications in quantum information processing. In most cases reported to date, the molecular complexes were optimized through the design for this purpose. In this work, we investigate the relaxation properties of vanadium(IV) ions incorporated in complexes with lanthanides using electron paramagnetic resonance (EPR). In all cases, the VO₆ moieties with no nuclear spins in the first coordination sphere are addressed. We develop and implement the approaches for facile diagnostics of relaxation characteristics in individual VO₆ moieties of such compounds. Remarkably, the estimated relaxation times are found to be close to those of other vanadium-based qubits obtained previously. In the future, a synergistic combination of qubit-friendly properties of vanadium ions with single-molecule magnetism and luminescence of lanthanides can be pursued to realize new functionalities of such materials.

Close Encounters of the Weak Kind: Investigations of Electron-Electron Interactions between Dissimilar Spins in Hybrid Rotaxanes

We report a family of hybrid [2]rotaxanes based on inorganic [Cr₇NiF₈(O₂C^tBu)₁₆]⁻ (“{Cr₇Ni}”) rings templated about organic threads that are terminated at one end with pyridyl groups. These rotaxanes can be coordinated to [Cu(hfac)₂] (where Hhfac = 1,1,1,5,5,5-hexafluoroacetylacetone), to give 1:1 or 1:2 Cu:{Cr₇Ni} adducts: {[Cu(hfac)₂](py-CH₂NH₂CH₂CH₂Ph)[Cr₇NiF₈(O₂C^tBu)₁₆]}, {[Cu(hfac)₂][py-CH₂NH₂CH₂CH₃][Cr₇NiF₈(O₂C^tBu)₁₆]}, {[Cu(hfac)₂]([py-CH₂CH₂NH₂CH₂C₆H₄SCH₃][Cr₇NiF₈(O₂C^tBu)₁₆])₂}, {[Cu(hfac)₂]([py-C₆H₄-CH₂NH₂(CH₂)₄Ph][Cr₇NiF₈(O₂C^tBu)₁₆])₂}, and {[Cu(hfac)₂]([3-py-CH₂CH₂NH₂(CH₂)₃SCH₃][Cr₇NiF₈(O₂C^tBu)₁₆])₂}, the structures of which have been determined by X-ray diffraction. The {Cr₇Ni} rings and Cu^II ions both have electronic spin S = 1/2, but with very different g-values. Continuous-wave EPR spectroscopy reveals the exchange interactions between these dissimilar spins, and hence the communication between the different molecular components that comprise these supramolecular systems. The interactions are weak such that we observe AX or AX₂ type spectra. The connectivity between the {Cr₇Ni} ring and thread terminus is varied such that the magnitude of the exchange interaction J can be tuned. The coupling is shown to be dominated by through-bond rather than through-space mechanisms.

The Second Quantum Revolution: Role and Challenges of Molecular Chemistry

Implementation of modern Quantum Technologies might benefit from the remarkable quantum properties shown by molecular spin systems. In this Perspective, we highlight the role that molecular chemistry can have in the current second quantum revolution, i.e., the use of quantum physics principles to create new quantum technologies, in this specific case by means of molecular components. Herein, we briefly review the current status of the field by identifying the key advances recently made by the molecular chemistry community, such as for example the design of molecular spin qubits with long spin coherence and the realization of multiqubit architectures for quantum gates implementation. With a critical eye to the current state-of-the-art, we also highlight the main challenges needed for the further advancement of the field toward quantum technologies development.

Synthetic Hilbert Space Engineering of Molecular Qudits: Isotopologue Chemistry

One of the most ambitious technological goals is the development of devices working under the laws of quantum mechanics. Among others, an important challenge to be resolved on the way to such breakthrough technology concerns the scalability of the available Hilbert space. Recently, proof-of-principle experiments were reported, in which the implementation of quantum algorithms (the Grover's search algorithm, iSWAP-gate, etc.) in a single-molecule nuclear spin qudit (with d = 4) known as ¹⁵⁹ TbPc₂ was described, where the nuclear spins of lanthanides are used as a quantum register to execute simple quantum algorithms. In this progress report, the goal of linear and exponential up-scalability of the available Hilbert space expressed by the qudit-dimension "d" is addressed by synthesizing lanthanide metal complexes as quantum computing hardware. The synthesis of multinuclear large-Hilbert-space complexes has to be carried out under strict control of the nuclear spin degree of freedom leading to isotopologues, whereby electronic coupling between several nuclear spin units will exponentially extend the Hilbert space available for quantum information processing. Thus, improved multilevel spin qudits can be achieved that exhibit an exponentially scalable Hilbert space to enable high-performance quantum computing and information storage.

