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.

The critical role of ultra-low-energy vibrations in the relaxation dynamics of molecular qubits

Improving the performance of molecular qubits is a fundamental milestone towards unleashing the power of molecular magnetism in the second quantum revolution. Taming spin relaxation and decoherence due to vibrations is crucial to reach this milestone, but this is hindered by our lack of understanding on the nature of vibrations and their coupling to spins. Here we propose a synergistic approach to study a prototypical molecular qubit. It combines inelastic X-ray scattering to measure phonon dispersions along the main symmetry directions of the crystal and spin dynamics simulations based on DFT. We show that the canonical Debye picture of lattice dynamics breaks down and that intra-molecular vibrations with very-low energies of 1-2 meV are largely responsible for spin relaxation up to ambient temperature. We identify the origin of these modes, thus providing a rationale for improving spin coherence. The power and flexibility of our approach open new avenues for the investigation of magnetic molecules with the potential of removing roadblocks toward their use in quantum devices.

Long-lived electronic spin qubits in single-walled carbon nanotubes

Electron spins in solid-state systems offer the promise of spin-based information processing devices. Single-walled carbon nanotubes (SWCNTs), an all-carbon one-dimensional material whose spin-free environment and weak spin-orbit coupling promise long spin coherence times, offer a diverse degree of freedom for extended range of functionality not available to bulk systems. A key requirement limiting spin qubit implementation in SWCNTs is disciplined confinement of isolated spins. Here, we report the creation of highly confined electron spins in SWCNTs via a bottom-up approach. The record long coherence time of 8.2 µs and spin-lattice relaxation time of 13 ms of these electronic spin qubits allow demonstration of quantum control operation manifested as Rabi oscillation. Investigation of the decoherence mechanism reveals an intrinsic coherence time of tens of milliseconds. These findings evident that combining molecular approaches with inorganic crystalline systems provides a powerful route for reproducible and scalable quantum materials suitable for qubit applications.

Towards Luminescent Vanadium(II) Complexes with Slow Magnetic Relaxation and Quantum Coherence

Molecular entities with doublet or triplet ground states find increasing interest as potential molecular quantum bits (qubits). Complexes with higher multiplicity might even function as qudits and serve to encode further quantum bits. Vanadium(II) ions in octahedral ligand fields with quartet ground states and small zero-field splittings qualify as qubits with optical read out thanks to potentially luminescent spin-flip states. We identified two V²⁺ complexes [V(ddpd)₂ ]²⁺ with the strong field ligand N,N'-dimethyl-N,N'-dipyridine-2-yl-pyridine-2,6-diamine (ddpd) in two isomeric forms (cis-fac and mer) as suitable candidates. The energy gaps between the two lowest Kramers doublets amount to 0.2 and 0.5 cm^-1 allowing pulsed EPR experiments at conventional Q-band frequencies (35 GHz). Both isomers possess spin-lattice relaxation times T₁ of around 300 μs and a phase memory time T_M of around 1 μs at 5 K. Furthermore, the mer isomer displays slow magnetic relaxation in an applied field of 400 mT. While the vanadium(III) complexes [V(ddpd)₂ ]³⁺ are emissive in the near-IR-II region, the [V(ddpd)₂ ]²⁺ complexes are non-luminescent due to metal-to-ligand charge transfer admixture to the spin-flip states.

Computational Insights into Electronic Excitations, Spin-Orbit Coupling Effects, and Spin Decoherence in Cr(IV)-Based Molecular Qubits

The great success of point defects and dopants in semiconductors for quantum information processing has invigorated a search for molecules with analogous properties. Flexibility and tunability of desired properties in a large chemical space have great advantages over solid-state systems. The properties analogous to point defects were demonstrated in the Cr(IV)-based molecular family, Cr(IV)(aryl)₄, where the electronic spin states were optically initialized, read out, and controlled. Despite this kick-start, there is still a large room for enhancing properties crucial for molecular qubits. Here, we provide computational insights into key properties of the Cr(IV)-based molecules aimed at assisting the chemical design of efficient molecular qubits. Using the multireference ab initio methods, we investigate the electronic states of Cr(IV)(aryl)₄ molecules with slightly different ligands, showing that the zero-phonon line energies agree with the experiment and that the excited spin-triplet and spin-singlet states are highly sensitive to small chemical perturbations. By adding spin-orbit interaction, we find that the sign of the uniaxial zero-field splitting (ZFS) parameter is negative for all considered molecules and discuss optically induced spin initialization via non-radiative intersystem crossing. We quantify (super)hyperfine coupling to the ⁵³Cr nuclear spin and to the ¹³C and ¹H nuclear spins, and we discuss electron spin decoherence. We show that the splitting or broadening of the electronic spin sub-levels due to superhyperfine interaction with ¹H nuclear spins decreases by an order of magnitude when the molecules have a substantial transverse ZFS parameter.

