BUSCADOR RECOLECTA

[EN] Heterometallic lanthanide [LnLn'] coordination complexes that are accessible thermodynamically are very scarce because the metals of this series have very similar chemical behaviour. Trinuclear systems of this category have not been reported. A coordination chemistry scaffold has been shown to produce molecules of type [LnLn'Ln] of high purity, i.e. exhibiting high metal distribution ability, based on their differences in ionic radius. Through a detailed analysis of density functional theory (DFT) based calculations, we discern the energy contributions that lead to the unparalleled chemical selectivity of this molecular system. Some of the previously reported examples are compared here with the newly prepared member of this exotic list, [Er2Pr(LA)(2)(LB)(2)(py)(H2O)(2)](NO3) (1) (H(2)LA and H2LB are two beta-diketone ligands). A magnetic analysis extracted from magnetization and calorimetry determinations identifies the necessary attributes for it to act as an addressable, conditional multiqubit spin-based quantum gate. Complementary ab initio calculations confirm the feasibility of these complexes as composite quantum gates, since they present well-isolated ground states with highly anisotropic and distinct g-tensors. The electronic structure of 1 has also been analyzed by EPR. Pulsed experiments have allowed the establishment of the quantum coherence of the transitions within the relevant spin states, as well as the feasibility of a coherent control of these states via nutation experiments., The authors thank the European Union's Horizon 2020 research and innovation programmes QUANTERA project SUMO and FET-OPEN grant 862893 FATMOLS (GA and OR) the Spanish AEI for grants PGC2018-098630-B-I00 (GA and DA), PID2020-118329RB-I00 (OR), PID2020-112806RB-I00 (DGR and CB) and the Severo Ochoa Excellence Accreditation 2020-2023 CEX2019-000925-S (DGR and CB), the ICIQ foundation (DGR and CB) and the Generalitat de Catalunya for a Beatriu de Pinos fellowship (2019-BP-00015, DR), an FI grant (also financed by AGAUR and the European Social Fund, DGR) an FI-SDUR Grant (2020-FISDU-00492, DM), the CERCA program and the ICREA Academia 2018 Prize (GA), as well as the National EPR Facility at the University of Manchester for support with the EPR measurements. This work used resources of the ALBA synchrotron through access to beamline 13-XALOC.

For the past three decades nanoscience has widely affected many areas in physics, chemistry and engineering, and has led to numerous fundamental discoveries, as well as applications and products. Concurrently, quantum science and technology has developed into a cross-disciplinary research endeavour connecting these same areas and holds burgeoning commercial promise. Although quantum physics dictates the behaviour of nanoscale objects, quantum coherence, which is central to quantum information, communication and sensing, has not played an explicit role in much of nanoscience. This Review describes fundamental principles and practical applications of quantum coherence in nanoscale systems, a research area we call quantum-coherent nanoscience. We structure this Review according to specific degrees of freedom that can be quantum-coherently controlled in a given nanoscale system, such as charge, spin, mechanical motion and photons. We review the current state of the art and focus on outstanding challenges and opportunities unlocked by the merging of nanoscience and coherent quantum operations., A.J.H. acknowledges financial support from the Institute for Basic Science under grant number IBS-R027-D1. W.D.O. received funding from the US Army Research Office under grant number W911WF-18-1-0116 and the National Science Foundation under grant number PHY-1720311. L.M.K.V. received funding from the European Research Council (grant number 882848) and A.A. from the UK Engineering and Physical Sciences Research Council (grant number EP/P000479/1) and the European Union’s Horizon 2020 research and innovation programme under grant agreement numbers 863098 and 862893. R.S. was funded by EU-H2020 research project number 862893. A.B.J. received funding from NSF award number QIS-1820938 and the NSF QLCI through grant number OMA-2016245. J.F.-R. was funded by Generalitat Valenciana funding Prometeo 2017/139 and MINECO-Spain (grant number PID2019-109539GB); A.L. by the UNSW Scientia Program; and A.M. by the Australian Research Council (grant numbers CE170100012 and DP180100969), the US Army Research Office (grant number W911NF-17-1-0200) and the Australian Department of Industry, Innovation and Science (grant number AUSMURI00002).

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., This work has received funding from the European Union's Horizon 2020 research and innovation programme (ERC Starting Grant 258060 FuncMolQIP, COST Action 15128 MOLSPIN, QUANTERA project SUMO, FET-OPEN grant 862893 FATMOLS), the Spanish MICINN (grants CTQ2015-68370-P, CTQ2015-64486-R, RTI2018-096075-B-C21, PCI2018-093116, PGC2018-098630-B-I00, MAT2017-86826-R), the Gobierno de Aragón (grants E09-17R-Q-MAD, and PLATON E31_17R), the Generalitat de Catalunya (ICREA Academia 2018 to GA), and from the Italian Ministry of Education and Research (MIUR) through the co-funding of SUMO and through the PRIN Project 2015 HYFSRT “Quantum Coherence in Nanostructures of Molecular Spin Qubits”. Institució Catalana de Recerca I Estudis Avançats: ICREA Academia Prize 2018., Peer reviewed

We combine top-down and bottom-up nanolithography to optimize the coupling of small molecular spin ensembles to 1.4 GHz on-chip superconducting resonators. Nanoscopic constrictions, fabricated with a focused ion beam at the central transmission line, locally concentrate the microwave magnetic field. Drops of free-radical molecules have been deposited from solution onto the circuits. For the smallest ones, the molecules were delivered at the relevant circuit areas by means of an atomic force microscope. The number of spins Neff effectively coupled to each device was accurately determined combining Scanning Electron and Atomic Force Microscopies. The collective spin-photon coupling constant has been determined for samples with Neff ranging between 2 × 106 and 1012 spins, and for temperatures down to 44 mK. The results show the well-known collective enhancement of the coupling proportional to the square root of Neff. The average coupling of individual spins is enhanced by more than 4 orders of magnitude (from 4 mHz up to above 180 Hz), when the transmission line width is reduced from 400 μm down to 42 nm, and reaches maximum values near 1 kHz for molecules located on the smallest nanoconstrictions., The authors acknowledge funding from the EU (COST Action 15128 MOLSPIN, QUANTERA SUMO and MICROSENSE projects, FET-OPEN Grant 862893 FATMOLS), the Spanish MICINN (Grants RTI2018-096075-B-C21, PCI2018-093116, MAT2017-89993-R, MAT2017-88358-C3-1-R, EUR2019-103823), the Gobierno de Aragón Ggrants E09-17R Q-MAD, E35-20R, BE and LMP55-18, FANDEPAM) and the BBVA foundation (Leonardo Grants 2018 and 2019)., Peer reviewed

Includes data relevant for publication with DOI 10.1038/s42004-020-00422-w plus a table with information about how the data were obtained and processed., Published in: Communications Chemistry: 3 pp. 176 1-11., FAult Tolerant MOlecular Spin processor. FET-OPEN project, Peer reviewed

