are essential. Spin states relax thermally, which is characterized by their longitudinal spin relaxation time T₁, while the loss of spin coherence is characterized by the transverse spin relaxation time T₂. In general T₁ > T₂; thus, T₁ places an upper bound on T₂. Designing molecules that feature quantum properties inherently involves synthetic control of coherence (Du et al. 1992; Eaton et al. 2001). The dominant contributions to decoherence are spin–spin coupling, either through electron spin–nuclear spin coupling or hyperfine interactions. Significant work has focused on understanding both T₁ and T₂, where increasing T₂ can be achieved by depleting the local nuclear spin environment (Canarie, Jahn, and Stoll 2020; Graham, Yu, et al. 2017; Krzyaniak et al. 2015; Morton et al. 2007; Wedge et al. 2012). Probing the impact of ligand nuclear spins on coherence properties requires the careful placement of nuclear spins within a system (Figure 2-1). For example, systematically varying the ligands decorating the periphery of a circular {Cr₇Ni} ring was used to identify the sources of decoherence. What these

for spin-based quantum information. One approach to creating molecular qubits is to develop molecular analogs of these systems. One of the most promising solid-state qubits is the anionic nitrogen-vacancy (NV) pair defect in diamond (Awschalom et al. 2018; Doherty et al. 2013). This system features optical readout of spin information, a powerful feature that coupled with its optical initialization makes it a core quantum technology. Indeed, initialization, which is essential and challenging for molecules, is a key feature. Initialization fundamentally means placing the system into a specific quantum state, which DiVincenzo (2000) highlighted as a fundamental requirement for a good qubit (see Box 2-2). The DiVincenzo criteria are difficult to fulfill using the Boltzmann populations of electron spin states in molecular systems because of the small energy gap between states. For example, achieving >95 percent spin polarization requires millikelvin temperatures. Even if strategies are used to increase the energy gap (e.g., manipulating Zeeman interactions), magnetic field strengths of ~10 T still require temperatures on the order of 4 K (Harvey and Wasielewski 2021).

Developing similar optical approaches for spin-based molecules is very powerful and enables molecules to feature initialization and optical readout. The combination of these properties enables molecules to be treated as quantum objects and to open up approaches to single-molecule readout, analogous to that of defect-based systems. The quantum properties of defects in semiconductors, such as the NV center, are exemplary, but their spatial properties are sub-optimal. Defects are generally synthesized through ion bombardment, which leads potentially to nanometer-scale control, not the sub-angstrom control desired for most quantum applications. Coupling the optical interface of diamond with a system featuring tunability and atomistic precision would enable the seamless integration of molecules with state-of-the-art readout technology. To access such a system, one has to probe the

With this design, we can move forward and consider the periphery of the system. Careful molecular design can use tethering groups, such as siloxanes, to bind to SiO₂ or imbue the molecule with water solubility. The ability to design a molecule from the ground up with the desired properties, couple it to other molecular centers, and tune its properties offers a tremendous opportunity for chemistry to target QIS applications.

It is important to note that this approach is broadly generalizable; by identifying the key quantum attributes that are desired, chemists can build from the ground up. While the example is based on transition metals, lanthanides and organic molecules offer similar versatility as illustrated below. Lanthanides offer narrow emission lines—for example, erbium emission lies in the telecom range—while organic molecules feature radicals with long coherence times that can be readily initialized using photophysical processes. With all of these systems, individual units can be connected via a building block approach into larger conjugates. Metal–organic frameworks, covalent organic frameworks, polymers, self-assembly chemistry, DNA origami, and other chemical strategies are well established for creating large architectures that prove useful for assembling multiqubit arrays.

2.2.2b Creating Optically Addressable Molecular Qubits: Lanthanides and Actinides

The study of quantum phenomena and the development of quantum information processing systems have benefited from understanding the chemistry and physics of f-electron compounds and materials. f-electron compounds include those formed from lanthanide or actinide elements. In this part of the periodic table, the f-electron subshells are filled from f⁰ to f¹⁴ electronic configurations for the tri-positive cations from La–Lu in the lanthanides and Ac–Lr in the actinides, respectively. Ions and materials with occupied (or partially occupied) f orbitals have provided key test beds for fundamental studies of a range of quantum phenomena including superconductivity (H. Wang et al. 2019), quantum tunneling of magnetization (Goodwin et al. 2017; Guo et al. 2018), quantum critical points (Kaluarachchi et al. 2018), time crystals (Zhang et al. 2017), teleportation (Olmschenk et al. 2009), optical cycling (Siyushev et al. 2014), clock transitions and others. Open-shell f-electron compounds provide unique electronic and magnetic properties derived from their quantum mechanical characteristics. Coupling of their spin and unquenched orbital angular momenta results in total angular momentum J states that are split by crystal field levels into M_j sublevels with large magneto-crystalline anisotropies.

Quantum information can be controlled in f-electron spins because of their inherently almost core-like, quantum mechanical characteristics (Cheisson and Schelter 2019). The 4f shells of lanthanides and the 5f shells of actinides are in the “Goldilocks zone” between valence and core (Jochen Autschbach, private communication, April 2019). The valence-like properties allow for f-shells to be tuned using chemical modifications in the first coordination sphere of the metal cations for QIS applications. Conversely, the core-like nature of f-electrons provides inherent protection against decoherence for local and emergent phenomena. f-element materials that display notable quantum phenomena include ceramics and intermetallics.

In an orthogonal area of research, molecular complexes of f-elements have also been explored extensively for quantum effects in dilute spin systems, such as single-molecule magnets. While single-molecule magnets operate on the opposite principles as qubits, the deep knowledge that has been developed about the electronic structure of such species can be repurposed toward QIS, in particular, based on understanding magnetic relaxation (Reta, Kragskow, and Chilton 2021). In these and related compounds, blocked magnetization is relaxed through quantum tunneling processes and one- (Orbach) or two-phonon (Raman) phonon scattering processes. Understanding how the factors that influence electronic spin and molecular vibration can benefit both fields (Chilton 2022). Additional understanding of electronic structure for quantum information can be gleaned from work on new modes of bonding in f-elements that can contribute to new types of strong spin polarization. Gould and colleagues (2022) have made progress in this regard with the isolation of a compound showing a 0.5-order metal–metal bond between two lanthanide centers. This new family of compounds—(CpⁱPr₅)₂Ln₂I₃—shows magnetic blocking below 65 K for Ln = Tb and 72 K for Ln = Dy, as well as unprecedented strong magnetic anisotropy owing to very strong magnetic exchange. Building off this foundation of knowledge to create molecules that are not magnetically bistable, such as single-molecule magnets, but can be manipulated in a superposition will benefit the emerging area of molecular QIS.

Furthermore, lanthanides have core electrons that are essentially protected from the environment; thus, this attribute can be used to increase coherence or maintain properties across a range of external environments.

