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As scientists living through a climate emergency, we have a responsibility to lead by example, or to at least be consistent with our understanding of the problem. This common goal of reducing the carbon footprint of our work can be approached through a variety of strategies. For theoreticians, this includes not only optimizing algorithms and improving computational efficiency but also adopting a frugal approach to modeling. Here we present and critically illustrate this principle. First, we compare two models of very different level of sophistication which nevertheless yield the same qualitative agreement with an experiment involving electric manipulation of molecular spin qubits while presenting a difference in cost of $>4$ orders of magnitude. As a second stage, an already minimalistic model of the potential use of single-ion magnets to implement a network of probabilistic p-bits, programmed in two different programming languages, is shown to present a difference in cost of a factor of $\simeq 50$. In both examples, the computationally expensive version of the model was the one that was published. As a community, we still have a lot of room for improvement in this direction.

Over the decades, the spin dynamics of a large set of lanthanide complexes have been explored. Lanthanide-based molecular nanomagnets are bistable spin systems, generally conceptualized as classical bits, but many lanthanide complexes have also been presented as candidate quantum bits (qubits). Here we offer a third alternative and model them as probabilistic bits (p-bits), where their stochastic behavior constitutes a computational resource instead of a limitation. We present a modelling tool for molecular spin p-bits, we demonstrate its capability to simulate bulk magnetic relaxation data and ac experiments and to simulate a minimal p-bit network under realistic conditions. Finally, we go back to a recent systematic data gathering and screen the best lanthanide complexes for p-bit behavior, lay out the performance of the different lanthanide ions and chemical families and offer some chemical design considerations.

Enhanced coherence in HoW$_{10}$ molecular spin qubits has been demonstrated by use of Clock Transitions (CTs). More recently it was shown that, while operating at the CTs, it was possible to use an electrical field to selectively address HoW$_{10}$ molecules pointing in a given direction, within a crystal that contains two kinds of identical but inversion-related molecules. Herein we theoretically explore the possibility of employing the electric field to effect entangling two-qubit quantum gates among two neighbouring CT-protected HoW$_{10}$ qubits within a diluted crystal. We estimate the thermal evolution of $T_1$, $T_2$, find that CTs are also optimal operating points from the point of view of phonons, and lay out how to combine a sequence of microwave and electric field pulses to achieve coherent control within a 2-qubit operating space that is protected both from spin-bath and from phonon-bath decoherence. Finally, we found a highly protected 1-qubit subspace resulting from the interaction between two clock molecules., Comment: updated with minor corrections

The ability to design quantum systems that decouple from environmental noise sources is highly desirable for development of quantum technologies with optimal coherence. The chemical tunability of electronic states in magnetic molecules combined with advanced electron spin resonance techniques provides excellent opportunities to address this problem. Indeed, so-called clock transitions (CTs) have been shown to protect molecular spin qubits from magnetic noise, giving rise to significantly enhanced coherence. Here we conduct a spectroscopic and computational investigation of this physics, focusing on the role of the nuclear bath. Away from the CT, linear coupling to the nuclear degrees of freedom causes a modulation and decay of electronic coherence, as quantified via electron spin echo signals generated experimentally and $\textit{in silico}$. Meanwhile, the effective hyperfine interaction vanishes at the CT, resulting in electron-nuclear decoupling and an absence of quantum information leakage to the nuclear bath, providing opportunities to characterize other decoherence sources., Comment: 29 pages, 4 figures

To design molecular spin qubits with enhanced quantum coherence, a control of the coupling between the local vibrations and the spin states is crucial, which could be realized in principle by engineering molecular structures via coordination chemistry. To this end, understanding the underlying structural factors that govern the spin relaxation is a central topic. Here, we report the investigation of the spin dynamics in a series of chemically-designed europium(II)-based endohedral metallofullerenes (EMFs). By introducing a unique structural difference, i.e. metal-cage binding site, while keeping other molecular parameters constant between different complexes, these manifest the key role of the three low energy metal-based vibrations in mediating the spin-lattice relaxation times (T1). The temperature dependence of T1 can thus be normalized by the frequencies of these low energy vibrations to show an unprecedentedly universal behavior for EMFs in frozen CS2 solution. Our theoretical analysis indicates that this structural difference determines not only the vibrational rigidity but also spin-vibration coupling in these EMF-based qubit candidates.