Includes data relevant for publication with DOI 10.1021/acsnano.0c03167 plus a table with information on how the data were obtained and processed., European Commission Grant Agreement no 862893., Peer reviewed

8 pags., 4 figs., Artificial magnetic molecules can contribute to progressing toward large scale quantum computation by (a) integrating multiple quantum resources and (b) reducing the computational costs of some applications. Chemical design, guided by theoretical proposals, allows embedding nontrivial quantum functionalities in each molecular unit, which then acts as a microscopic quantum processor able to encode error protected logical qubits or to implement quantum simulations. Scaling up even further requires “wiring-up” multiple molecules. We discuss how to achieve this goal by the coupling to on-chip superconducting resonators. The potential advantages of this hybrid approach and the challenges that still lay ahead are critically reviewed., We acknowledge funding from the European Union’s Horizon 2020 research and innovation programme (QUANTERA project SUMO, FET-OPEN Grant No. 862893 FATMOLS), the Spanish MICINN (Grant Nos. RTI2018-096075-B-C21, PCI2018-093116, and PGC2018-094792-B-I00), the Gobierno de Aragon (Grant No. E09-17R-Q-MAD), the CSIC Quantum Technology Platform PT-001, and the Italian Ministry of Education and Research (MIUR) through the co-funding of SUMO., Peer reviewed

Bound states in superconductors are expected to exhibit a spatially resolved electron-hole asymmetry which is the hallmark of their quantum nature. This asymmetry manifests as oscillations at the Fermi wavelength, which is usually tiny and thus washed out by thermal broadening or by scattering at defects. Here we demonstrate theoretically and confirm experimentally that, when coupled to magnetic impurities, bound states in a vortex core exhibit an emergent axial electron-hole asymmetry on a much longer scale, set by the coherence length. We study vortices in 2H-NbSe2 and in 2H-NbSe1.8S0.2 with magnetic impurities, characterizing these with detailed Hubbard-corrected density functional calculations. We find that the induced electron-hole imbalance depends on the band character of the superconducting material. Our results show that coupling between quantum bound states in superconductors is remarkably robust and has a strong influence in tunneling measurements., This work was supported by the EU (ERC-StG-679080, ERC-2015-AdG-694097, ERC AdG Mol-2D 788222 and H2020-MSCAIF-2016-751047), by FET-OPEN program (grant 862893 FATMOLS), by EU program Cost (NANOCOHYBRI CA16218 and MOLSPIN CA15128), by Graphene Flagship (grant agreement number 881603, GrapheneCore3 project), by the Spanish State Research Agency (FIS2017-84330-R, RYC-2014-15093, MDM-2014-0377, MDM-2015-0538, CEX2018-000805-M, MAT2017-89993-R co-financed by FEDER and CEX2019-000919-M), by the Deutsche Forschungsgemeinschaft (DFG) under under Germany’s Excellence Strategy— Cluster of Excellence Advanced Imaging of Matter (AIM) EXC 2056—390715994, RTG 1995 and GRK 2247, by the Comunidad de Madrid through program NANOMAGCOST-CM (Program No.S2018/NMT-4321) and by the Generalitat Valenciana (Grupos Consolidados IT1249-19, Prometeo Programme, iDiFEDER/2018/061 and CDEIGENT/2019/022). J.J.B. acknowledges the Marie Curie Fellowship program (H2020-MSCA-IF-2016-751047). A.R. acknowledges support by the MPI-New York City Center for Non-Equilibrium Quantum Phenomena. H.S., E.H., and I.G. acknowledge SEGAINVEX at UAM for design and construction of STM cryogenic equipment. S.P. acknowledges Banco Santander - María de Maeztu program (ref. 102I0112)., Peer reviewed

The design of dissymmetric organic ligands featuring combinations of 1,3-diketone and 2,6-diacetylpyridine coordination pockets has been exploited to produce dinuclear and trinuclear lanthanide-based coordination compounds. These molecules exhibit two or more non-equivalent Ln ions, most remarkably enabling the access to well-defined heterolanthanide compositions. The site-selective disposition of each metal ion within the molecular entities allows the study of each centre individually as a spin-based quantum bit, affording unparalleled versatility for quantum gate design. The inherent weak interaction between the Ln ions permits the performance of multi-qubit quantum logical operations realized through their derived magnetic states, or implementing quantum-error correction protocols. The different studies performed to date on these systems are revised, showing their vast potential within spin-based quantum information processing., The authors thank the European Union's Horizon 2020 research and innovation programmes, the QUANTERA project SUMO, the FET-OPEN FATMOLS (grant 862893; GA and OR), the Spanish MICINN (grants PGC2018-098630-B-I00 (GA and DA) and MAT2017-86826-R (OR)) and the Generalitat de Catalunya for the ICREA Academia 2018 Prize (GA).

This article is part of the themed collection: 2021 Chemical Science HOT Article Collection., 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., This work was supported by funds from the EU (COST Action 15128 MOLSPIN, QUANTERA project SUMO, FET-OPEN grant 862893 FATMOLS), the Spanish MICINN (grants RTI2018-096075-B-C21 and PCI2018-093116) and the Gobierno de Aragón (grant E09-17R-Q-MAD). TM thanks the IUF (Institut Universitaire de France) for financial support., Peer reviewed

The critical current of a superconducting nanostructure can be suppressed by applying an electric field in its vicinity. This phenomenon is investigated throughout the fabrication and electrical characterization of superconducting tungsten-carbon (W-C) nanostructures grown by Ga+ focused ion beam induced deposition (FIBID). In a 45 nm-wide, 2.7 μm-long W-C nanowire, an increasing side-gate voltage is found to progressively reduce the critical current of the device, down to a full suppression of the superconducting state below its critical temperature. This modulation is accounted for by the squeezing of the superconducting current by the electric field within a theoretical model based on the Ginzburg–Landau theory, in agreement with experimental data. Compared to electron beam lithography or sputtering, the single-step FIBID approach provides with enhanced patterning flexibility and yields nanodevices with figures of merit comparable to those retrieved in other superconducting materials, including Ti, Nb, and Al. Exhibiting a higher critical temperature than most of other superconductors, in which this phenomenon has been observed, as well as a reduced critical value of the gate voltage required to fully suppress superconductivity, W-C deposits are strong candidates for the fabrication of nanodevices based on the electric field-induced superconductivity modulation., P.O. acknowledges Aragón Government for funding. The project that gave rise to these results received the support of a fellowship from “la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/PR19/11700008. Funding from the the European Union’s Horizon 2020 research and innovation programme under the grant agreement No 892427 has been received. Authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through Projects MAT2018-102627-T and MAT2017-82970-C2-2-R, from CSIC through project PIE202060E187, and from the Aragón Regional Government (Construyendo Europa desde Aragón) through Project E13_20R, with European Social Fund funding. The microscopy works have been conducted in the Laboratory for Advanced Microscopies (LMA), at the Institute of Nanoscience and Materials of Aragón (INMA)—University of Zaragoza. Authors acknowledge the LMA for offering access to their instruments and expertise. Authors acknowledge the use of the Physical Measurements Service in Servicio General de Apoyo a la Investigación (SAI)—University of Zaragoza. This work has been supported by projects H2020 (FATMOLSproject) and Excellence Unit María de Maeztu (CEX2019-000919-M). This work has been supported by the German Research Foundation (DFG) project #FO 956/6-1 (Germany) and European Cooperation in Science and Technology—COST Action #CA16218 (NANOCOHYBRI). V.M.F. acknowledges partial support from the MEPhI (Russia)., Peer reviewed