Certain lanthanides have the capability to interconvert light and spin information (i.e., to perform quantum transduction). Laorenza and Freedman (2022) demonstrated in one example the potential to interface molecular spins with telecommunication photons by designing systems with Er³⁺ emitters. These molecules have a natural transition at ~1540 nm and, when placed in the appropriate crystal field environment, function as an effective S = ½ ground state. This attractive optical transition has sparked impressive work with Er³⁺ dopants in yttrium orthosilicate, including single-spin control, quantum nondemolition measurements, and coherent control of multiple spin centers at the same optical spot. Molecular approaches provide a complementary approach to designing Er³⁺ emitters. As the natural 4f-4f transition at ~1540 nm is largely insensitive to the surrounding ligand environment, bottom-up design may be used to place nuclear spin memories on the surrounding ligands while developing the principles to achieve long spin coherence in systems that interface with telecom fibers.

A second example is the potential for molecular lanthanides to serve as optical memories. According to Laorenza and Freedman (2022), “These dopants in solid-state hosts (e.g., yttrium aluminum garnet, yttrium orthosilicate, yttrium orthovanadate, lithium niobate) have emerged as a valuable platform for quantum optical networking with single-spin optical readout of Ce³⁺, Pr³⁺, Nd³⁺, Yb³⁺, and Er³⁺, as well as long-lived quantum optical memories for ensembles of rare-earth ions. The trivalent ions Pr³⁺, Eu³⁺, and Tm³⁺ have demonstrated an optically addressable ground-state nuclear spin. Furthermore, the ¹⁵¹Eu³⁺ spins have been shown to exhibit coherence times as long as six hours. These dopants offer 4f-4f transitions that are highly shielded from their environment. This level of shielding gives rise to a narrow homogenous linewidth, Γ_hom, for spin initialization, long optical coherence time, T_{2, opt}, and high quantum yields. Translating these features into molecular Eu³⁺ spins, recent work from Kumar and colleagues (2021) demonstrated optical initialization and readout of the ground-state nuclear spin in dinuclear Eu³⁺ molecules, [Eu₂Cl₆(4-picoline N-oxide)₄(μ₂-4-picoline N-oxide)₂]·2H₂O, denoted (Eu₂) (Figure 2-3a). The narrow Γ_hom of 22 MHz enabled all optical initialization of the ground state. Using a mononuclear Eu³⁺ complex, [piperidin-1-ium][Eu(benzoylacetonate)₄], Serrano and colleagues (2022) demonstrated a three-order-of-magnitude improvement with a recorded Γ_hom value of 13 kHz (Figure 2-3b).” The energy diagram shown in Figure 2-3c illustrates the relevant transitions that are occurring in the dopant to allow for selective optical pumping; > 95% of the spin population is was initialized into the F₀M₁ = ± ½ sublevels.

Complexes of actinide elements, where the 5f- and/or 6d-valence electron shells are partially filled, also present unique opportunities related to the manipulation of electron spins for QIS applications. One possible advantage of actinides is that the larger radial extent of the 5f principal quantum shell and indirect relativistic effects that are operative for these heavy atoms render the 5f electrons more accessible for covalent bonding interactions and modifications through structural chemistry, as compared to the 4f electrons. Thus, actinide elements, especially early actinides, have been considered as conferring advantages that derive from a chemical behavior intermediate between lanthanide and d-block transition metal systems. While thus far the work on understanding the electronic

structure of these systems focused on uranium, significant space exists beyond that element. All isotopes of the actinides are radioactive, but hazards associated with the radioactivity of uranium, primarily ²³⁸U, and thorium, primarily ²³²Th, are relatively minor such that those isotopes can be handled in a conventional laboratory setting. Rinehart and Long (2009) reported the first uranium single-molecule magnet with a spin relaxation barrier of 20 cm⁻¹: U(Ph₂BPz₂)₃, [Ph₂BPz₂]⁻ = diphenylbis(pyrazolyl)borate. Evidently, the complex valence electronic structures of uranium complexes comprising strongly mixed electronic wave functions and a tendency for those to couple strongly with molecular vibrations undermines better performance. Deconvoluting these competing interactions represents an opportunity for creating a first generation of molecular qubits based off of these ideas that were developed for single-molecule magnets (Escalera-Moreno et al. 2019).

For actinide elements heavier than uranium, so-called transuranic elements, there are notable opportunities but also practical challenges due to radiotoxicity and requirements for the safe handling of these isotopes. The key isotopes of interest are ²³⁷Np and ²³⁹Pu, whose non-integer nuclear spins could couple with the valence electron spin. To advance the study of transuranic QIS systems, it is necessary to have access to rare isotopes and to the infrastructure to be able to synthesize and study them safely. Engineering or administrative requirements for sample handling of transuranics typically demand only small amounts of material (e.g., <5 mg of the isotope). This situation has resulted in the reporting of only a small number of Np and Pu complexes. Within the United States, much of the work on transuranic isotopes is performed in the national laboratory system, largely due to both safety and security concerns; a smaller fraction is performed in certain academic laboratories where the necessary licensing and infrastructure are in place.

2.2.2c Creating Optically Addressable Molecular Qubits: Organic Multispin Qubits

Molecular qubit design principles based on fully organic systems utilize the premise that decoherence in metal-based molecular spin-qubits has significant contributions from spin–orbit coupling and ZFS. In organic systems, spin–orbit coupling is generally very weak, leading to the prediction of longer decoherence times in organic spin-based molecular qubit architectures. Contributions to spin relaxation T₁ are dominated by the direct process, spin–phonon coupling, and the Raman process, while contributions to T₂ decoherence times are dominated by hyperfine coupling to spin ½ hydrogen atoms within organic molecules (Canarie, Jahn, and Stoll 2020).

Based on the DiVincenzo criteria (Box 2-2), a critical requirement for a physical qubit is the preparation of a pure initial state. In addition, the preparation of two-qubit entangled states is necessary to execute fundamental quantum gate operations. The primary challenge is generating well-defined initial quantum states of the system that maintain their spin coherence for times long enough to permit a useful number of spin manipulations to carry out quantum gate operations. As was discussed earlier in Section 2.2.2, these criteria are difficult to fulfill using the Boltzmann populations of the electron spin states due to the small energy gap between the states (Harvey and Wasielewski 2021).

Electron spins are good qubits because their two spin states constitute the quintessential two-level quantum system, in which the two states can exist in a superposition. In addition, coupling two or more spins via the spin-spin exchange (J) and/or dipolar (D) interactions results in rich spin physics that allow for quantum entanglement as well as implementation of two-qubit gates essential for quantum gate operations. Sub-nanosecond photodriven electron transfer (ET) from a molecular electron donor (D) to an acceptor (A) can generate two spatially separated, entangled electron spins that function as a spin qubit pair (SQP) in a well-defined pure initial singlet quantum state even at ambient temperature (Figure 2-4a) (Closs, Forbes, and Norris 1987; Thurnauer and Norris 1980).

Generally, the two spins of the SQP experience different magnetic environments as a consequence of differing electron-nuclear hyperfine interactions in D^•+ and A^•−, as well as differing spin–orbit interactions in each radical leading to different electronic g-factors. This results in coherent spin evolution from the singlet to the triplet SQP state (Closs, Forbes, and Norris 1987; Hore et al. 1987). In the absence of SQP spin relaxation or recombination, the spin coherence between the singlet and triplet SQP states can persist indefinitely.