Quantum tunnelling of the magnetisation in single-molecule magnet isotopologue dimers

Quantum tunnelling of the magnetisation plays a major role in the magnetic properties of lanthanide Single-Molecule Magnets: while it is considered a problem for data storage device applications since it leads to information loss, it is an essential pre-requisite for the read-out and manipulation of the nuclear states in Quantum Information Processing schemes. Here we describe two isotopologue dysprosium dimers, i.e. [(163Dy(tmhd)3)2(bpym)] and [(164Dy(tmhd)3)2(bpym)] (tmd = tris(tetramethylheptanedionato) and bpym = bipyrimidine), where the nuclear spin presence or absence clearly affects the magnetic properties of the systems. Through μ-SQUID studies at milli-Kelvin temperatures and alternating current magnetic measurements, we find significant differences in the magnetic behaviour of both complexes. While simulation of the hysteresis loops at 30 mK reveals that the presence of nuclear spin does not influence the tunnelling rate, we find that it facilitates the coupling to the phonon bath enhancing the direct relaxation process; an observation reflected in the temperature and field dependence of the relaxation rates.

Molecular spins for quantum computation

Spins in solids or in molecules possess discrete energy levels, and the associated quantum states can be tuned and coherently manipulated by means of external electromagnetic fields. Spins therefore provide one of the simplest platforms to encode a quantum bit (qubit), the elementary unit of future quantum computers. Performing any useful computation demands much more than realizing a robust qubit-one also needs a large number of qubits and a reliable manner with which to integrate them into a complex circuitry that can store and process information and implement quantum algorithms. This 'scalability' is arguably one of the challenges for which a chemistry-based bottom-up approach is best-suited. Molecules, being much more versatile than atoms, and yet microscopic, are the quantum objects with the highest capacity to form non-trivial ordered states at the nanoscale and to be replicated in large numbers using chemical tools.

Enabling single qubit addressability in a molecular semiconductor comprising gold-supported organic radicals

A bis(dithiolene)gold complex is presented as a model for an organic molecular electron spin qubit attached to a metallic surface that acts as a conduit to electrically address the qubit. A two-membered electron transfer series is developed of the formula [AuIII(adt)2]1-/0, where adt is a redox-active dithiolene ligand that is sequentially oxidized as the series is traversed while the central metal ion remains AuIII and steadfastly square planar. One-electron oxidation of diamagnetic [AuIII(adt)2]1- (1) produces an S = 1/2 charge-neutral complex, [AuIII(adt2 3-˙)] (2) which is spectroscopically and theoretically characterized with a near negligible Au contribution to the ground state. A phase memory time (T M) of 21 μs is recorded in 4 : 1 CS2/CCl4 at 10 K, which is the longest ever reported for a coordination complex possessing a third-row transition metal ion. With increasing temperature, T M dramatically decreases becoming unmeasurable above 80 K as a consequence of the diminishing spin-lattice (T 1) relaxation time fueled by spin-orbit coupling. These relaxation times are 1-2 orders of magnitude shorter for the solid dilution of 2 in isoelectronic [Ni(adt)2] because this material is a molecular semiconductor. Although the conducting properties of this material provide efficient pathways to dissipate the energy through the lattice, it can also be used to electrically address the paramagnetic dopant by tapping into the mild reduction potential to switch magnetism "on" and "off" in the gold complex without compromising the integrity of its structure. These results serve to highlight the need to consider all components of these spintronic assemblies.

Counterion influence on dynamic spin properties in a V(iv) complex

Using transition metal ions for spin-based applications, such as electron paramagnetic resonance imaging (EPRI) or quantum computation, requires a clear understanding of how local chemistry influences spin properties. Herein we report a series of four ionic complexes to provide the first systematic study of one aspect of local chemistry on the V(iv) spin - the counterion. To do so, the four complexes (Et3NH)2[V(C6H4O2)3] (1), (n-Bu3NH)2[V(C6H4O2)3] (2), (n-Hex3NH)2[V(C6H4O2)3] (3), and (n-Oct3NH)2[V(C6H4O2)3] (4) were probed by EPR spectroscopy in solid state and solution. Room temperature, solution X-band (ca. 9.8 GHz) continuous-wave electron paramagnetic resonance (CW-EPR) spectroscopy revealed an increasing linewidth with larger cations, likely a counterion-controlled tumbling in solution via ion pairing. In the solid state, variable-temperature (5-180 K) X-band (ca. 9.4 GHz) pulsed EPR studies of 1-4 in o-terphenyl glass demonstrated no effect on spin-lattice relaxation times (T 1), indicating little role for the counterion on this parameter. However, the phase memory time (T m) of 1 below 100 K is markedly smaller than those of 2-4. This result is counterintuitive, as 2-4 are relatively richer in 1H nuclear spin, hence, expected to have shorter T m. Thus, these data suggest an important role for counterion methyl groups on T m, and moreover provide the first instance of a lengthening T m with increasing nuclear spin quantity on a molecule.