Spin-lattice relaxation is a key open problem to understand the spin dynamics of single-molecule magnets and molecular spin qubits. While modelling the coupling between spin states and local vibrations allows to determine the more relevant molecular vibrations for spin relaxation, this is not sufficient to explain how energy is dissipated towards the thermal bath. Herein, we employ a simple and efficient model to examine the coupling of local vibrational modes with long-wavelength longitudinal and transverse phonons in the clock-like spin qubit [Ho(W$_5$O$_{18}$)$_2$]$^{9-}$. We find that in crystals of this polyoxometalate the vibrational mode previously found to be vibronically active at low temperature does not couple significantly to lattice phonons. This means that further intramolecular energy transfer via anharmonic vibrations is necessary for spin relaxation in this system. Finally, we discuss implications for the spin-phonon coupling of [Ho(W$_5$O$_{18}$)$_2$]$^{9-}$ deposited on a MgO (001) substrate, offering a simple methodology that can be extrapolated to estimate the effects on spin relaxation of different surfaces, including 2D materials.

Three decades of research in molecular nanomagnets have raised their magnetic memories from liquid helium to liquid nitrogen temperature thanks to a wise choice of the magnetic ion and coordination environment. Still, serendipity and chemical intuition played a main role. In order to establish a powerful framework for statistically driven chemical design, we collected chemical and physical data for lanthanide-based nanomagnets, catalogued over 1400 published experiments, developed an interactive dashboard (SIMDAVIS) to visualise the dataset, and applied inferential statistical analysis. Our analysis showed that the Arrhenius energy barrier correlates unexpectedly well with the magnetic memory, as both Orbach and Raman processes can be controlled by vibronic coupling. Indeed, only bis-phthalocyaninato sandwiches and metallocenes, with rigid ligands, consistently present magnetic memory up to high temperature. Analysing magnetostructural correlations, we offer promising strategies for improvement, in particular for the preparation of pentagonal bipyramids, where even "softer" complexes are protected against molecular vibrations., Comment: 109 pages (main text+SI). Related content: interactive dashboard for Single Ion Magnet Data Visualization. https://go.uv.es/rosaleny/SIMDAVIS

Molecular vibrations play a key role in magnetic relaxation processes of molecular spin qubits as they couple to spin states, leading to the loss of quantum information. Direct experimental determination of vibronic coupling is crucial to understand and control the spin dynamics of these nano-objects, which represent the limit of miniaturization for quantum devices. Herein, we measure the vibrational properties of the molecular spin qubit $[$Ho(W$_5$O$_{18}$)$_2]^{9-}$ by means of magneto-infrared spectroscopy. Our results allow us to unravel the vibrational decoherence pathways in combination with $ab$ $initio$ calculations including vibronic coupling. We observe field-induced spectral changes near 63 and 370 cm$^{-1}$ that are modeled in terms of $f$-manifold crystal field excitations activated by odd-symmetry vibrations. The overall extent of vibronic coupling in this system is limited by a transparency window in the phonon density of states that acts to keep the intramolecular vibrations and $M_J$ levels apart. These findings advance the understanding of vibronic coupling in molecular magnets, place significant constraints on the pattern of crystal field levels in these systems, and provide a strategy for designing molecular spin qubits with improved coherence lifetimes., Comment: 22 pages

Electrical control of spins at the nanoscale offers significant architectural advantages in spintronics, because electric fields can be confined over shorter length scales than magnetic fields. Thus, recent demonstrations of electric-field (E-field) sensitivities in molecular spin materials are tantalising, raising the viability of the quantum analogues of macroscopic magneto-electric devices.However, the E-field sensitivities reported so far are rather weak, prompting the question of how to design molecules with stronger spin-electric couplings. Here we show that one path is to identify an energy scale in the spin spectrum that is associated with a structural degree of freedom with a significant electrical polarisability. We study an example of a molecular nanomagnet in which a small structural distortion establishes clock transitions (i.e. transitions whose energy is to first order independent of magnetic field) in the spin spectrum; the fact that this distortion is associated with an electric dipole allows us to control the clock transition energy to an unprecedented degree. We demonstrate coherent electrical control of the quantum spin state and exploit it to manipulate independently the two magnetically-identical but inversion-related molecules in the unit cell of the crystal. Our findings pave the way for the use of molecular spins in quantum technologies and spintronics.