However, spin decoherence and charge recombination to either the ground state via the singlet channel or a neutral molecular triplet via the triplet channel can occur. Upon application of a magnetic field that is much larger than J, D, and the electron-nuclear hyperfine interactions, only the S and T₀ states mix (Figure 2-4b), which results

in their overpopulation and produces strong spin-polarization that can be observed readily using time-resolved electron paramagnetic resonance (EPR) or optically detected magnetic resonance (ODMR) spectroscopies (Figure 2-4c) to yield detailed data on the spin dynamics of the system. For example, using time-resolved EPR techniques, researchers have shown that photogenerated SQPs can polarize a third spin (Colvin et al. 2013; Horwitz et al. 2016, 2017; Mi et al. 2006), engage in quantum–spin state teleportation (Rugg et al. 2019), and serve as a controlled NOT (CNOT) gate (Nelson et al. 2020). In addition, using g-factor engineering, individual spin qubit addressability can be achieved within an SQP system (Fernandez et al. 2015; Nakazawa et al. 2012; Olshansky et al. 2020).

A second promising approach to photoinitialized molecular qubits having optical pumping and addressability properties similar to those of NV centers uses photoexcited, covalently linked chromophore-stable radical (C-R•) systems. Three-spin systems that produce photogenerated molecular quartet states were first observed using porphyrin and fullerene chromophores connected to stable nitroxide radicals via covalent or coordination bonds (Corvaja et al. 1995; Fujisawa et al. 2001; Ishii et al. 1996, 1998; Mizuochi, Ohba, and Yamauchi 1997). Metalloporphyrins with paramagnetic metals also exhibit photogenerated quartet spin states (Gouterman 1970; Kandrashkin, Asano, and van der Est 2006a, 2006b; Poddutoori et al. 2019). In addition, more recent work has demonstrated that robust perylenediimide chromophores linked to nitroxide radicals can produce quartet states that subsequently spin-polarize the nitroxide doublet ground state (Giacobbe et al. 2009; Maylaender et al. 2021).

Figure 2-5 depicts a typical photophysical pathway for a C-R^• molecule. Upon photoexcitation, the chromophore of the doublet ground state (D₀) is optically pumped to its first excited state (D₁), followed by enhanced

intersystem crossing driven by the exchange interaction between the two spin-paired electrons on ^1*C with the unpaired electron on R^• to generate ^3*C-²R^•. The resulting three-spin system is best described at high magnetic fields as an excited doublet state (D₂) and a quartet state (Q). The D₂ and Q states are separated by an energy difference of 3J_TR, and Q is typically lower in energy than D₂. Since D₁ and D₂ have the same spin multiplicity, the transition from D₁ to D₂ is more rapid than to Q, which is populated by intersystem crossing from the D₂ state driven by the ZFS (Kandrashkin and van der Est 2003, 2004; Teki 2020).

Furthermore, the decay of Q to D₀ is spin forbidden, allowing a sufficiently long lifetime to probe and manipulate Q using resonant microwave pulses. The three electronic spins in the photogenerated quartet state Q form a multilevel qubit, or qudit, which can reduce the number of entangled gates required in quantum algorithms, thus improving algorithmic efficiency (Wang et al. 2020). C-R^• molecules also offer the possibility of optical readout of spin information using ODMR because the chromophore excited state is usually photoluminescent or has readily observable excited-state absorptions.

2.2.2d Creating Optically Addressable Molecular Qubits: Optical Cycling Centers

Optically active molecules could provide high-fidelity quantum-state initialization through optical pumping and high-fidelity qubit readout through the detection of laser-induced fluorescence. Both purposes require the ability to scatter optical photons repeatedly (i.e., optically cycle). To achieve optical cycling, it is necessary to find an optically closed molecular system, in which a molecule excited by resonant laser light returns to its initial quantum state with high probability and continues to interact with the laser light. Although optical cycling is routinely achieved in neutral atoms and atomic ions, it is challenging to realize in molecular systems owing to the large number of quantum states. A successful approach has been to engineer atom-like electronic structures—for example, using the NV centers or designer organometallic qubits discussed above. A complementary approach has been to identify molecules in the gas phase that permit optical cycling even in isolation. These molecules, known as optical cycling centers (OCCs), have electronic transitions with nearly diagonal Franck–Condon factors that enable the optical cycling of large numbers of photons.

The topic of OCCs lies at the intersection of atomic physics and chemistry. Initial work with small OCCs started when the first candidates were identified in the early 2000s (Di Rosa 2004; Stuhl et al. 2008). Early work had been pursued by the atomic physics community, whose motivation was to extend established optical techniques of atomic control such as optical pumping and laser-cooling to molecules. This effort has been successful. To date, many OCCs have been identified; experimentally, a variety of optical control techniques in atoms have been extended to OCCs. Notably, diatomic OCCs have emerged as promising molecular qubit candidates. As first proposed by DeMille (2002), trapped polar molecules can be a promising platform for quantum computation. A qubit can be encoded in two rotational states of a polar molecule to provide long-lived and highly coherent quantum memory. By relying on the long-ranged electric dipolar interaction that couples rotational states of a molecule, two-qubit gates sufficient for universal quantum computation can be naturally implemented (Ni, Rosenband, and Grimes 2018). On this frontier, recent work with laser-cooled OCCs trapped in programmable optical tweezer arrays has demonstrated the DiVincenzo criteria (see Box 2-2) required for quantum computing (Bao et al. 2022; Holland, Lu, and Cheuk 2022). In particular, the abilities to prepare and detect defect-free arrays of single molecules with high fidelity were demonstrated; coherent electric dipolar interactions and on-demand creation of entangled pairs of molecules were also shown. These results establish optical tweezer arrays of molecular OCCs as a promising new platform for quantum science. Notably, such molecular tweezer arrays are scalable to hundreds of qubits in the near term and naturally offer the ability for single-qubit resolved addressing (Figure 2-6).

We note in passing that another successful approach to creating ultracold and trapped molecules is coherently assembling them from samples of ultracold atoms, which can be routinely created and controlled to a high precision in the laboratory. These molecules, however, do not permit the high degree of optical cycling necessary for direct fluorescent detection, quantum-state preparation, or laser cooling. Nevertheless, assembled molecules are also being pursued successfully for quantum science applications. In fact, coherent electric dipolar interactions between polar molecules were first observed using assembled ⁴⁰K⁸⁷Rb molecules trapped in an optical lattice

(Yan et al. 2013). Recently, single ²³Na¹³³Cs molecules have also been successfully created and fully controlled in optical tweezer traps (Liu et al. 2018).