Understanding the effect of structural changes on slow magnetic relaxation in mononuclear octahedral copper(II) complexes

Current advances in molecular magnetism are aimed at the construction of molecular nanomagnets and spin qubits for their utilization as high-density data storage materials and quantum computers. Mononuclear coordination compounds with low spin values of S = ½ are excellent candidates for this endeavour, but knowledge of their construction via rational design is limited. This particularly applies to the single copper(II) spin center, having been only recently demonstrated to exhibit slow relaxation of magnetisation in the appropriate octahedral environment. We have thus prepared a unique organic scaffold that would allow one to gain in-depth insight into how purposeful structural differences affect the slow magnetic relaxation in monometallic, transition metal complexes. As a proof-of-principle, we demonstrate how one can construct two, structurally very similar complexes with isolated Cu(II) ions in an octahedral ligand environment, the magnetic properties of which differ significantly. The differences in structural symmetry effects and in magnetic relaxation are corroborated with a series of experimental techniques and theoretical approaches, showing how symmetry distortions and crystal packing affect the relaxation behaviour in these isolated Cu(II) systems. Our unique organic platform can be efficiently utilized for the construction of various transition-metal ion systems in the future, effectively providing a model system for investigation of magnetic relaxation via targeted structural distortions.

Mechanism of Electron Spin Decoherence in a Partially Deuterated Glassy Matrix

Long electron spin coherence lifetimes are essential for applications in quantum information science and electron paramagnetic resonance, for instance, for nanoscale distance measurements in biomolecular systems using double electron-electron resonance. We experimentally investigate the decoherence dynamics under the Hahn echo sequence of the organic radical d₁₈-TEMPO in a variably deuterated frozen water:glycerol matrix. The coherence time (phase memory time) T_M scales with proton concentration as [¹H]^-0.65. For selectively deuterated matrices, decoherence is accelerated in the presence of proton clustering, that is, substantial short-range density in the proton-proton radial distribution functions (<3 Å). Simulations using molecular dynamics and many-body spin quantum dynamics show excellent agreement with experiment and show that geminal proton pairs such as CH₂ and OH₂ groups are major decoherence drivers. This provides a predictive tool for designing molecular systems with long electron spin coherence times.

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.

Near Isotropic D4d Spin Qubits as Nodes of a Gd(III)-Based Metal-Organic Framework

Embedding coherent spin motifs in reproducible molecular building blocks is a promising pathway for the realization of quantum technologies. Three-dimensional (3D) MOFs are a versatile platform for the rational design of extended structures employing coordination chemistry. Here, we report the synthesis and characterization of a gadolinium(III)-based MOF, [Gd(bipyNO)₄](TfO)₃·xMeOH (bipyNO = bipyridine,N,N'-dioxide; TfO = triflate; and MeOH = methanol) (quMOF-1), which presents a unique coordination geometry that leads to a tiny magnetic anisotropy (in terms of D, an equivalent zero-field splitting would be achieved by D = 0.006 cm^-1) even compared with regular Gd(III) complexes. Pulsed electron paramagnetic resonance experiments on its magnetically diluted samples confirm the preservation of quantum coherence of single Gd(III) qubit units in this 3D extended molecular architecture (T₂ = 612 ns and T₁ = 66 μs at 3.5 K), which allows for the detection of Rabi oscillations at 40 K.