Relevance of Dzyaloshinskii–Moriya spectral broadenings in promoting spin decoherence: a comparative pulsed-EPR study of two structurally related iron(iii) and chromium(iii) spin-triangle molecular qubits

Two related iron(iii) and chromium(iii) spin-triangle molecular qubits show coherent driving of their spins, and decoherence that is not significantly affected by Dzyaloshikskii–Moriya spectral broadenings.

Quantum chemistry on quantum computers: quantum simulations of the time evolution of wave functions under the S2 operator and determination of the spin quantum number S

Quantum computers have an enormous impact on quantum chemical calculations. Approaches to calculate the energies of atoms and molecules on quantum computers by utilizing quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) have been well documented, and dozens of methodological improvements to decrease computational costs and to mitigate errors have been reported until recently. However, the possible methodological implementation of observables on quantum computers such as calculating the spin quantum numbers of arbitrary wave functions, which is a crucial issue in quantum chemistry, has been discussed less. Here, we propose a quantum circuit to simulate the time evolution of wave functions under an S2 operator, exp(-iS2t)|Ψ, and integrate it into the QPE circuit enabling us to determine the spin quantum number of the arbitrary wave functions. We demonstrate that the spin quantum numbers of up to three spins can be determined by only one qubit measurement in QPE.

Tunable Spin-Superconductor Coupling of Spin 1/2 Vanadyl Phthalocyanine Molecules

Atomic-scale magnetic moments in contact with superconductors host rich physics based on the emergence of Yu-Shiba-Rusinov (YSR) magnetic bound states within the superconducting condensate. Here, we focus on a magnetic bound state induced into Pb nanoislands by individual vanadyl phthalocyanine (VOPc) molecules deposited on the Pb surface. The VOPc molecule is characterized by a spin magnitude of 1/2 arising from a well-isolated singly occupied d _xy-orbital and is a promising candidate for a molecular spin qubit with long coherence times. X-ray magnetic circular dichroism (XMCD) measurements show that the molecular spin remains unperturbed even for molecules directly deposited on the Pb surface. Scanning tunneling spectroscopy and density functional theory (DFT) calculations identify two adsorption geometries for this "asymmetric" molecule (i.e., absence of a horizontal symmetry plane): (a) oxygen pointing toward the vacuum with the Pc laying on the Pb, showing negligible spin-superconductor interaction, and (b) oxygen pointing toward the Pb, presenting an efficient interaction with the Pb and promoting a Yu-Shiba-Rusinov bound state. Additionally, we find that in the first case a YSR state can be induced smoothly by exerting mechanical force on the molecules with the scanning tunneling microscope (STM) tip. This allows the interaction strength to be tuned continuously from an isolated molecular spin case, through the quantum critical point (where the bound state energy is zero) and beyond. DFT indicates that a gradual bending of the VO bond relative to the Pc ligand plane promoted by the STM tip can modify the interaction in a continuously tunable manner. The ability to induce a tunable YSR state in the superconductor suggests the possibility of introducing coupled spins on superconductors with switchable interaction.

A Rare Example of Four-Coordinate Nonoxido Vanadium(IV) Alkoxide in the Solid State: Structure, Spectroscopy, and Magnetization Dynamics

The distorted tetrahedral [V(OAd)₄] alkoxide (OAd = 1-adamantoxide, complex 1) is the first homoleptic, mononuclear vanadium(IV) alkoxide to be characterized in the solid state by X-ray diffraction analysis. The compound crystallizes in the cubic P4̅3 n space group with two highly disordered, crystallographically independent molecules in the asymmetric unit. Spin Hamiltonian parameters extracted from low temperature X- and Q-band electron paramagnetic resonance (EPR) experiments performed for polycrystalline samples of 1, both in the concentrated (bulk) form and diluted in the diamagnetic [Ti(OAd)₄] analogue, reveal a fully axial system with g _z < g _x, g _y and A _z ≫ A _x, A _y. Complex 1 has also been characterized by alternate current susceptometry with varying temperature (3-30 K) and static magnetic field (up to 8.5 T), showing field-induced slow relaxation of the magnetization with relaxation times ranging from ca. 3 ms at 3 K to 0.02-0.03 ms at 30 K, in line with relevant results described recently for other potential molecular quantum bits. Pulsed EPR measurements, in turn, disclosed long coherence times of ca. 4 μs at temperatures lower than 40 K, despite the presence of the H-rich ligands. The slow spin relaxation in 1 is the first observed for a tetracoordinate nonoxido vanadium(IV) complex, and results are compared here to those generated by square-pyramidal V^IV(O)²⁺ and trigonal prismatic V⁴⁺ with oxygen donor atom sets. Considering that the number of promising d¹ complexes investigated in detail for slow magnetization dynamics is still small, the present work contributes to the establishment of possible structural/electronic correlations of interest to the field of quantum information processing.