Parallel to the successful developments with diatomic OCCs, efforts to find larger OCCs have significantly increased. Larger OCCs could open up new areas in quantum science and chemistry, including new schemes for processing quantum information (Yu et al. 2019), enhanced sensitivity to new fundamental physics (Augenbraun et al. 2020; Hutzler 2020; Kozyryev, Lasner, and Doyle 2021), and new methods to control and witness molecular dynamics (Zhu et al. 2022). Recent proposals have also envisioned OCCs that are chemically bonded to surfaces, which could give rise to practical quantum devices such as sensors and quantum interconnects (Guo et al. 2021).

The chemistry community is increasingly interested in the topic of larger OCCs. Since the general structural and chemical principles that determine optical cycling properties are not fully known, identifying larger molecular candidates for OCCs remains challenging. Nevertheless, progress has been rapid, and varied organic molecules functionalized with OCCs have been discovered. Notably, recent collaborations between chemists and physicists have identified new chemical principles that determine optical properties in certain classes of organic molecules (Dickerson, Guo, Shin, et al. 2021; Lao et al. 2022; Zhu et al. 2022). These chemical principles, and those yet to be discovered, could guide the identification of candidate OCCs. Below, we describe in more detail some OCCs that have been explored to date.

Diatomic OCCs are well explored. Early work identified optical cyclable diatomic OCCs and an optical cycling scheme with only a few lasers (Di Rosa 2004; Stuhl et al. 2008). Initial experimental work with diatomic OCCs has focused on ²S radicals such as SrF (Barry et al. 2014), CaF (Anderegg et al. 2017; Truppe et al. 2017), and YO

(Collopy et al. 2018). With successful optical cycling, many atomic techniques such as sub-Doppler laser cooling and conservative trapping have been extended to diatomic OCCs to bring them into the ultracold quantum regime (Anderegg et al. 2018; Cheuk et al. 2018; Williams et al. 2018). Notably, single OCCs have been trapped and detected with high fidelity in optical tweezer arrays (Anderegg et al. 2019). Recent work in this platform has created additional defect-free arrays of OCCs, achieved programmable control over individual OCCs, and demonstrated an entangling two-qubit gate sufficient for universal quantum computation (Holland, Lu, and Cheuk 2022). These results establish OCCs as a viable platform for quantum simulation, quantum computing, and quantum-enhanced sensing, and further open the door to exploring chemistry with entangled matter.

In addition to ²S OCCs, a new class of ¹S molecules such as aluminum monofluoride (Hofsäss et al. 2021) and aluminum monochloride (Daniel et al. 2021) has garnered increased attention over the past few years. These molecules have favorable properties, such as magnetic insensitivity and narrow optical transitions, useful for highly coherent qubits and new types of molecular clocks but that come at the expense of ultraviolet transitions that are technically difficult to work with. Whether ¹S OCCs with convenient optical transitions exist remains an open question.

Motivated by the alkaline-earth fluorine motif in diatomic OCCs of CaF and SrF, initial work on polyatomic OCCs extended to alkaline earth alkoxide, M-O-R, molecules, such as SrOH, CaOH, and CaOCH₃ (Kozyryev et al. 2019; Kozyryev, Baum, Matsuda, et al. 2016; Kozyryev, Baum, Matsuda, and Doyle 2016). These efforts have been successful, and optical control over CaOH and CaOCH₃ is rapidly approaching that in diatomic OCCs (Hallas et al. 2023; Mitra et al. 2020; Vilas et al. 2022). Recent work has extended this idea to a large class of M-O-R molecules (Dickerson et al. 2022; Dickerson, Guo, Shin, et al. 2021; Dickerson, Guo, Zhu, et al. 2021; Mitra et al. 2022; Zhu et al. 2022). Notably, in M-O-R aromatic compounds functionalized with OCCs, the transition energies and degree of optical cycling have been found to correlate well with the electron-withdrawing strength of the R-ligand (Dickerson, Guo, Shin, et al. 2021; Zhu et al. 2022), providing a new chemical design principle to tune Franck–Condon factors. While the M-O-R appears to be a very successful design principle, whether other motifs for creating larger OCCs exist remains an open question. In a closely related area, recent work has also explored small molecules with multiple OCCs, where the OCCs could be separately optimized for different purposes such as sensitivity to new physics and favorable laser-cooling and optical trapping properties (Ivanov, Gulania, and Krylov 2020; Kłos and Kotochigova 2020; Yu et al. 2022).

2.2.2e Exploiting Entanglement and Quantum Transduction

The coherent coupling of molecules with photons allows long-range entanglement of qubits, opening up the possibility of quantum networks over long distances. Separately, photons can be used as a universal bus to transduce quantum information to other physical platforms such as superconducting qubits. In addition, visible to near-infrared (NIR) radiation provides a wealth of information through its frequency, phase, and polarization; it also provides spatial information down to the length scale of the optical wavelength or below using near-field techniques. Optical detection ensures that molecular qubits can be addressed on a single-molecule level. In particular, the strong interaction of molecular chromophores with light (visible to NIR radiation) provides a facile means of optically addressing spins to initialize and read out qubit states (Awschalom et al. 2018). Strategies for optically gating spin states on the single-molecule level have been demonstrated in transition metal-semiquinone complexes via photoisomerization-induced spin-charge excited states (Paquette et al. 2018) and single-photon-induced spin polarization strategies (Kirk, Shultz, Reddy Marri, et al. 2022).

According to Mao (2023), a promising route for realizing multiple-spin entangled molecular qubits in organic systems is the photogeneration of high spin states in organic semiconductors using singlet fission (SF). In SF, the absorption of a single photon generates a triplet-pair multiexciton, ¹(T₁T₁), in an initial singlet state (S = 0). (Smith and Michl 2010). If the ¹(T₁T₁) lifetime is sufficiently long, spin evolution can occur to produce the quintet (S = 2) state, ⁵(T₁T₁), before the spins decohere (Figure 2-7) (Basel et al. 2017; Kumarasamy et al. 2017; Matsuda, Oyama, and Kobori 2020; Sakai et al. 2018; Tayebjee et al. 2017; Weiss et al. 2017). The ⁵(T₁T₁) state comprises four entangled spins that can be initialized into a pure, well-defined quantum state by optical pumping. In addition, the spins of the ⁵(T₁T₁) state can be addressed and manipulated using pulsed microwaves to execute quantum gate

operations, the results of which can be read out using either pulse-EPR or ODMR spectroscopies (Smyser and Eaves 2020; Tayebjee et al. 2017; Weiss et al. 2017; Yunusova et al. 2020). The latter potentially offers single-spin sensitivity and is possible because the decay of the ⁵(T₁T₁) state by triplet-triplet annihilation produces delayed fluorescence. Using entangled photons to produce the ⁵(T₁T₁) state may also enable transduction between photons and spins that serve as propagating and stationary qubits, respectively.