The decay of the refocused Hahn echo in double electron-electron resonance (DEER) experiments

Double electron-electron resonance (DEER) is a pulse electron paramagnetic resonance (EPR) technique that measures distances between paramagnetic centres. It utilizes a four-pulse sequence based on the refocused Hahn spin echo. The echo decays with increasing pulse sequence length 2 ( τ 1 + τ 2 ) , where τ 1 and τ 2 are the two time delays. In DEER, the value of τ 2 is determined by the longest inter-spin distance that needs to be resolved, and τ 1 is adjusted to maximize the echo amplitude and, thus, sensitivity. We show experimentally that, for typical spin centres (nitroxyl, trityl, and Gd(III)) diluted in frozen protonated solvents, the largest refocused echo amplitude for a given τ 2 is obtained neither at very short τ 1 (which minimizes the pulse sequence length) nor at τ 1 = τ 2 (which maximizes dynamic decoupling for a given total sequence length) but rather at τ 1 values smaller than τ 2 . Large-scale spin dynamics simulations based on the coupled cluster expansion (CCE), including the electron spin and several hundred neighbouring protons, reproduce the experimentally observed behaviour almost quantitatively. They show that electron spin dephasing is driven by solvent protons via the flip-flop coupling among themselves and their hyperfine couplings to the electron spin.

Magnetic anisotropy in YbIII complex candidates for molecular qubits: a theoretical analysis

The magnetic properties of mononuclear YbIII complexes have been explored by using multiconfigurational CASPT2/RASSI calculations. Such complexes, in particular the case of [Yb(trensal)] complex, have been proposed as molecular qubits due to their spin dynamics properties. We have verified the accuracy of the theoretical approach to study such systems by comparing with experimental magnetic data. In order to have a wide overview of the magnetic properties of mononuclear YbIII complexes, we have considered simple charged and neutral models, [Yb(H2O)n]3+ and [Yb(OH)3(H2O)n-3], for many coordination modes. Thus, the results for more than 100 models allow extraction of some conclusions about the best ligand distributions in the coordination sphere to tailor the magnetic properties. Some low coordination, between 3 and 5, complexes that have no experimental magnetic data have been studied computationally to check if they can present high magnetic anisotropy.

Exploring the Organometallic Route to Molecular Spin Qubits: The [CpTi(cot)] Case

The coherence time of the 17-electron, mixed sandwich complex [CpTi(cot)], (η⁸ -cyclooctatetraene)(η⁵ -cyclopentadienyl)titanium, reaches 34 μs at 4.5 K in a frozen deuterated toluene solution. This is a remarkable coherence time for a highly protonated molecule. The intramolecular distances between the Ti and H atoms provide a good compromise between instantaneous and spin diffusion sources of decoherence. Ab initio calculations at the molecular and crystal packing levels reveal that the characteristic low-energy ring rotations of the sandwich framework do not yield a too detrimental spin-lattice relaxation because of their small spin-phonon coupling. The volatility of [CpTi(cot)] and the accessibility of the semi-occupied, non-bonding d z 2 orbital make this neutral compound an ideal candidate for single-qubit addressing on surface and quantum sensing in combination with scanning probe microscopy.

Unusually large hyperfine structure of the electron spin levels in an endohedral dimetallofullerene and its spin coherent properties

We report the synthesis, ESR spectroscopic and spin coherent properties of the dimetallofullerene Sc2@C80(CH2Ph). The single-electron metal-metal bond of the Sc2 dimer inside the fullerene's cage is stabilized with the electron spin density being fully localized at the metal bond. This results in an extraordinary strong hyperfine interaction of the electron spin with the 45Sc nuclear spins with a coupling constant a = 18.2 mT (∼510 MHz) and yields a fully resolved hyperfine-split ESR spectrum comprising 64 lines. The splitting is present even at low temperatures where the molecular dynamics are completely frozen. The large extent and the robustness of the hyperfine-split spectra enable us to identify and control the well-defined transitions between specific electron-nuclear quantum states. This made it possible to demonstrate in our pulse ESR study the remarkable spin coherent dynamics of Sc2@C80(CH2Ph), such as the generation of arbitrary superpositions of the spin states in a nutation experiment and the spin dephasing times above 10 μs at temperatures T < 80 K reaching the value of 17 μs at T ≤ 20 K. These observations suggest Sc2@C80(CH2Ph) as an interesting qubit candidate and motivate further synthetic efforts to obtain fullerene-based systems with superior spin properties.