A system of magnetic molecules coupled to microwave cavities (LC resonators) undergoes the equilibrium superradiant phase transition. The transition is experimentally observable. The effect of the coupling is first illustrated by the vacuum-induced ferromagnetic order in a quantum Ising model and then by the modification of the magnetic phase diagram of Fe8 dipolar crystals, exemplifying the cooperation between intrinsic and photon-induced spin-spin interactions. Finally, a transmission experiment is shown to resolve the transition, measuring the quantum electrodynamical control of magnetism., The authors acknowledge funding from the EU (COST Action 15128 MOLSPIN, QUANTERA SUMO and FET-OPEN Grant No. 862893 FATMOLS), the Spanish
MICINN (Grants No. MAT2017-88358-C3-1-R, No. RTI2018-096075-B-C21, No. PCI2018-093116, and No. EUR2019-103823), the Gobierno de Aragón (Grant No. E09-17R Q-MAD), and the BBVA foundation (Leonardo Grants 2018)., Peer reviewed

We study the physics of a magnetic molecule described by a "giant"spin with multiple (d>2) spin states interacting with the quantized cavity field produced by a superconducting resonator. By means of the input-output formalism, we derive an expression for the output modes in the dispersive regime of operation. It includes the effect of magnetic anisotropy, which makes different spin transitions addressable. We find that the measurement of the cavity transmission allows us to uniquely determine the spin state of the qudits. We discuss, from an effective Hamiltonian perspective, the conditions under which the qudit readout is a nondemolition measurement and consider possible experimental protocols to perform it. Finally, we illustrate our results with simulations performed for realistic models of existing magnetic molecules., We acknowledge funding from Spanish MCIN/AEI/10.
13039/501100011033/ Grants No. PGC2018-094792-BI00, No. RTI2018-096075-B-C21, and No. PCI2018-093116 (MCIU/AEI/FEDER, UE), the European Union’s Horizon 2020 research and innovation programme (QUANTERA project SUMO and FET-OPEN project
FATMOLS, Grant No. 862893), the Gobierno de Aragón Grant No. E09-17R-Q-MAD, and the CSIC Quantum Technology Platform PT-001., Peer reviewed

A designed dimetallic Fe(II) helicate made with biphenylene-bridged bispyrazolylpyridine ligands and exhibiting a process of spin crossover at temperatures above ambient is shown to encapsulate an S = 5/2 tris-oxalato Fe(III) ion. The spin relaxation dynamics of this guest are strongly reduced upon encapsulation., This research was supported by Spanish MINECO (PGC2018-098630-B-I00, PID2020-118329RB-I00), the Aragón government (E31_20R PLATON) and the EU (FET-OPEN grant 862893 FATMOLS), and used resources of the ALBA synchrotron and of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DEAC02-05CH11231. G. A. thanks the Generalitat de Catalunya for the prize ICREA Academia 2018., Peer reviewed

We demonstrate, numerically, the possibility of manipulating the spin states of molecular nanomagnets with shaped microwave pulses designed with quantum optimal control theory techniques. The state-to-state or full gate transformations can be performed in this way in shorter times than using simple monochromatic resonant pulses. This enhancement in the operation rates can therefore mitigate the effect of decoherence. The optimization protocols and their potential for practical implementations are illustrated by simulations performed for a simple molecular cluster hosting a single Gd3+ ion. Its eight accessible levels (corresponding to a total spin S=7/2) allow encoding an eight-level qudit or a system of three coupled qubits. All necessary gates required for universal operation can be obtained with optimal pulses using the intrinsic couplings present in this system. The application of optimal control techniques can facilitate the implementation of quantum technologies based on molecular spin qudits., We acknowledge financial support from Grants No. FIS2017-82426-P, No. PGC2018-094792-B-I00, No. RTI2018-096075-B-C21, and No. PCI2018-093116 funded by MCIN/AEI/ 10.13039/501100011033 and ERDF “A way of making Europe,” Grant No. PID2020115221GB-C41/AEI/10.13039/501100011033, the European Union’s Horizon 2020 research and innovation programme (QUANTERA project SUMO and FET-OPEN project FATMOLS, Grant No. 862893), the Gobierno de Aragón Grant No. E09-17R-Q-MAD, and the CSIC Quantum Technology Platform PT-001., Peer reviewed

Classical machine learning has succeeded in the prediction of both classical and quantum phases of matter. Notably, kernel methods stand out for their ability to provide interpretable results, relating the learning process with the physical order parameter explicitly. Here we exploit quantum kernels instead. They are naturally related to the fidelity, and thus it is possible to interpret the learning process with the help of quantum information tools. In particular, we use a support vector machine (with a quantum kernel) to predict and characterize second-order quantum phase transitions. We explain and understand the process of learning when the fidelity per site (rather than the fidelity) is used. The general theory is tested in the Ising chain in transverse field. We show that for small-sized systems, the algorithm gives accurate results, even when trained away from criticality. Besides, for larger sizes we confirm the success of the technique by extracting the correct critical exponent ν. Finally, we present two algorithms, one based on fidelity and one based on the fidelity per site, to classify the phases of matter in a quantum processor., Funding from the EU (COST Action 15128 MOLSPIN, QUANTERA SUMO, and FET-OPEN Grant No. 862893 FATMOLS), the Spanish MICINN, PID2020115221GBC41/AEI/10.13039/501100011033, the Gobierno de Aragón (Grant E09-17R Q-MAD), and the CSIC Quantum Technologies Platform PTI-001., Peer reviewed

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., This research was supported by Spanish MINECO (PGC2018-098630-B-I00, PID2020-118329B-I00), the Aragón government (E31_20R PLATON) and the EU (FET-OPEN grant 862893 FATMOLS). G. A. thanks the Generalitat de Catalunya for the prize ICREA Academia 2018. B. L. G. thanks French GENCI/IDRIS-CINES centers for high-performance computing resources. The authors thank the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DEAC02-05CH11231, and the Zeiss Foundation for funding., Peer reviewed