Several other qualities of the ⁵(T₁T₁) state also make it attractive for QIS applications. Due to its high spin multiplicity, ⁵(T₁T₁) can be utilized as a five-level qudit, which may facilitate greater storage and processing of quantum information compared to systems with fewer accessible quantum levels (Moreno-Pineda et al. 2018; Wang et al. 2020; Wang et al. 2021). The ⁵(T₁T₁) state can also be photogenerated on demand at arbitrary locations, potentially with high spatial resolution using nanophotonic architectures (Kauranen and Zayats 2012; Tame et al. 2013). Investigations of the ⁵(T₁T₁) state for QIS applications have been limited, despite its intriguing potential properties (Bayliss, Weiss, et al. 2020; Jacobberger et al. 2022; Weiss et al. 2017). Consequently, the mechanisms of its decoherence are not well understood, and strategies for extending coherence lifetimes have not been fully realized. While SF has been observed in molecular dimers or higher aggregates of more than 200 chromophores in both solution and the solid state (Casillas et al. 2020), ⁵(T₁T₁) has been relatively elusive in solid-state materials (Bae et al. 2020; Lubert-Perquel et al. 2018; Matsuda, Oyama, and Kobori 2020; Pace et al. 2020; Weiss et al. 2017). To realize a long-lived ⁵(T₁T₁) state, the interchromophore electronic coupling must be sufficiently strong to allow efficient SF and prevent dissociation of the triplet-pair state, yet sufficiently weak to minimize triplet-triplet annihilation and triplet diffusion. Striking this delicate balance is a major challenge. In the solid state, electronic coupling is dictated by the crystal structure, which is difficult to predict or rationally control due to the various weak intermolecular forces that govern molecular packing (Corpinot and Bucar 2019; Day and Cooper 2018). Recently, the advantages of engineering the crystal morphology of tetracene (Bayliss, Weiss, et al. 2020; Jacobberger et al. 2022; Weiss et al. 2017) and related polyacene (Rugg et al. 2022) single crystals were demonstrated. In the most favorable case reported, the lifetime of ⁵(T₁T₁) is 130 μs, and the spin coherence lifetime is 3 μs at 5 K with ⁵(T₁T₁) readily observable even at room temperature (Jacobberger et al. 2022). The single crystals spatially align and organize the interchromophore spacing to optimize the electronic coupling needed to achieve favorable ⁵(T₁T₁) properties. This enables more facile identification of decoherence sources, facilitating the development of guidelines to further tailor the ⁵(T₁T₁) state for QIS applications.

2.2.3 Beyond Qubits: Multispin Systems, Error-Mitigation and Error-Correction

While challenges in the field of molecular QIS have focused primarily on the structural factors that control decoherence rates and relaxation dynamics of qubit states, significant challenges remain in the design and control

of both superposition and entangled states critical for quantum algorithms, error correction and error mitigation, communications, and sensing. Within this context, “qudits,” or d-dimensional quantum systems, have significant advantages over simple two-state qubits. The entangled quantum states lead to the implementation of quantum algorithms with decreased code sizes and fewer entangling gates, quantum error correction, and robust quantum cryptography protocols. For example, the possibility of information-scrambling protocols with entangled qutrits (d = 3) leads to effective teleportation protocols that can be carried out even in the presence of decoherence and decreased fidelities for quantum communication (Blok et al. 2021). The physical realization of qudits and qutrits has been explored within the physics realm through entangled superconducting qubits and entangled photons (Kues et al. 2017; Sciara et al. 2021).

Early approaches to quantum control in molecular systems involved the physical realization of single-qubit, two-qubit, and qudit logical gates within rotational and vibrational states of a diatomic molecule with operations via resonant Raman transitions (Shapiro et al. 2003). The entanglement of nuclear and electron spin or electron and electron spin that is possible in molecular systems allows the discovery of molecular qudits. Unlike two-level qubits (b = 2), qudits are quantum systems with dimensional space (d > 2) (Moreno-Pineda et al. 2018). The entanglement among electron-nuclear (S + I), nuclear-nuclear (I + I), and electron-electron (S + S) states in molecular dimers or trimers allows the development of strategies to uniquely address error correction and mitigation critical to quantum computation, sensing, and telecommunication protocols.

Within the context of information theory, a question arises as to how one transmits information reliably over a noisy channel. Shannon (1948) developed the noisy channel coding theorem, which quantifies the upper limit of information transmitted over a noisy channel and provides an upper limit to the protection afforded by error-correcting codes. Although no analog to Shannon’s noisy channel theorem exists for quantum information, the theory of quantum error-correcting codes is sufficiently developed to allow for reliable communication over noisy quantum channels. The basic theory of quantum error correction protects quantum information against noise by encoding quantum states with redundant information, followed by decoding to recover the original state. Within the theory of quantum error-correcting codes, there are those derived from classical theory of linear codes that give rise to the Calderbank–Shor–Steane codes and stabilizer codes (Nielsen and Chuang 2009). Effective quantum error correction need not assume that encoding and decoding of quantum states must be carried out without error, as would be the case for noisy quantum gates. Fault tolerance, as defined by the threshold theorem for quantum computation, allows for a certain degree of faulty gates. The theorem roughly states that if the noise in individual quantum gates is below a certain constant threshold, it is possible to efficiently perform an arbitrarily large quantum computation. In current practice, quantum error correction is a method of protecting quantum information from errors by encoding that quantum information in quantum error-correcting codes and making a series of measurements (known as syndrome measurements) in order to determine whether errors have caused the quantum information to leave the valid space of those codes. Unlike classical information, quantum information cannot be copied (i.e., the no-cloning theorem); thus, simply redundant encodings like a repetition code are not possible. Instead, quantum error-correction codes often encode logical information in global properties of quantum states, such as in topological features of the states. Critically, quantum error correction involves extraction of entropy from the quantum state through a series of syndrome measurements and a reset of qubits.

It is important to distinguish between error correction and error mitigation. Error mitigation is a loosely defined set of techniques that are usually algorithmic in nature and that aim to extract not-noisy quantities from noisy quantum computers. Usually error mitigation does not include techniques that extract entropy from the system via reset and measurement (such techniques are in the realm of error correction). As a result of this, error mitigation often amounts to making a trade-off between the number of circuit repetitions and the fidelity of the estimated quantities. For example, many methods of error mitigation aspire to the limit of perfect error detection and postselection. Although error-mitigation techniques are generally not scalable in the way that error-correction techniques are, all state-of-the-art noisy intermediate-scale quantum (NISQ) experiments leverage some form of error mitigation. However, error mitigation is often used as a catchall and includes techniques like spin echoes, which are basically pulses equating to the identity gate that are used to decorrelate errors.

A critical figure of merit for error mitigation is quantum fidelity, which is the squared overlap of some errant quantum state with the intended (error-free) quantum state. If the fidelity of a quantum system is 1, then there

were no errors and the output state is perfect. If the fidelity is 0, then there was an error and the states are now orthogonal. The infidelity is just 1 − fidelity.