A central goal in quantum technologies is to maximize $G$T$_{2}$, where $G$ stands for the coupling of a qubit to control and readout signals and T$_{2}$ is the qubit's coherence time. This is challenging, as increasing $G$ (e.g. by coupling the qubit more strongly to external stimuli) often leads to deleterious effects on T$_{2}$. Here, we study the coupling of pure and magnetically diluted crystals of HoW$_{10}$ magnetic clusters to microwave superconducting coplanar waveguides. Absorption lines give a broadband picture of the magnetic energy level scheme and, in particular, confirm the existence of level anticrossings at equidistant magnetic fields determined by the combination of crystal field and hyperfine interactions. Such 'spin clock transitions' are known to shield the electronic spins against magnetic field fluctuations. The analysis of the microwave transmission shows that the spin-photon coupling becomes also maximum at these transitions. The results show that engineering spin-clock states of molecular systems offers a promising strategy to combine sizeable spin-photon interactions with a sufficient isolation from unwanted magnetic noise sources., Comment: 11 pages, 10 figures

The anchoring of the molecule to an electrode is known to be a key factor in single-molecule spintronics experiments. Likewise, a relaxation down to the most stable geometry is a critical step in theoretical simulations of transport through single-molecule junctions. Herein we present a set of calculations designed to analyze and compare the effect of different anchoring points and the effect of perturbations in the molecular geometry and interelectrode distance. As model system we chose the [V($\alpha$-C$_3$S$_5$)$_3$]$^{2-}$ complex connecting two Au(111) electrodes in a slightly compressed geometry. In our calculations, the attachment happens through an S-Au bond, a common anchoring strategy in molecular spintronics experiments. Our results confirm that small alterations in the molecular geometry have important effects in the conductance. We were able to compare these effects with the ones arising from changing the anchoring position with a constant molecular geometry. Unexpectedly, we demonstrate that the anchoring position has only a lesser relevance in the spintronic behavior of the device, as long as all other parameters are kept frozen. As a consequence, we predict that for experimentalists aiming for reproducibility, the molecular design of rigid linkers is more relevant than the design of univocal anchoring positions.

Chiral-induced spin selectivity (CISS), whereby helical molecules polarize the spin of electrical current, is an intriguing effect with potential applications in nanospintronics. In this nascent field, the study of paramagnetic chiral molecules, which could introduce another degree of freedom in the control of the spin transport, remains so far unexplored. To address this challenge, herein, we propose the use of self-assembled monolayers of helical lanthanide-binding peptides. In order to elucidate the effect of the paramagnetic nuclei, monolayers of the peptide coordinating paramagnetic or diamagnetic ions are prepared. By means of spin-dependent electrochemistry, CISS effect is demonstrated by cyclic voltammetry and impedance measurements for both samples. Additionally, an implementation of the standard liquid-metal drop electron transport setup has been carried out, demonstrating their suitability for solid-state devices. Remarkably, the inclusion of a paramagnetic center in the peptide increases the spin polarization as independently proved by different techniques. These findings permit the inclusion of magnetic biomolecules in the CISS field, paving the way to their implementation in a new generation of spintronic nanodevices.

Molecular vibrations are increasingly seen as a key factor for spin dynamics in single-ion magnets and molecular spin qubits. Herein we show how an inexpensive combination of molecular dynamics calculations and a crystal field analysis can be employed to obtain a dynamical picture of the crystal field splitting. We present the time evolution during 500ns of the spin energy levels in a terbium complex. Our calculations evidence an amplitude of up to tens of cm-1 for the oscillations of the spin energy levels in the fs time scale. Crucially, we also see that the oscillations average out and practically disappear at longer time scales, thus ruling out the modulation of spin energy levels by phonons of long wavelength $\lambda$ (up to $\lambda$ = 40 $\AA$). Our results are compatible with the common approximation of focusing on local vibrations, but at the same time highlight the risks of assuming that the spin energy levels are time- and temperature-independent., Comment: 5 pages main text + 7 pages Supporting Information

The role of Chemistry in the road towards quantum devices is the design of elementary pieces with a built-in function. A brilliant example is the use of molecular transistors as nuclear spin detectors, which, up to now, has been implemented only on [TbPc$_2$]$^-$. We argue that this is an artificial constraint and critically discuss the limitations of current theoretical approaches to assess the potential of molecules for their use in spintronics. In connection with this, we review the recent progress in the preparation of highly coherent spin qubits based on vanadium dithiolate complexes and argue that the use of vanadyl dithiolates as single molecule transistors to read and control a triple nuclear spin qubit could give rise to new phenomena, notably including a low-current nuclear spin detection scheme by means of a spin valve effect.