Nuclear spins are candidates to encode qubits or qudits due to their isolation from magnetic noise and potentially long coherence times. However, their weak coupling to external stimuli makes them hard to integrate into circuit quantum electrodynamics architectures, the leading technology for solid-state quantum processors. Here, we study the coupling of 173Yb(III) nuclear spin states in an [Yb(trensal)] molecule to superconducting cavities. Experiments have been performed on magnetically dilute single crystals placed on the inductors of lumped-element LC superconducting resonators with characteristic frequencies spanning the range of nuclear and electronic spin transitions. We achieve a high cooperative coupling to all electronic and most nuclear [173Yb(trensal)] spin transitions, a necessary ingredient for the implementation of qudit protocols with molecular spins using a hybrid architecture., This work has been funded by the European Union Horizon 2020 research and innovation program through FET-OPEN grant FATMOLS-No 862893 and the QUANTERA project SUMO. It was also supported by the Spanish Ministry of Science and Innovation under grants RT2018-096075-B-C21, PCI2018-093116, PID2019-105552RB-C41 and C-44, PID2020- 115221GB-C41/AEI/10.13039/501100011033, and Grant SEV-2016-0686 (MCIU/AEI/FEDER, UE) and by Novo Nordisk Foundation grant NNF20OC0065610. The SUMO project was also co-funded by the Italian Ministry of University and Research. We also acknowledge financial support from the Gobierno de Aragón grant E09-17R-Q-MAD, from CSIC Research Platform PTI-001, and from ONR-Global through Grant DEFROST N62909-19-1-2053., Peer reviewed

The anion SiF62− exerts a strong template effect, driving the exclusive assembly of two different bispyridylpyrazolyl ligands into a triple stranded Fe(II) dinuclear heteroleptic helicate, engendering a new class within the large family of coordination helicates., This research was supported by Spanish MINECO (PGC2018-098630-B-I00, PID2020-118329RB-I00 and PID2020-117803GB-I00), the Aragón government (E31_20R PLATON), the EU (FET-OPEN grant 862893 FATMOLS) and the IN2UB (Masters Fellowship to NC), and used resources of the ALBA synchrotron and of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DEAC02-05CH11231. G. A. thanks the Generalitat de Catalunya for the prize ICREA Academia 2018., Peer reviewed

Quantum optics with giant emitters has shown a new route for the observation and manipulation of non-Markovian properties in waveguide QED. In this paper we extend the theory of giant atoms, hitherto restricted to the perturbative light-matter regime, to deal with the ultrastrong-coupling regime. Using static and dynamical polaron methods, we address the low-energy subspace of a giant atom coupled to an Ohmic waveguide beyond the standard rotating-wave approximation. We analyze the equilibrium properties of the system by computing the atomic frequency renormalization as a function of the coupling characterizing the localization-delocalization quantum phase transition for a giant atom. We show that virtual photons dressing the ground state are nonexponentially localized around the contact points but decay as a power law. The dynamics of an initially excited giant atom is studied, pointing out the effects of ultrastrong coupling on the Lamb shift and the spontaneous emission decay rate. Finally, we comment on the existence of the so-called oscillating bound states beyond the rotating-wave approximation., The authors acknowledge funding from the EU (COST Action 15128 MOLSPIN, QUANTERA SUMO, and FET-OPEN Grant No. 862893 FATMOLS), the Spanish Government Grants No. MAT2017-88358-C3-1-R and No. PID2020-115221GBC41/AEI/10.13039/501100011033, the Gobierno de Aragón (Grant No. E09-17R Q-MAD), and the CSIC Quantum Technologies Platform PTI-001. C.A.G.-G. acknowledges funding from the program Acciones de Dinamización “Europa Excelencia” Grant No. EUR2019-103823. F.N. was supported in part by Nippon Telegraph and Telephone Corporation Research, the Japan Science and Technology Agency (via the Quantum Leap Flagship Program and the Moonshot R&D Grant No. JPMJMS2061), the Japan Society for the Promotion of Science (via the Grants-in-Aid for Scientific Research Grant No. JP20H00134), the Army Research Office (Grant No.W911NF-18-1-0358), the Asian Office of Aerospace Research and Development (via Grant No.FA2386-20-1-4069), and the Foundational Questions Institute Fund via Grant No.FQXi-IAF19-06., Peer reviewed

Heterometallic lanthanide [LnLn′] coordination complexes that are accessible thermodynamically are very scarce because the metals of this series have very similar chemical behaviour. Trinuclear systems of this category have not been reported. A coordination chemistry scaffold has been shown to produce molecules of type [LnLn′Ln] of high purity, i.e. exhibiting high metal distribution ability, based on their differences in ionic radius. Through a detailed analysis of density functional theory (DFT) based calculations, we discern the energy contributions that lead to the unparalleled chemical selectivity of this molecular system. Some of the previously reported examples are compared here with the newly prepared member of this exotic list, [Er2Pr(LA)2(LB)2(py)(H2O)2](NO3) (1) (H2LA and H2LB are two β-diketone ligands). A magnetic analysis extracted from magnetization and calorimetry determinations identifies the necessary attributes for it to act as an addressable, conditional multiqubit spin-based quantum gate. Complementary ab initio calculations confirm the feasibility of these complexes as composite quantum gates, since they present well-isolated ground states with highly anisotropic and distinct g-tensors. The electronic structure of 1 has also been analyzed by EPR. Pulsed experiments have allowed the establishment of the quantum coherence of the transitions within the relevant spin states, as well as the feasibility of a coherent control of these states via nutation experiments., The authors thank the European Union's Horizon 2020 research and innovation programmes QUANTERA project SUMO and FET-OPEN grant 862893 FATMOLS (GA and OR) the Spanish AEI for grants PGC2018-098630-B-I00 (GA and DA), PID2020-118329RB-I00 (OR), PID2020-112806RB-I00 (DGR and CB) and the Severo Ochoa Excellence Accreditation 2020–2023 CEX2019-000925-S (DGR and CB), the ICIQ foundation (DGR and CB) and the Generalitat de Catalunya for a Beatriu de Pinós fellowship (2019-BP-00015, DR), an FI grant (also financed by AGAUR and the European Social Fund, DGR) an FI-SDUR Grant (2020-FISDU-00492, DM), the CERCA program and the ICREA Academia 2018 Prize (GA), as well as the National EPR Facility at the University of Manchester for support with the EPR measurements. This work used resources of the ALBA synchrotron through access to beamline 13-XALOC., Peer reviewed

Cavity photons and ferromagnetic spin excitations can exchange information coherently in hybrid architectures, at speeds set by their mutual coupling strength. Speed enhancement is usually achieved by optimizing the geometry of the electromagnetic cavity. Here we show that the geometry of the ferromagnet also plays a role, by setting the fundamental frequency of the magnonic resonator. Using focused ion-beam patterning, we vary the aspect ratio of different Permalloy samples reaching operation frequencies above 10 GHz while working at low external magnetic fields. Additionally, we perform broadband ferromagnetic resonance measurements and cavity experiments that demonstrate that the light-matter coupling strength can be estimated using either open transmission lines or resonant cavities, yielding very good agreement. Finally, we describe a simple theoretical framework based on electromagnetic and micromagnetic simulations that successfully accounts for the experimental results. This approach can be used to design hybrid quantum systems exploiting magnetostatic mode excited in ferromagnets of arbitrary size and shape and to tune their operation conditions., This work is partly funded and supported by the spanish MCIN/AEI/10.13039/501100011033/ and FEDER (Una manera de hacer Europa) through GrantNo. RTI2018-096075-B-C21, the BBVA Foundation through “Beca Leonardo a Investigadores y Creadores Culturales 2019”, the Aragón Regional Government through project E09_20R (Construyendo Europa desde Aragón), CSIC through project 202160I034, the CSIC program for the Spanish Recovery, Transformation and Resilience Plan funded by the Recovery and Resilience Facility of the European Union, established by the Regulation (EU) 2020/2094, the EU through FET-OPEN (862893FATMOLS) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (948986QFaST). S.M.-L.R. acknowledges a FPI grant from the spanish MCIN. Authors would like to acknowledge the use of Servicio General de Apoyo a la Investigación-SAI, Universidad de Zaragoza., Peer reviewed