In the context of error correction, the gates need to have a certain fidelity in order to be below the threshold for an error-correcting code to work. When describing the fidelity of a gate, if this gate is applied to an input state, what is the output fidelity (squared overlap) of the state that is produced with respect to the state it is supposed to produce (i.e., a measure of the error rate of the gate)? If that gate has 0.99, or “two nines,” fidelity, that means an error will occur roughly 1 in 100 times that we apply the gate. The threshold error rate for gates in the surface code is approximately 0.999. For error correction, an important goal is to develop a scalable system that includes a sufficient number of physical qubits with gate fidelities below the threshold (e.g., 0.99). In such a case, the probability of an error occurring at all is suppressed exponentially, leading to a scalable system in which error correction can be carried out. However, if the intention is to run a NISQ algorithm with error mitigation only and without error correction, in order to detect one correct output, the number of times the circuit must be run to see one correct output (the number of gates × gate fidelity) is very large and is not scalable.

The physical implementation toward error correction and error mitigation on the molecular level relies primarily on single-ion magnets, in which the electron and nuclear spin states are weakly anisotropic and exchange coupled, leading to multiple spin states that are entangled. For suitably engineered systems, unequal energy spacings allow addressing via microwave resonant pulses (Aguilà, Roubeau, and Aromí 2021; Chicco et al. 2021; Ferrando-Soria et al. 2016; Gimeno et al. 2021; Godfrin et al. 2017; Hussain et al. 2018; Jenkins et al. 2017; Luis et al. 2011, 2020; Moreno-Pineda et al. 2017, 2018). Molecular electron-nuclear spin-based qudits have been utilized for implementation of quantum error-mitigation codes, where the molecular systems function as NISQ systems in which the electronic structure is “protected from decoherence” (Chiesa et al. 2020; Macaluso et al. 2020). Taking advantage of nuclear spin structures that function as qudits leads to the possibility for long coherence times due to isolation of the system from the environment but, consequently, long manipulation times. Strategies to shorten the manipulation times for gate operations involve taking advantage of electron–nuclear coupling (hyperfine interactions) to perform operations on nuclear spin states at rates much shorter than the decoherence times (Castro et al. 2022; Chizzini et al. 2022; Hussain et al. 2018).

Coherent control of molecular qudit states requires magnetic dilution in order to minimize dipolar coupling, alignment of molecular axes within the crystalline environment (such as can be obtained in a diamagnetic crystalline host environment or encapsulation), and sufficient magnetic anisotropy to address each transition independently. The magnetic anisotropy, however, needs to be small enough to access transitions within experimentally accessible microwave frequencies. As mentioned in Section 2.2.2b, lanthanides with zero spin–orbit coupling (e.g., orbitally quenched f⁷ configuration) and small ZFS, such as Gd (S = 7/2), have been investigated extensively (Jenkins et al. 2017). X-band pulsed EPR experiments allow independent addressing of observed transitions, with little dependence of T₁ and T₂ on each transition and the observation of Rabi oscillations. Implementation of a universal Toffoli gate (controlled-controlled NOT [CCNOT] gate) in the GdPOM system was demonstrated, as well as implementation of the Deutsch (Kiktenko et al. 2015), Grover (Godfrin et al. 2017), and quantum phase estimation algorithms (Wang et al. 2020) with shorter times and fewer operations than on qubit (S = ½) systems. The implementation of quantum error-correction gates and algorithms successfully with microwave or radiowave pulse control in molecular qudits suggests feasibility for coherence control of entangled states, making this a promising area for new breakthroughs in the molecular sciences.

Molecular electron-nuclear spin-based qudits have been utilized for implementation of quantum error-mitigation codes, in which the electronic structure is “protected from decoherence” and the molecular systems can function as NISQ systems. The dominant source of decoherence or error can be ascribed to the electron spin units’ high susceptibility to interact with the nuclear spin bath (typically 10² spins) via hyperfine interactions, which results in a nonexponential decay behavior. Simulations of the coupled system-bath dynamics predict that while the squared fidelity of the recovered state is above 0.9 for up to ~30 μs for S = ½, the recovered fidelity is above this value for 40–300 μs for qudit spin states of S > ½ (3/2 and 9/2, respectively) (Petiziol et al. 2021). The high fidelity accompanied by long evolution times is significant for the development of quantum error-correction and error-mitigation schemes, in which the long implementation operations (encoding, detection, recovery) must fall within the evolution time. The assumption here is that the electronic qudit energy gaps are larger than the energy

gaps in the nuclear spin bath or, for a nuclear qudit, than the hyperfine interaction (electron-nuclear interaction). Hussain and colleagues (2018) showed that coupling of a nuclear qudit to an S = ½ electronic spin ancillae (d = 2) offers the combination of the long decoherence times associated with nuclear degrees of freedom and the large reduction of nutation time induced by electron-nuclear (hyperfine) mixing to enable coherent control of a qudit by radiofrequency pulses. For systems in which the electron-nuclear transitions are well resolved, coherence times are longer than operation times, and coherent control of dynamics allows for implementation of simple gates; qubit–qudit systems can be exploited to implement quantum error-mitigation and quantum simulation algorithms (Chicco et al. 2021; Hussain et al. 2018).

Large energy splitting between M_J states, large g-factors that provide polarizability for better initialization capacities, and electron-nuclear transitions present in lanthanides make this class of molecular candidates attractive as targets for error-correction and error-mitigation codes. Quantum transitions can be driven coherently (coherent manipulations) via electromagnetic pulses, and the time that quantum coherence is maintained is the phase memory time T_m. Decoherence occurs predominantly through dipole-dipole interactions and hyperfine coupling (all spin-spin interactions) versus T₁ spin relaxation, in which spin-phonon, Orbach (one-phonon), and Raman (two-phonon) mechanisms play a role. By applying spin-echo pulse sequences in GdW₃₀ systems S = 7/2, 99 percent fidelity can be achieved in less than 10 ns (much shorter than T₂ ≃ 2 μs); reaching the same result with a sequence of monochromatic pulses would take more than 1 μs (Castro et al. 2022). Such strategies allow reaching a high fidelity of the outcome wave function with a single control pulse.

Challenges in implementation arise due to the limited frequencies of EPR spectrometers, giving rise to electrically gated techniques that may offer additional possibilities for pulse manipulation (Castro et al. 2022). The multilevel structure of the qudit [Yb(trensal)] (I = 5/2) can encode a d = 6 qudit that can be exploited for coherent control of the nuclear-spin degrees of freedom by nuclear magnetic resonance and via hyperfine coupling to electron spin (S = ½) (Hussain et al. 2018). A minimal code protecting against amplitude or phase shift errors can be implemented within a Gd-oxalate complex, suggesting a strategy for error mitigation with reasonably high fidelity. The possibility of carrying out error-correction protocols in heterobimetallic lanthanide complexes (LnLn) has been demonstrated (Aguilà, Roubeau, and Aromí 2021). As shown in Figure 2-8, a LnLnLn trimer [000]-[111]

was investigated, and the quantum error code for a three-qubit phase-flip repetition code was successfully carried out by resonant pulses. The first step (encoding) involved two CNOT operations; followed by a πi/2 pulse; followed by T_m, with a reverse step (decoding); followed by the correction step, consisting of a CCNOT operation with the central qubit as the target and the two ancillae as control qubits. Time-dependent numerical simulations were performed to assess the protocol, and the error was found to be efficiently reduced by the correction code, allowing implementation of 50–100 gates before repetition was required (Aguilà, Roubeau, and Aromí 2021).