Spin dynamics in single-molecule magnets and molecular qubits

Over recent decades, much effort has been made to lengthen spin relaxation/decoherence times of single-molecule magnets and molecular qubits by following different chemical design rules such as maximizing the total spin value, controlling symmetry, enhancing the ligand field or inhibiting key vibrational modes. Simultaneously, electronic structure calculations have been employed to provide an understanding of the processes involved in the spin dynamics of molecular systems and served to refine or introduce new design rules. This review focuses on contemporary theoretical approaches focused on the calculation of spin relaxation/decoherence times, highlighting their main features and scope. Fundamental aspects of experimental techniques for the determination of key Single Molecule Magnet/Spin Qubit properties are also reviewed.

Quantitative Structure-Based Prediction of Electron Spin Decoherence in Organic Radicals

The decoherence, or dephasing, of electron spins in paramagnetic molecules limits sensitivity and resolution in electron paramagnetic resonance spectroscopy, and it represents a challenge for utilizing paramagnetic molecules as qubit units in quantum information devices. For organic radicals in dilute frozen aqueous solution at cryogenic temperatures, electron spin decoherence is driven by neighboring nuclear spins. Here, we show that this nuclear-spin-driven decoherence can be quantitatively predicted from the molecular structure and solvation geometry of the radicals. We use a fully deterministic quantum model of the electron spin and up to 2000 neighboring protons with a static spin Hamiltonian that includes nucleus-nucleus couplings. We present experiments and simulations of two nitroxide radicals and one trityl radical, which have decoherence time scales of 4-5 μs below 60 K. We show that nuclei within 12 Å of the electron spin contribute to decoherence, with the strongest impact from protons 4-8 Å away.

Decoherence in Molecular Electron Spin Qubits: Insights from Quantum Many-Body Simulations

Quantum states are described by wave functions whose phases cannot be directly measured but which play a vital role in quantum effects such as interference and entanglement. The loss of the relative phase information, termed decoherence, arises from the interactions between a quantum system and its environment. Decoherence is perhaps the biggest obstacle on the path to reliable quantum computing. Here we show that decoherence occurs even in an isolated molecule, although not all phase information is lost, via a theoretical study of a central electron spin qubit interacting with nearby nuclear spins in prototypical magnetic molecules. The residual coherence, which is molecule-dependent, provides a microscopic rationalization for the nuclear spin diffusion barrier proposed to explain experiments. The contribution of nearby molecules to the decoherence has a nontrivial dependence on separation, peaking at intermediate distances. Molecules that are far away affect only the long-time behavior. Because the residual coherence is simple to calculate and correlates well with the coherence time, it can be used as a descriptor for coherence in magnetic molecules. This work will help establish design principles for enhancing coherence in molecular spin qubits and serve to motivate further theoretical work.

Nuclear-spin-pattern control of electron-spin dynamics in a series of V(iv) complexes

Achieving control of phase memory relaxation times (T m) in metal ions is an important goal of molecular spintronics. Herein we provide the first evidence that nuclear-spin patterning in the ligand shell is an important handle to modulate T m in metal ions. We synthesized and studied a series of five V(iv) complexes with brominated catecholate ligands, [V(C6H4-n Br n O2)3]2- (n = 0, 1, 2, and 4), where the 79/81Br and 1H nuclear spins are arranged in different substitutional patterns. High-field, high-frequency (120 GHz) pulsed electron paramagnetic resonance spectroscopic analysis of this series reveals a pattern-dependent variation in T m for the V(iv) ion. Notably, we show that it is possible for two molecules to have starkly different (by 50%) T m values despite the same chemical composition. Nuclear magnetic resonance analyses of the protons on the ligand shell suggest that relative chemical shift (δ), controlled by the patterning of nuclear spins, is an important underlying design principle. Here, having multiple ligand-based protons with nearly identical chemical shift values in the ligand shell will, ultimately, engender a short T m for the bound metal ion.

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.