Starting from a general material system of N particles coupled to a cavity, we use a coherent-state path integral formulation to produce a non-perturbative effective theory for the material degrees of freedom. We tackle the effects of image charges, the A2 term and a multimode arbitrary-geometry cavity. The resulting (non-local) action has the photonic degrees of freedom replaced by an effective position-dependent interaction between the particles. In the large-N limit, we discuss how the theory can be cast into an effective Hamiltonian where the cavity induced interactions are made explicit. The theory is applicable, beyond cavity QED, to any system where bulk material is linearly coupled to a diagonalizable bosonic bath. We highlight the differences of the theory with other well-known methods and numerically study its finite-size scaling on the Dicke model. Finally, we showcase its descriptive power with three examples: photon condensation, the 2D free electron gas in a cavity and the modification of magnetic interactions between molecular spins; recovering, condensing and extending some recent results in the literature., Funding from the EU (QUANTERASUMO andFET-OPEN Grant 862893 FATMOLS) the Spanish Government Grant PID2020-115221GBC41/AEI/10.13039/501100011033, the Gobierno de Aragón (GrantE09-17RQ-MAD) and the CSIC Quantum Technologies Platform PTI-001., Peer reviewed

Resumen del trabajo presentado al APS March Meeting, celebrado en Chicago (USA) del 14 al 18 de marzo de 2022., We report the results of ac susceptibility measurements performed down to very low temperatures (T > 13 mK) on thin layers of asymmetric Dy2 molecular coordination complexes that have been proposed as candidates for the realization of 2-qubit quantum gates. The molecules are integrated into a ¿-SQUID susceptometer by means of Dip Pen Nanolithography. Frequency-dependent susceptibility data measured on 5 and 20 molecular layers thick films are compared with similar results obtained for bulk polycrystalline samples. The results show that the molecular Dy2 units largely remain intact at the surface. Low-nuclearity lanthanide magnetic clusters might then provide suitable building blocks for the development of a scalable quantum architecture., We acknowledge support from EU (FATMOLS FETOPEN-RIA-2019-01 862893, QUANTERA SUMO), Spanish MCIU/AEI/FEDER (RTI-2018-096075-B-C21, PCI2018-093116) and CSIC Research Platform PTI-001., No

Resumen del trabajo presentado al APS March Meeting, celebrado en Chicago (USA) del 14 al 18 de marzo de 2022., This work was supported by funds from the EU (COST Action 15128 MOLSPIN, QUANTERA project SUMO, FET-OPEN grant 862893 FATMOLS), the
Spanish MICINN (grants CTQ2015- 68370-P, CTQ2015-64486-R, RTI2018-096075-B-C21, PCI2018- 093116, PGC2018-098630-B-I00, MAT2017-86826-R)
and the Gobierno de Aragón (grants E09-17R-Q-MAD, E31_17R PLATON)., No

Resumen del trabajo presentado al APS March Meeting, celebrado en Chicago (USA) del 14 al 18 de marzo de 2022., We acknowledge support from EU (FATMOLS FETOPEN-RIA-2019-01 862893, QUANTERA SUMO), Spanish MCIU/AEI/FEDER (RTI-2018-096075-B C21, PCI2018-093116) and CSIC Research Platform PTI-001, No

Open data set supporting figures of the related publication., European Commission:
FATMOLS - FAult Tolerant MOLecular Spin processor (862893), Peer reviewed

A quantum spin liquid (QSL) is an elusive state of matter characterized by the absence of long-range magnetic order, even at zero temperature, and by the presence of exotic quasiparticle excitations. In spite of their relevance for quantum communication, topological quantum computation and the understanding of strongly correlated systems, like high-temperature superconductors, the unequivocal experimental identification of materials behaving as QSLs remains challenging. Here, we present a novel 2D heterometallic oxalate complex formed by high-spin Co(II) ions alternating with diamagnetic Rh(III) in a honeycomb lattice. This complex meets the key requirements to become a QSL: a spin ½ ground state for Co(II), determined by spin–orbit coupling and crystal field, a magnetically-frustrated triangular lattice due to the presence of antiferromagnetic correlations, strongly suppressed direct exchange interactions and the presence of equivalent interfering superexchange paths between Co centres. A combination of electronic paramagnetic resonance, specific heat and ac magnetic susceptibility measurements in a wide range of frequencies and temperatures shows the presence of strong antiferromagnetic correlations concomitant with no signs of magnetic ordering down to 15 mK. These results show that bimetallic oxalates are appealing QSL candidates as well as versatile systems to chemically fine tune key aspects of a QSL, like magnetic frustration and superexchange path geometries., This work has received support from grants RTI2018-096075-A-C22 (FL), PID2021-124796OB-I00 (JRGM), CEX2019-000919-M (CMG, SMV, EC), RTI2018-096075-B-C21 (EB), PID2020-118117RB-I00 (CMG), PID2021-124734OB-C21 (ME), RYC2019-028429-I (EB), and the María de Maeztu Excellence Seal CEX2018-000805-M (EB) funded by MCIN/AEI/10.13039/501100011033, ERDF A way of making Europe and ESF Investing in your future. We also acknowledge funding from the European Union Horizon 2020 research and innovation program through FET-OPEN grant FATMOLS-No 862893 and ERC Advanced grant Mol-2D-No788222, the Gobierno de Aragón grant E09-17R-Q-MAD and CSIC Research Platform PTI-001. ICIQ acknowledges the Ministerio de Ciencia e Innovación for support through Severo Ochoa Excellence Accreditation 2020–2023 (CEX2019-000925-S, MIC/AEI)., Peer reviewed