Purely organic molecular qudits can be designed by coupling an organic stable radical with a doublet state (S = ½, qubit) to a photoexcited organic triplet state (S = 1, qutrit) to generate spin-polarized quartet and doublet states, depending on the sign and magnitude of exchange coupling between the triplet and doublet states (Qiu et al. 2022; Wang et al. 2021). As organic systems have extremely small spin–orbit coupling, the decoherence times are long (10–100 μs) and dominated by hyperfine and/or dipolar coupling to the nuclear spin bath. As optical initialization is possible in these systems, future work in this area may involve the demonstration of simple gates with high fidelities, which have so far not been implemented for organic qudit states. Alternate strategies for optical initialization of molecular organic qudit systems have been demonstrated in semiquinone radical–cobalt complexes (Kirk, Shultz, Hewitt, et al. 2022; Paquette et al. 2018). Critical to this strategy is an understanding of the sign and magnitude of exchange in the excited state, the quantum yield of excited-state (triplet) population, and the competition between rates of excited-state relaxation to the ground state versus decoherence of the resultant spin states. Ultimately, in order to implement desired quantum protocols, the excited-state relaxation must be at least an order of magnitude slower than the decoherence rate.

2.3 INVESTIGATING THE INTERACTIONS OF MOLECULAR QUBITS WITH THEIR ENVIRONMENTS

Synthetic chemistry can play a key role in the design of molecules with so-called clock transitions (Gaita-Ariño et al. 2019), which can lead to enhanced coherence by providing protection from various environmental decoherence sources. Spin clock transitions are found at avoided level crossings associated with the Zeeman splitting of qubit states in a magnetic field. Named after the principle that gives atomic clocks their exceptional temporal stability, spin clock transitions provide an optimal operating point at which the qubit resonance frequency, f, becomes insensitive to the local magnetic field (B₀) fluctuations (i.e., df/dB₀ = 0) (Brantley et al. 2022). In this way, a molecular clock qubit is immune to magnetic noise. There are several synthetic strategies for generating molecular clock transitions. The key requirement is a term in the spin Hamiltonian that does not commute with the Zeeman interaction. For molecules containing metal centers with integer total spin (an even number of unpaired electrons), the crystal or ligand field interaction can do the job: clock transitions can be tuned through manipulation of the coordination environment around the metal center (Giménez-Santamarina et al. 2020; Sørensen et al. 2017). In the first molecular example, a crystal-field clock transition was demonstrated for a Ho(III) ion ([Xe]4f¹⁰ electronic configuration) encapsulated within a polyoxometalate moiety, resulting in a significant enhancement in spin coherence for a crystal rich in fluctuating electron and nuclear spins (Kundu et al. 2023; Shiddiq et al. 2016) (Figure 2-9).

An alternative strategy that works for molecules possessing half-integer spin states (i.e., an odd number of unpaired electrons) involves the electron-nuclear hyperfine interaction, which has been employed widely in trapped-ion quantum devices (Wright et al. 2019). Crucially, in the molecular case, the hyperfine interaction can again be controlled using coordination chemistry to maximize unpaired electron spin density at the relevant nuclear site. Recent examples have shown that this is possible by varying the degree of s-orbital mixing into a spin-bearing molecular d-orbital (Kundu et al. 2022; McInnes 2022; Zadrozny et al. 2017). This has the added advantage of increasing the s-orbital character, which reduces spin–orbit coupling and suppresses spin-lattice relaxation.

Additional recent work (Bayliss et al. 2022) has demonstrated that by tuning the local structural environment in the vicinity of a molecular qubit, it is possible to prepare molecules with clock-like transitions operating in practical frequency ranges that were not possible in their native structures (Figure 2-10). The potential to access these transitions lies not only within the molecular design but within instrumental flexibility. Using cavity-based EPR spectrometers, one is often limited by the field and frequency range that can be accessed (see Section 3.4 of this report). Moving to custom broadband microwave resonators analogous to the types that are used to probe

defects in semiconductors enables access to much wider field/frequency combinations. The potential for tuning into clock-like transitions with a more flexible operating field and frequency could enable more robust transitions to be harnessed for future devices.

Clock transitions can also be incorporated into frameworks, bringing together concepts of arrays of spins with structurally induced control over coherence time. Figure 2-11 illustrates one such example: by tuning the interaction of nuclear and electronic spins, a clock-like transition was engineered within a metal–organic framework (MOF).

With the discovery that scalable quantum applications are possible using single-photon sources, linear optical elements, and single-photon detectors, approaches based on cluster states or error encoding have made all-optical architectures promising targets for the ultimate goal of large-scale quantum devices. Challenges involve the development of high-efficiency sources of indistinguishable single photons, scalable optical circuits, high-efficiency single-photon detectors, and low-loss interfacing of these components (O’Brien 2007). Controlled coupling of organic chromophores to photonic structures has led to the development of single-photon quantum emitters, which could play a key role in molecular systems for QIS. With transition linewidths of ~10 MHz at low temperatures, organic chromophores can function as single-photon sources with long coherence times that are scalable and compatible with diverse integrated platforms. In addition, such chromophores can be used as transducers for the optical readout of electrical and/or magnetic fields and material properties for quantum sensing with single-quantum resolution (Dickerson, Guo, Zhu, et al. 2021; Toninelli et al. 2021; Wang, Kelkar, et al. 2019). Recent strategies for the generation and manipulation of organic-based photonic qubits require the generation of polarized light emission (or absorption) by utilizing the crystallographic order of the resulting environment.

Organic crystals doped with organic emitters function as single quantum emitters with a well-defined polarization relative to the crystal axes, making them amenable to alignment with optical nanostructures. The radiative lifetime and saturation intensity varies little within the crystalline environment, and a large fraction of these emitters can be excited more than 10¹² times without photobleaching (Polisseni et al. 2016).

2.4 DESIGNING MOLECULAR STRUCTURES WITH INTEGRATED CHIRALITY-INDUCED SPIN SELECTIVITY EFFECTS

Chirality is a key property of molecules important in many chemical and nearly all biological processes. Recent observations have shown that electron transport through chiral molecules attached to solid electrodes can induce high spin polarization even at room temperature (Wasielewski 2023). The ability to produce highly spin-polarized electrons at ambient temperatures is potentially important for developing room-temperature quantum devices.

Electrons with their spin aligned parallel or antiparallel to the ET displacement vector are preferentially transmitted depending on the chirality of the molecular system, resulting in chirality-induced spin selectivity (CISS). The first evidence of the relationship between chirality and electron motion dates back to 1999 when Naaman and Waldeck (2012) observed a large asymmetry in the transmission of oppositely spin-polarized electrons by thin films of chiral molecules. The coupling of orbital angular momentum to spin angular momentum in directional ET processes can provide a method of manipulating spin polarization. However, little is known about the interplay of CISS with the spin dynamics of molecular ET processes. One way to address this question is to explore the ET dynamics of covalently linked donor–chiral bridge–acceptor (D-Bχ-A) molecules following photoexcitation. The CISS effect on the coherent spin dynamics of photogenerated radical pairs in these systems depends on competing photophysical processes, most of which operate on a sub-nanosecond timescale (Aiello et al. 2022; Harvey and Wasielewski 2021). While femtosecond and nanosecond transient absorption spectroscopies can be used to probe the ET dynamics of D-Bχ-A molecules, time-resolved EPR spectroscopy is essential to elucidating their spin dynamics.

