Chapter 309 - Lanthanide molecules for spin-based quantum technologies
Introduction
Quantum technologies are one of the most relevant scientific developments to shape the (near) future. Indeed, they will be key to expand and/or have a huge impact on many branches of science and technology. Some examples are the understanding of quantum systems including the design of pharmaceutical therapies through quantum simulation [1], the area of data protection and secure communications thanks to quantum cryptography [2], or the processing of information through quantum computing [3]. The central concept of quantum computing is that the information can be encoded and manipulated on quantum bits (or qubits), which may consist of any physical system able to generate quantum superpositions of two basis states (see Fig. 1). This implies a natural computational gain, since the size of the computer register required for an N-component quantum problem grows only linearly with N, instead of the exponential growth occurring when using classical bits [3]. In addition, the manipulation of quantum information allows the exploitation of resources inherent to quantum mechanics, not available for classic computation. These are, for example, quantum superposition and quantum entanglement, which enable the development and use of algorithms that take advantage of quantum parallelism, thus providing avenues to solve information problems currently intractable.
The other key hardware ingredients are universal quantum logic gates that involve wiring various qubits (directly or through quantum buses) in such a way that they realize elaborate quantum operations [4]. There are many proposed candidates to embody qubits in quantum gates [5], among which superconducting circuits [6], trapped ions [7] and spins in solids [8], [9], [10] or molecules [11], [12]. Each has advantages and weaknesses, but there are two key challenges that are common to all of them. One is the extreme fragility and associated loss of the quantum information due to interactions of the qubits with their environment, which is called decoherence [13]. And the other, termed scalability, is the necessity to have a large number of robust and reliable qubit identical replicas available [4]. Molecular spins could arguably be in a very advantageous position to meet both requirements. Compared with superconducting circuits and trapped ions, which are two of the most advanced qubit realizations, spins in general are relatively insensitive to electric field fluctuations. Thus spin qubits may have excellent coherence, if protected from what they are sensitive to, which are magnetic field fluctuations. This has been largely demonstrated with the very long coherence times observed for the spins of NV (Nitrogen Vacancy) centers in diamond [14] or phosphorous atom impurities in silicon [15]. Spins in molecules are comparatively less robust in terms of coherence, but it has been shown that synthetic strategies may open the way to reach sufficiently enhanced quantum coherence [11]. On the other hand, the ability of chemistry to produce large amounts of identical molecular qubit replicas or to build multi-qubit molecules able to encode a specific gate operation [16], [17], while also controlling their organization, offers unique solutions to the scalability challenge.
Both, organic radicals and paramagnetic coordination compounds of first-raw transition metal ions have been studied as potential qubits [11], [12], [18], [19], [20], [21]. The latter in particular have witnessed a significant improvement of their quantum coherence, with record values of the phase-memory or dephasing time Tm (see Fig. 1) reaching tens and even hundreds of microseconds for specific and optimized conditions [22], [23]. Rare-earth (RE) complexes have been considered later, but have offered additional options that could prove very useful for quantum computation [24]. These are the possibility to encode multiple qubits in one sole ion [25], and the increased size of the quantum basis states by coupling the electronic spin with the nuclear spin [26]. So far, relatively few studies of lanthanide and actinide spin qubits exist. Some recent contributions are, however, particularly relevant for the development of quantum technologies, such as the proof-of-concept realization of quantum algorithms [27], [28]. This chapter is an attempt to review the most significant contributions made recently on the preparation, physical description and implementation of lanthanide (Ln) and actinide (Ac) containing molecules for the advancement of quantum computing. We start with a non-exhaustive qualitative view on the spin dynamics of RE molecules in general (Section 2), with a more detailed focus on those for which quantum coherence has been evaluated (Section 3). Then, the use of lanthanide-based molecules for the realization of multiqubit quantum gates (Section 4) and quantum algorithms (Section 5) is described, followed by some conclusions and outlook on the future of RE molecular materials in the development of spin-based quantum technologies.
Section snippets
Spin relaxation dynamics of lanthanide and actinide single-molecule magnets (SMMs)
In order to embody a spin qubit, any specific paramagnetic molecule requires that its electronic scheme provides a suitable basis for the qubit states, i.e., that under certain conditions it can be pictured as a two-level quantum system. This is obviously the case of a pure S = 1/2 spin, but the peculiar magnetism of lanthanide ions also makes them, in most cases, good qubit candidates. On the one hand, the ground state J multiplet is in general sufficient to accurately describe their magnetic
Causes of decoherence for spin qubits
The main criterion to evaluate the goodness of a quantum system as qubit is its figure of merit, QM = ΩRT2, which measures the number of feasible qubit operations before decoherence results in the loss of quantum information. ΩR is the Rabi oscillation frequency and T2 the spin-spin relaxation time. In Bloch equations, T2 is defined as the time constant for dephasing in the xy plane of the Bloch sphere, and corresponds to the rate of mutual electron spin flips. There are, however, other factors
A dinuclear Tb(III) complex as prototype of a two-qubit quantum gate
A universal set of quantum gates (qugates) is a small group of elementary quantum logic operations that, combined, can lead to any possible quantum algorithm, independently of its complexity. Two of the most paradigmatic universal gates are the controlled-NOT (CNOT) and the √ SWAP (Fig. 10) operations. Both operate on two coupled qubits, therefore, on a computational basis formed by four double-qubit states; |↑, ↑⟩, |↑, ↓⟩, |↓, ↑⟩ and |↓, ↓⟩ (or | 1, 1⟩, | 1, 0⟩, | 0, 1⟩ and | 0, 0⟩). In these | a, b
Lanthanide electronic spins as qudits
The minimum possible amount of quantum information is encoded by a qubit, which exhibits two well defined quantum states. However, if the n accessible states of a given system is larger than two, it can be exploited as a higher dimension qudit. In this sense, the GdW30 “three-qubit” system described in the previous section constitutes a qudit of dimension d = 8. Exploiting large dimension qudits enhances the density of quantum information that can be handled within a physical system, allowing it
Conclusions
In this review, we have shown the fast progress made by chemists and physicist in the race for turning molecules containing one or a few lanthanide ions into useful hardware (the qubits, qudits and qugates) for the future quantum processing of information. We have proposed that this process has benefited initially from the strong wave of research that emerged following the discovery of SMMs. Ten years after the advent of SMM research, lanthanide complexes were recognized as very good candidates
Acknowledgments
The authors thank financial support from Spanish Government through projects MAT2017-86826-R (O.R.) and PGC2018-098630-B-I00 (G.A.), QUANTERA for project SUMO (through Spanish PCI2018-093106), the Aragón government (DGA) through a consolidated group PLATON E31_17R (O.R.) and the Generalitat de Catalunya for an ICREA-Academia 2018 Prize (G.A.).
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