The implementation of a universal quantum processor still poses fundamental issues related to error mitigation and correction, which demand to investigate also platforms and computing schemes alternative to the main stream. A possibility is offered by employing multi-level logical units (qudits), naturally provided by molecular spins. Here we present the blueprint of a Molecular Spin Quantum Processor consisting of single Molecular Nanomagnets, acting as qudits, placed within superconducting resonators adapted to the size and interactions of these molecules to achieve a strong single spin to photon coupling. We show how to implement a universal set of gates in such a platform and to readout the final qudit state. Single-qudit unitaries (potentially embedding multiple qubits) are implemented by fast classical drives, while a novel scheme is introduced to obtain two-qubit gates via resonant photon exchange. The latter is compared to the dispersive approach, finding in general a significant improvement. The performance of the platform is assessed by realistic numerical simulations of gate sequences, such as Deutsch-Josza and quantum simulation algorithms. The very good results demonstrate the
feasibility of the molecular route towards a universal quantum processor., This work has received funding from the European Unions Horizon 2020 research and innovation programme (FET-OPEN project FATMOLS) under Grant Agreement No. 862893. Project also funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.3 – Call for Tender No. 341 of 15/03/2022 of Italian Ministry of University and Research funded by the European Union – NextGenerationEU, Award No. PE0000023, Concession Decree No. 1564 of 11/10/2022 adopted by the Italian Ministry of University and Research, CUP D93C22000940001, Project title “National Quantum Science and Technology Institute” (NQSTI). The authors also acknowledge support from Fondazione Cariparma, from Novo Nordisk foundation grant NNF21OC0070832 in the call “Exploratory Interdisciplinary Synergy Programme 2021,” from Grants No PID2020-115221GB-C41/AEI/10.13039/501100011033, No. TED2021-131447B-C21, and No. TED2021-131447BC22 funded by the Spanish MCIN/AEI/10.13039/ 501100011033 and by the EU “NextGenerationEU”/PRTR, from the Gobierno de Aragón (Grant No. E09-17R QMAD) and from the CSIC Quantum Technologies Platform PTI-001. A.G.L. acknowledges support from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 899354 (SuperQuLAN), and from CSIC Interdisciplinary Thematic Platform (PTI+) on Quantum Technologies (PTI-QTEP+)., Peer reviewed

A biphenyl-spaced bis-pyrazolylpyridine ligand interacts with ferrous ions to engender a dimetallic helical coordination cage that encapsulates an Fe3+ tris-anilate complex. The host–guest interaction breaks the symmetry of the Fe2+ centers causing a differential spin crossover behavior in them that can be followed in great detail crystallographically., This research was supported by the Spanish Ministry of Science (TED2021-129214B-I00, PID2020-118329RB-I00, PRTR-C17.I1), the Aragón government (E31_20R PLATON) and the EU (FET-OPEN grant 862893 FATMOLS and Next Generation EU), and used resources of the ALBA synchrotron and of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DEAC02-05CH11231. G. A. thanks the Generalitat de Catalunya for Next Generation co-fund (PRTR-C17.I1) and the prize ICREA Academia 2018., Peer reviewed

In this paper, we analyze the circuit complexity for preparing ground states of quantum many-body systems. In particular, how this complexity grows as the ground state approaches a quantum phase transition. We discuss different definitions of complexity, namely the one following the Fubini-Study metric or the Nielsen complexity. We also explore different models: Ising, ZZXZ or Dicke. In addition, different forms of state preparation are investigated: analytic or exact diagonalization techniques, adiabatic algorithms (with and without shortcuts), and Quantum Variational Eigensolvers. We find that the divergence (or lack thereof) of the complexity near a phase transition depends on the non-local character of the operations used to reach the ground state. For Fubini-Study based complexity, we extract the universal properties and their critical exponents. In practical algorithms, we find that the complexity depends crucially on whether or not the system passes close to a quantum critical point when preparing the state. For both VQE and Adiabatic algorithms, we provide explicit expressions and bound the growth of complexity with respect to the system size and the execution time, respectively., The authors acknowledge funding from the EU (QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS), the Spanish Government Grants PID2020-115221GB-C41/AEI/10.13039/501100011033 and TED2021-131447B-C21 funded by MCIN/AEI/10.13039/501100011033 and the EU “NextGenerationEU”/PRTR, the Gobierno de Aragón (Grant E09-17R Q-MAD) and the CSIC Quantum Technologies Platform PTI001. This work has been financially supported by the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through the QUANTUM ENIA project call - Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda. J R-R acknowledges support from the Ministry of Universities of the Spanish Government through the grant FPU2020-07231. S. R-J. acknowledges financial support from Gobierno de
Aragón through a doctoral fellowship., Peer reviewed

Molecular spins are considered as the quantum hardware to build hybrid quantum processors in which coupling to superconducting devices would provide the means to implement the necessary coherent manipulations. As an alternative to large magnetically-dilute crystals or concentrated nano-scale deposits of paramagnetic molecules that have been studied so far, the use of pre-formed sub-micronic spherical particles of a doped Gd@Y hydroxycarbonate is evaluated here. Particles with an adjustable number of spin carriers are prepared through the control of both particle size and doping. Bulk magnetic properties and continuous wave and time-domain-EPR spectroscopy show that the Gd spins in these particles are potential qubits with robust quantum coherence. Monolayers of densely-packed particles are then formed interfacially and transferred successfully to the surface of Nb superconducting resonators. Alternatively, these particles are disposed at controlled localizations as isolated groups of a few particles through Dip-Pen Nanolithography using colloidal organic dispersions as ink. Altogether, this study offers new material and methodologies relevant to the development of viable hybrid quantum processors., This research was supported by the European Union Horizon 2020 research and innovation program through FET-OPEN grant FATMOLS-No 862893, the Spanish MCIN/AEI/10.13039/501100011033 and ERDF “A way of making Europe” (projects PID2019-105881RB-I00 and PID2020-1183294RB-I00), the Aragón government (PLATON E31_20R and E31_23R, doctoral grant to IT), and the CSIC Research Platform PTI-001 QUTEP., Peer reviewed

We present an exact analytical solution for quantum strong long-range models in the canonical ensemble by extending the classical solution proposed in Campa et al. [J. Phys. A: Math. Gen. 36, 6897 (2003)]. Specifically, we utilize the equivalence between generalized Dicke models and interacting quantum models as a generalization of the Hubbard-Stratonovich transformation. To demonstrate our method, we apply it to the Ising chain in transverse field and discuss its potential application to other models, such as the Fermi-Hubbard model, combined short- and long-range models, and models with antiferromagnetic interactions. Our findings indicate that the critical behavior of a model is independent of the range of interactions, within the strong long-range regime, and the dimensionality of the model. Moreover, we show that the order-parameter expression is equivalent to that provided by mean-field theory, thus confirming the exactness of the latter. Finally, we examine the algebraic decay of correlations and characterize its dependence on the range of interactions in the full phase diagram., The authors acknowledge funding from the EU (QUANTERA SUMO and FET-OPEN Grant No. 862893 FATMOLS), the Spanish Government Grants No. PID2020-115221GB-C41/AEI/10.13039/501100011033
and No. TED2021-131447B-C21 funded by MCIN/AEI/10.13039/501100011033 and the EU
“NextGenerationEU”/PRTR, the Gobierno de Aragón (Grant No. E09-17R Q-MAD) and the CSIC Quantum
Technologies Platform PTI-001. This work has been financially supported by the Ministry of Economic Affairs
and Digital Transformation of the Spanish Government through the QUANTUM ENIA project call Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan NextGenerationEU within the framework of the “Digital Spain 2026 Agenda.” J.R.-R. acknowledges support from the Ministry of Universities of the Spanish Government through the Grant No. FPU2020-07231., Peer reviewed