Polynuclear magnetic molecules often present dense transmission spectra with many overlapping conduction spin channels. Single-metal complexes display a sparser density of states, which in the presence of a fixed external magnetic field makes them interesting candidates for spin filtering. Here we perform a DFT study of a family of bis- and tris-dithiolate vanadium complexes sandwiched between Au(111) electrodes and demonstrate that [VO(C$_3$S$_4$O)$_2$]$^{2-}$ can behave as a dual spin filter. This means that an external stimulus can switch between the selective transmission of spin-up and spin-down carriers. By using an electrostatic gate as external stimulus we show that the onset for the spin-up conductance is at a voltage V$_g$ = -0.51 V but a small shift to V$_g$ = -0.63 V is capable of activating spin-down transport. For both cases, we estimate a large low-bias conductance (approx. 2 {\mu}S at Vbias < 50 mV) with excellent spin selectivity (> 99.5%). We conclude by commenting on the general molecular requirements for the chemical design of further such spintronics components.

In this work we showcase the potential of peptides as versatile scaffolds for quantum computing and molecular spintronics. In particular, we focus on lanthanide-binding tags, which were originally developed in the field of biotechnology for the study of protein structure and dynamics. Firstly, we demonstrate quantum coherent oscillations in a Neodymium peptidic qubit. Then, employing bacterial biosynthesis, we investigate the possibility of increasing the number of qubits in the same molecular system, with the case studies being a double spin qubit with two distinct coordination environments, and an asymmetric chain of 9 spin qubits with a spin-spin separation of about 2 nm and in an arbitrarily chosen sequence of coordination environments. Finally, we take advantage of biochemical modification for the preparation of paramagnetic, chiral, Self-Assembled Monolayers (SAMs) on Au(111).Our experimental and theoretical characterization shows that this is a promising structure for spintronic applications, and in particular to improve on two state-of-the-art approaches to molecular spin qubits. We conclude with an overview of the challenges and new opportunities opened by this emerging field., Comment: Manuscript from LaTeX, Supporting Information as pdf from LibreOffice. Update: minor changes, improved figures

In a straightforward application of molecular nanospintronics to quantum computing, single-molecule spin transistors can be used to measure and control nuclear spin qubits. A jump in the conductance occurs when the electronic spin inverts its polarization, and this happens at a so-called anticrossing between energy levels, which in turn only takes place at a specific magnetic field determined by the nuclear spin state. So far, this procedure has only been implemented for the terbium(III) bis(phthalocyaninato) complex. Here we explore theoretically whether a similar behavior is expected for a highly stable molecular spin qubit, the vanadium tris-dithiolate complex [V$^{IV}$({\alpha}-C$_3$S$_5$)]${^2-}$. We consider such molecule sandwiched into a two-terminal device and determine the spin-dependent conductance. We verify that the transport channel at minimal bias voltage does not overlap with the occupied spin orbitals, indicating that the spin states may survive in the conduction regime. We estimate some physical parameters to guide the experiments, and verify the robustness of the theoretical methodology by applying it to two chemically related vanadium complexes.

We show that several qubits can be integrated in a single magnetic ion, using its internal electronic spin states with energies tuned by a suitably chosen molecular environment. This approach is illustrated with a nearly-isotropic Gd(III) ion entrapped in a polyoxometalate molecule. Experiments with microwave technologies, either three dimensional cavities or quantum superconducting circuits, show that this magnetic molecule possesses the number of spin states and the set of coherently addressable transitions connecting these states that are needed to perform a universal three-qubit processor or, equivalently, a d=8-level 'qudit'. Our findings open prospects for developing more sophisticated magnetic molecules which can result in more powerful and noise resilient quantum computation schemes., Comment: 5 pages, 4 figures

Herein we develop a simple first-principles methodology to determine the modulation that vibrations exert on spin energy levels, a key for the rational design of high-temperature molecular spin qubits and single-molecule magnets. This methodology is demonstrated by applying it to [Cu(mnt)$_2$]$^{2-}$ (mnt$^{2-}$ = 1,2-dicyanoethylene-1,2-dithiolate), a highly coherent complex, using DFT to calculate the normal vibrational modes and wave-function based theory calculations to estimate the spin energy level structure. By theoretically identifying the most relevant vibrational modes, we are able to offer general strategies to chemically design more resilient magnetic molecules, where the qubit energy is not coupled to local vibrations.

This is the installation and used manual of SIMPRE 1.1, a fortran77 code that can implement different effective electrostatic models based on point charges around a rare earth ion. The program calculates the full set of crystal field parameters, Stark energy levels and spin wave functions, as well as magnetic properties. The package has already been successfully applied to several mononuclear systems with single-molecule magnetic behavior. The determination of effective point charge parameters in these studies facilitates its application to new systems. SIMPRE was presented in DOI:10.1002/jcc.23341, and SIMPRE 1.1, an updated version that follows the prevailing Crystal Field conventions was presented in DOI:10.1002/jcc.23699.