Chromium(iii)-based potential molecular quantum bits with long coherence times

Molecular quantum bits based on copper(ii) or vanadium(iv) have been shown to possess long coherence times on multiple occasions. In contrast, studies in which non-spin-½ ions are employed are relatively scarce. High-spin ions provide additional states that can be used to encode further quantum bits. Furthermore, an optical rather than a microwave readout of molecular quantum bits is highly desirable, because in principle it could allow addressing at the single quantum bit level. The chromium(iii) complex [Cr(ddpd)2]3+ (ddpd = N,N'-dimethyl-N,N'-dipyridine-2-yl-pyridine-2,6-diamine) combines both the large spin (S = 3/2) and optical activity (strong, long lived luminescence). Here we demonstrate that the compound possesses coherence times of up to 8.4(1) μs, which are much longer (at least three times) than those for other chromium(iii)-based compounds. On the other hand, it is proved to be impossible to read out or influence the quantum state by optical means, underlining that further work is needed in this direction.

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.

Measurement of quantum coherence in thin films of molecular quantum bits without post-processing

A novel Fabry–Pérot pulsed EPR resonator with very good microwave magnetic field homogeneity allows facile measurement of thin films of molecular quantum bits.

A graphene-based hybrid material with quantum bits prepared by the double Langmuir–Schaefer method

The scalability and stability of molecular qubits deposited on surfaces is a crucial step for incorporating them into upcoming electronic devices. Herein, we report on the preparation and characterisation of a molecular quantum bit, copper(ii)dibenzoylmethane [Cu(dbm)₂], deposited by a modified Langmuir–Schaefer (LS) technique onto a graphene-based substrate. A double LS deposition was used for the preparation of a few-layer-graphene (FLG) on a Si/SiO₂ substrate with subsequent deposition of the molecules. Magnetic properties were probed by high-frequency electron spin resonance (HF-ESR) spectroscopy and found maintained after deposition. Additional spectroscopic and imaging techniques, such as Raman spectroscopy (RS), X-ray photoelectron spectroscopy (XPS), atomic force microscopy (AFM), and scanning electron microscopy (SEM) were performed to characterise the deposited sample. Our approach demonstrated the possibility to utilise a controlled wet-chemistry protocol to prepare an array of potential quantum bits on a disordered graphene-based substrate. The deployed spectroscopic techniques showed unambiguously the robustness of our studied system with a potential to fabricate large-scale, intact, and stable quantum bits.

Graphene-based hybrid material with array of copper(ii)-based quantum bits was prepared by a wet-chemistry protocol and characterised by HF-ESR, XPS, Raman, and AFM.

Ligand Radicals as Modular Organic Electron Spin Qubits

The intrinsic redox activity of the dithiolene ligand is presented here as the novel spin host in the design of a prototype molecular electron spin qubit, where the traditional roles of the metal and ligand components in coordination complexes are inverted. A series of paramagnetic bis(dithiolene) complexes with group 10 metals-nickel, palladium, platinum-provides a backdrop to investigate the spin dynamics of the organic ligand radical using pulsed EPR spectroscopy. The temperature dependence of the phase memory time (T_M ) is shown to be dependent on the identity of the diamagnetic metal ion, with the short times recorded for platinum a consequence of a diminishing spin-lattice (T₁ ) relaxation time driven by spin-orbit coupling. The utility of the radical ligand spin center is confirmed when it delivers one of the longest phase memory times ever recorded for a molecular two-qubit prototype.

Spin states, vibrations and spin relaxation in molecular nanomagnets and spin qubits: a critical perspective

Very recently the closely related fields of molecular spin qubits, single ion magnets and single atom magnets have been shaken by unexpected results. We have witnessed a jump in the phase memory times of spin qubits from a few microseconds to almost a millisecond in a vanadium complex, magnetic hysteresis up to 60 K in a dysprosium-based magnetic molecule and magnetic memory up to 30 K in a holmium atom deposited on a surface. With single-molecule magnets being more than two decades old, this rapid improvement in the physical properties is surprising and its explanation deserves urgent attention. The general assumption of focusing uniquely on the energy barrier is clearly insufficient to model magnetic relaxation. Other factors, such as vibrations that couple to spin states, need to be taken into account. In fact, this coupling is currently recognised to be the key factor that accounts for the slow relaxation of magnetisation at higher temperatures. Herein we will present a critical perspective of the recent advances in molecular nanomagnetism towards the goal of integrating spin-phonon interactions into the current computational methodologies of spin relaxation. This presentation will be placed in the context of the well-known models developed in solid state physics, which, as we will explain, are severely limited for molecular systems.