A central goal in quantum technologies is to maximize GT2, where G stands for the coupling of a qubit to control and readout signals and T2 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 T2. Here, we study the coupling of pure and magnetically diluted crystals of HoW10 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 also becomes maximum at these transitions. The results show that engineering spin-clock states of molecular systems offers a promising strategy to combine sizable spin-photon interactions with a sufficient isolation from unwanted magnetic noise sources., This work has received support from Grants No. RTI2018-096075-A-C21, No. PID2019-105552RB-C41, No. PID2019-105552RB-C44, No. P2018/NMT-4291 No. TEC2SPACE-CM, No. TED2021-131447B-C21, No. TED2021-131447B-C22, No. CEX2019-000919-M and No. CEX2020-001039-S, funded by MCIN/AEI/10.13039/501100011033, ERDF “A way of making Europe” and ESF “Investing in your future”, and from the Gobierno de Aragón Grant No. E09-17R-Q-MAD. We also acknowledge funding from the European Union Horizon 2020 research and innovation programme through FETOPEN Grant No. FATMOLS-No862893, ERC advanced Grant No. Mol-2D-No788222, ERC consolidator Grant No. DECRESIM-No647301, and HORIZON-MSCA2021 Grant No. HyQuArch-No101064707. This study also forms part of the Advanced Materials and Quantum Communication programmes, supported by MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1), by Gobierno de Aragón, by Generalitat Valenciana, and by CSIC (PTI001). S.H. acknowledges support from the US Department of Energy (DE-SC0020260). Work done at the National High Magnetic Field Laboratory is supported by the US National Science Foundation (DMR-1644779 and DMR-2128556) and the State of Florida., With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2019-000919-M)., With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2020-001039-S)., Peer reviewed

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., This work has received funding from the European Union Horizon 2020 research and innovation programme (FET-OPEN project FATMOLS) under Grant Agreement No. 862893, and from Fondazione Cariparma. It was also funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.3—Call for tender No. 341 of 15/03/2022 of Italian Ministry of University and Research funded by the European Union—NextGenerationEU, award number PE0000023, Concession Decree No. 1564 of 11/10/2022 adopted by the Italian Ministry of University and Research, CUP D93C22000940001, Project title 'National Quantum Science and Technology Institute' (NQSTI). FL acknowledges support from grant TED2021-131447B-C21 and from "Plan Complementario en Comunicación Cuántica", funded by MCIN/AEI/10.13039/501100011033, the European Union NextGenerationEU/PRTR and the CSIC Quantum Technology Platform PT-001. We also acknowledge the European Research Council-Synergy project CASTLE (Project No. 101071533) funded by the Horizon Europe Programme., Peer reviewed

We present a comprehensive investigation into the learning capabilities of a simple d-level system (qudit). Our study is specialized for classification tasks using real-world databases, specifically the Iris, breast cancer, and MNIST datasets. We explore various learning models in the metric learning framework, along with different encoding strategies. In particular, we employ data re-uploading techniques and maximally orthogonal states to accommodate input data within low-dimensional systems. Our findings reveal optimal strategies, indicating that when the dimension of input feature data and the number of classes are not significantly larger than the qudit’s dimension, our results show favorable comparisons against the best classical models. This trend holds true even for small quantum systems, with dimensions d < 5 and utilizing algorithms with a few layers (L = 1,2). However, for high-dimensional data such as MNIST, we adopt a hybrid approachinvolving dimensional reduction through a convolutional neural network. In this context, we observe that small quantum systems often act as bottlenecks, resulting in lower accuracy compared to their classical counterparts., The authors acknowledge funding from the EU (QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS), the Spanish Government Grants PID2020-115221GB-C41/AEI/10.13039/501100011033 and TED2021-131447B-C21 funded by MCIN/AEI/10.13039/501100011033 and the EU 'NextGenerationEU'/PRTR, the Gobierno de Aragón (Grant E09-17R Q-MAD)and the CSIC Quantum Technologies Platform PTI-001. This work has been financially supported by the Ministry of Economic Affairs and Digital Transformation of the Spanish Government through the QUANTUM ENIA project call—Quantum Spain project, and by the European Union through the Recovery, Transformation and Resilience Plan—NextGenerationEU within the framework of the 'Digital Spain 2026 Agenda'. J R R acknowledges support from the Ministry of Universities of the Spanish Government through the grant FPU2020-07231. S R J acknowledges financial support from Gobierno de Aragón through a doctoral fellowship., Peer reviewed

Artificial magnetic molecules can host several spin qubits, which could then implement small-scale algorithms. In order to become of practical use, such molecular spin processors need to increase the available computational space and warrant universal operations. Here, we design, synthesize and fully characterize dissymetric molecular dimers hosting either one or two Gadolinium(III) ions. The strong sensitivity of Gadolinium magnetic anisotropy to its local coordination gives rise to different zero-field splittings at each metal site. As a result, the [LaGd] and [GdLu] complexes provide realizations of distinct spin qudits with eight unequally spaced levels. In the [Gd2] dimer, these properties are combined with a Gd-Gd magnetic interaction, sufficiently strong to lift all level degeneracies, yet sufficiently weak to keep all levels within an experimentally accessible energy window. The spin Hamiltonian of this dimer allows a complete set of operations to act as a 64-dimensional all-electron spin qudit, or, equivalently, as six addressable qubits. Electron paramagnetic resonance experiments show that resonant transitions between different spin states can be coherently controlled, with coherence times TM of the order of 1 µs limited by hyperfine interactions. Coordination complexes with embedded quantum functionalities are promising building blocks for quantum computation and simulation hybrid platforms., This work was supported by funds from the EU (ERC Starting Grant 258060 FuncMolQIP, COST Action 15128 MOLSPIN, QUANTERA project SUMO, FET-OPEN grant 862893 FATMOLS), the Spanish MICINN (grants CTQ2015–68370-P, CTQ2015–64486-R, RTI2018–096075-B-C21, PCI2018–093116, PGC2018–098630-B-I00, MAT2017–86826-R) and the Gobierno de Aragón (grants E09–17R-Q-MAD, E31_17R PLATON). G.A. thanks to the Generalitat de Catalunya for the prize ICREA Academia 2018., Peer reviewed

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

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.

Heterometallic lanthanide [LnLn′] coordination complexes that are accessible thermodynamically are very scarce because the metals of this series have very similar chemical behaviour. Trinuclear systems of this category have not been reported. A coordination chemistry scaffold has been shown to produce molecules of type [LnLn′Ln] of high purity, i.e. exhibiting high metal distribution ability, based on their differences in ionic radius. Through a detailed analysis of density functional theory (DFT) based calculations, we discern the energy contributions that lead to the unparalleled chemical selectivity of this molecular system. Some of the previously reported examples are compared here with the newly prepared member of this exotic list, [Er2Pr(LA)2(LB)2(py)(H2O)2](NO3) (1) (H2LA and H2LB are two β-diketone ligands). A magnetic analysis extracted from magnetization and calorimetry determinations identifies the necessary attributes for it to act as an addressable, conditional multiqubit spin-based quantum gate. Complementary ab initio calculations confirm the feasibility of these complexes as composite quantum gates, since they present well-isolated ground states with highly anisotropic and distinct g-tensors. The electronic structure of 1 has also been analyzed by EPR. Pulsed experiments have allowed the establishment of the quantum coherence of the transitions within the relevant spin states, as well as the feasibility of a coherent control of these states via nutation experiments.