The bias energies of various two-level systems (TLSs) and their strengths of interactions with the strain are calculated for Ar:N$_2$ glass. Unlike the case in KBr:CN, a distinct class of TLSs having weak interaction with the strain and untypically small bias energies is not found. The addition of CO molecules introduces CO flips which form such a class of weakly interacting TLSs, albeit at much lower coupling than are typically observed in solids. We conclude that because of the absence of a distinct class of weakly interacting TLSs, Ar:N$_2$ is a non-universal glass, the first such system in three dimensions and in ambient pressure. Our results further suggest that Ar:N$_2$:CO may show universal properties, but at temperatures lower than $\approx 0.1$ K, much smaller than typical temperature $\approx 3$ K associated with universality, because of the untypical softness of this system. Our results thus shed light on two long standing questions regarding low temperature properties of glasses: the necessary and sufficient conditions for quantitative universality of phonon attenuation, and what dictates the energy scale of $\approx 3$ K below which universality it typically observed., Comment: 6 pages, 2 figures, results for excitations densities of states are added, presentation improved

Quantum algorithms often assume independent spin qubits to produce trivial $|\uparrow\rangle=|0\rangle$, $|\downarrow\rangle=|1\rangle$ mappings. This can be unrealistic in many solid-state implementations with sizeable magnetic interactions. Here we show that the lower part of the spectrum of a molecule containing three exchange-coupled metal ions with $S=1/2$ and $I=1/2$ is equivalent to nine electron-nuclear qubits. We derive the relation between spin states and qubit states in reasonable parameter ranges for the rare earth $^{159}$Tb$^{3+}$ and for the transition metal Cu$^{2+}$, and study the possibility to implement Shor's Quantum Error Correction code on such a molecule. We also discuss recently developed molecular systems that could be adequate from an experimental point of view., Comment: 5 pages, 3 figures, 2 tables

Polyoxometalate single ion magnet [GdW$_{30}$P$_5$O$_{110}$]$^{12-}$ has been studied by generalized Rabi oscillation experiments. It was possible to increase the number of coherent rotations tenfold through matching the Rabi frequency with the frequency of the proton. Achieving high coherence with polyoxometalate chemistry, we show their excellent potential not only for the storage of quantum information but even for the realization of quantum algorithms., Comment: 4 pages, 3 figures, published in Chem. Commun., 2013 as Advance Article: http://pubs.rsc.org/en/content/articlelanding/2013/cc/c3cc44838j

Tunneling two level systems (TLSs) are believed to be the source of phenomena such as the universal low temperature properties in disordered and amorphous solids, and $1/f$ noise. The existence of these phenomena in a large variety of dissimilar physical systems testifies for the universal nature of the TLSs, which however, is not yet known. Following a recent suggestion that attributes the low temperature TLSs to inversion pairs [M. Schechter and P.C.E. Stamp, arXiv:0910.1283.] we calculate explicitly the TLS-phonon coupling of inversion symmetric and asymmetric TLSs in a given disordered crystal. Our work (a) estimates parameters that support the theory in M. Schechter and P.C.E. Stamp, arXiv:0910.1283, in its general form, and (b) positively identifies, for the first time, the relevant TLSs in a given system., Comment: minor modifications, published version

This introductory review discusses the main problems facing the attempt to build quantum information processing systems (like quantum computers) from spin-based qubits. We emphasize 'bottom-up' attempts using methods from chemistry. The essentials of quantum computing are explained, along with a description of the qubits and their interactions in terms of physical spin qubits. The main problem to be overcome is decoherence - how this works is described, along with ways to suppress contributions from spin bath and oscillator bath environments, and from dipolar interactions. Finally we discuss various strategies for making chemistry-based spin qubits, using both magnetic molecules and magnetic ions., Comment: 13 pages, 10 figures

Spin qubits offer one of the most promising routes to the implementation of quantum computers. Very recent results in semiconductor quantum dots show that electrically-controlled gating schemes are particularly well-suited for the realization of a universal set of quantum logical gates. Scalability to a larger number of qubits, however, remains an issue for such semiconductor quantum dots. In contrast, a chemical bottom-up approach allows one to produce identical units in which localized spins represent the qubits. Molecular magnetism has produced a wide range of systems with tailored properties, but molecules permitting electrical gating have been lacking. Here we propose to use the polyoxometalate [PMo12O40(VO)2]q-, where two localized spins-1/2 can be coupled through the electrons of the central core. Via electrical manipulation of the molecular redox potential, the charge of the core can be changed. With this setup, two-qubit gates and qubit readout can be implemented., Comment: 9 pages, 6 figures, to appear in Nature Nanotechnology