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.

Artificial magnetic molecules can host several spin qubits, which could then implement small-scale algorithms. In order to become of practical use, such molecular spin processors need to increase the available computational space and warrant universal operations. Here, we design, synthesize and fully characterize dissymetric molecular dimers hosting either one or two Gadolinium(III) ions. The strong sensitivity of Gadolinium magnetic anisotropy to its local coordination gives rise to different zero-field splittings at each metal site. As a result, the [LaGd] and [GdLu] complexes provide realizations of distinct spin qudits with eight unequally spaced levels. In the [Gd2] dimer, these properties are combined with a Gd-Gd magnetic interaction, sufficiently strong to lift all level degeneracies, yet sufficiently weak to keep all levels within an experimentally accessible energy window. The spin Hamiltonian of this dimer allows a complete set of operations to act as a 64-dimensional all-electron spin qudit, or, equivalently, as six addressable qubits. Electron paramagnetic resonance experiments show that resonant transitions between different spin states can be coherently controlled, with coherence times TM of the order of 1 µs limited by hyperfine interactions. Coordination complexes with embedded quantum functionalities are promising building blocks for quantum computation and simulation hybrid platforms.

A system of magnetic molecules coupled to microwave cavities (LC resonators) undergoes the equilibrium superradiant phase transition. The transition is experimentally observable. The effect of the coupling is first illustrated by the vacuum-induced ferromagnetic order in a quantum Ising model and then by the modification of the magnetic phase diagram of Fe8 dipolar crystals, exemplifying the cooperation between intrinsic and photon-induced spin-spin interactions. Finally, a transmission experiment is shown to resolve the transition, measuring the quantum electrodynamical control of magnetism.

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.

Starting from the paradigmatic spin-boson model (SBM), we investigate the static and dynamical properties of a system of two distant two-level emitters coupled to a one-dimensional Ohmic waveguide beyond the rotating wave approximation. Employing static and dynamical polaron Ansätze we study the effects of finite separation distance on the behavior of the photon-mediated Ising-like interaction, qubit frequency renormalization, ground-state magnetization, and entanglement entropy of the two-qubit system. Based on previous works we derive an effective approximate Hamiltonian for the two-impurity SBM that preserves the excitation-number and thus facilitates the analytical treatment. In particular, it allows us to introduce non-Markovianity arising from delay-feedback effects in two distant emitters in the so-called ultrastrong coupling (USC) regime. We test our results with numerical simulations performed over a discretized circuit-QED model, finding perfect agreement with previous results, and showing interesting dynamical effects arising in ultrastrong waveguide QED with distant emitters. In particular, we revisit the Fermi two-atom problem showing that, in the USC regime, initial correlations yield two different evolutions for symmetric and antisymmetric states even before the emitters become causally connected. Finally, we demonstrate that the collective dynamics, e.g., superradiance or subradiance, are affected not only by the distance between emitters, but also by the coupling, due to significant frequency renormalization. This constitutes another dynamical consequence of the USC regime.

Heterometallic lanthanide LnLn'] coordination complexes that are accessible thermodynamically are very scarce because the metals of this series have very similar chemical behaviour. Trinuclear systems of this category have not been reported. A coordination chemistry scaffold has been shown to produce molecules of type LnLn'Ln] of high purity, i.e. exhibiting high metal distribution ability, based on their differences in ionic radius. Through a detailed analysis of density functional theory (DFT) based calculations, we discern the energy contributions that lead to the unparalleled chemical selectivity of this molecular system. Some of the previously reported examples are compared here with the newly prepared member of this exotic list, Er2Pr(LA)2(LB)2(py)(H2O)2](NO3) (1) (H2LA and H2LB are two ß-diketone ligands). A magnetic analysis extracted from magnetization and calorimetry determinations identifies the necessary attributes for it to act as an addressable, conditional multiqubit spin-based quantum gate. Complementary ab initio calculations confirm the feasibility of these complexes as composite quantum gates, since they present well-isolated ground states with highly anisotropic and distinct g-tensors. The electronic structure of 1 has also been analyzed by EPR. Pulsed experiments have allowed the establishment of the quantum coherence of the transitions within the relevant spin states, as well as the feasibility of a coherent control of these states via nutation experiments. © 2022 The Royal Society of Chemistry

Classical machine learning has succeeded in the prediction of both classical and quantum phases of matter. Notably, kernel methods stand out for their ability to provide interpretable results, relating the learning process with the physical order parameter explicitly. Here we exploit quantum kernels instead. They are naturally related to the fidelity, and thus it is possible to interpret the learning process with the help of quantum information tools. In particular, we use a support vector machine (with a quantum kernel) to predict and characterize second-order quantum phase transitions. We explain and understand the process of learning when the fidelity per site (rather than the fidelity) is used. The general theory is tested in the Ising chain in transverse field. We show that for small-sized systems, the algorithm gives accurate results, even when trained away from criticality. Besides, for larger sizes we confirm the success of the technique by extracting the correct critical exponent ¿. Finally, we present two algorithms, one based on fidelity and one based on the fidelity per site, to classify the phases of matter in a quantum processor.

We demonstrate, numerically, the possibility of manipulating the spin states of molecular nanomagnets with shaped microwave pulses designed with quantum optimal control theory techniques. The state-to-state or full gate transformations can be performed in this way in shorter times than using simple monochromatic resonant pulses. This enhancement in the operation rates can therefore mitigate the effect of decoherence. The optimization protocols and their potential for practical implementations are illustrated by simulations performed for a simple molecular cluster hosting a single Gd3+ ion. Its eight accessible levels (corresponding to a total spin S=7/2) allow encoding an eight-level qudit or a system of three coupled qubits. All necessary gates required for universal operation can be obtained with optimal pulses using the intrinsic couplings present in this system. The application of optimal control techniques can facilitate the implementation of quantum technologies based on molecular spin qudits.

Nuclear spins are candidates to encode qubits or qudits due to their isolation from magnetic noise and potentially long coherence times. However, their weak coupling to external stimuli makes them hard to integrate into circuit quantum electrodynamics architectures, the leading technology for solid-state quantum processors. Here, we study the coupling of 173Yb(III) nuclear spin states in an [Yb(trensal)] molecule to superconducting cavities. Experiments have been performed on magnetically dilute single crystals placed on the inductors of lumped-element LC superconducting resonators with characteristic frequencies spanning the range of nuclear and electronic spin transitions. We achieve a high cooperative coupling to all electronic and most nuclear [173Yb(trensal)] spin transitions, a necessary ingredient for the implementation of qudit protocols with molecular spins using a hybrid architecture.