Photoexcitation of the donor (D) or acceptor (A) in a molecular electron donor-bridge-acceptor (D-B-A) system can result in the formation of a D^•+-B-A^•− quantum-entangled electron SQP initially in a pure singlet state. Coherent spin evolution of this system results in a partial triplet character, which results in strong electron spin polarization that can be observed by time-resolved EPR or ODMR spectroscopies. If charge transfer occurs through a chiral bridge, such as in D-Bχ-A, the CISS effect induces a spin polarization that depends on the chirality and the direction of the ET (Aiello et al. 2022). As a result, only one of the four two-spin states is populated in the weak-coupling limit, with vanishing entanglement between the hole and electron spin (Figure 2-12a) (Luo and Hore 2021). Moreover, CISS retards the rate of the radical pair recombination reaction, making it possible to extend the radical pair lifetime (Hafner et al. 2018).

Pulse-EPR techniques have been used to obtain detailed information about magnetic exchange (J) and dipolar coupling (D) for photogenerated spin-correlated radical pairs in a variety of systems, where D gives detailed distance and structural information, as well as to provide a direct probe of spin coherence in the radical pair (Aiello et al. 2022). For example, if photogeneration of the radical pair is followed by a microwave Hahn echo pulse sequence, π/2 - τ - π - τ - echo (Figure 2-12b), and the time delay τ is scanned, coherent oscillations between ∣Φ_A〉 and ∣Φ_B〉that are related to both J and D result in modulation of the spin echo amplitude (Figure 2-12c) (Tang, Thurnauer, and Norris 1994; Thurnauer and Norris 1980). When this experiment is performed on spin-coherent radical pairs, the echo appears “out of phase” (i.e., in the detection channel in quadrature to the one in which it is expected) and is therefore termed out-of-phase electron-spin-echo envelope modulation (OOP-ESEEM). Recent theory has shown that based on the phase relationships of these coherent oscillations, OOP-ESEEM can be used to detect the CISS effect on the spin coherence of spin-correlated radical pairs (Chizzini, et al. 2021; Fay 2021; Fay and Limmer 2021).

Across a range of 1.2–2.5 nm, von Kugelgen and colleagues (2021) found that the electron spins have a negligible effect on coherence times, a finding attributed to the distinct resonance frequencies. Coherence times are governed, instead, by the distance to nuclear spins on the other qubit’s ligand framework (Figure 2-14). This finding offers guidance for the design of spectrally addressable molecular qubits. This work lays the foundation for the bottom-up integration of multiple qubits with distinct functions into custom quantum systems, such as pairs of anisotropic quantum sensors for mapping vector fields in three-dimensional or multifunctional arrays featuring unique quantum elements designed for initialization, sensing, storage, and readout.

Moving beyond individual units, there has been a significant amount of work on bimetallic systems, which are powerful because they allow simple operations with the potential for frequency addressability that will enable their use as quantum gates. Some notable examples include bimetallic lanthanide systems (Aguilà et al. 2014) and systems connecting two wheel-like species featuring ground states amenable to quantum manipulation (Ardavan et al. 2015). In each of these approaches, the modularity of the linker enables

control over both the separation between the spins and potentially the form of gate that could be executed; for example, an electrochemical gate could be enabled by a redox switchable linker (Figure 2-15) (Walsh and Freedman 2016).

Moving from discrete systems to arrays of spins requires careful design of the local spin, linker, and interaction of molecular systems with both interfaces and the environment. There are numerous examples of MOFs featuring spins; however, there are a limited number of studies in which the coherence properties of an array of spins have been examined. Arguably, the first such example of a fully concentrated array is a two-dimensional porphyrin lattice, which was measured to have spin coherence (Urtizberea et al. 2018) and is an important achievement. A second approach to measuring coherence in a lattice of spins connects clock transitions with the design of individual moieties that form the building blocks of an MOF (Zadrozny et al. 2017). Clock transitions were described earlier in this chapter and are particularly powerful for mitigating the effects of magnetically noisy environments such as those found in MOFs.

MOFs provide many new opportunities for QIS. For example, a recent result demonstrates using an MOF for lithium-ion sensing (Figure 2-16) (Sun et al. 2022). Specifically, the porosity of MOFs can be harnessed for sensing intercalated analytes. In a different approach, one could envision using a two-dimensional MOF to position individual quantum units, such as sensors, onto an analyte with a great level of precision. Using framework chemistry, it is possible to employ a building block approach for positioning quantum units. The linking unit tunes both the distance and the strength of magnetic coupling between the quantum units, while different quantum units can be positioned within the framework. Significant work has focused on creating MOFs with designer topologies and incorporating different core units (Figure 2-17). The molecular nature of MOFs also enables tuning their phonon spectrum, which determines the interaction of qubits with the thermal energy of the environment and, therefore, T₁.

or molecular); augment the existing functionality to incorporate initialization, addressability, and/or readout in hybrid architectures; and can be used to control and mitigate primary decoherence channels. The synthesis, fabrication, characterization, and understanding of nanomaterials are mature with applications ranging from microelectronics and energy storage to renewable energy—these same approaches can be used to accelerate discovery and control in molecular qubits systems. Chemical inclusion or covalent design of qubits/qudits into carbon-based materials offer promising strategies for development of multiqubit arrays. By taking advantage of delocalization in carbon frameworks, strategies for generating structurally well-defined nanographene biradicals (Lombardi et al. 2019) and porphyrin tapes (Van Raden et al. 2022) reveal long memory times with well-defined environments for inclusion into graphene-based nanostructures. Chapters 3 and 4 will discuss more general applications of inorganic materials for qubits.

One of the challenges within a variety of wide-bandgap semiconductor materials is maintaining stable charge defect states of spin qubits for quantum computing, communication, and sensing. In particular, charged vacancy and divacancy states in these systems are often subject to noise-driven ionization processes and surface potential fluctuations that rapidly decohere and ultimately destroy the quantum state, typically requiring re-initializing/resetting the system. These unstable and often random dynamics limit the stability of qubits for broad applications in QIS and engineering. There is a high-impact opportunity for chemical passivation and controlled electronic stabilization of these materials that would enable longer coherent lifetimes and a subsequent increase in quantum sensing sensitivity and readout fidelity. A combination of theoretical modeling of electronic structure and targeted chemical synthesis offers a promising solution to this problem while providing fundamentally new mechanisms of quantum-state stability and control (see below).

Atomic-scale gating and local doping would accelerate the engineering of scalable semiconductor qubits and controlled entanglement of both electronic and nuclear spin states. Moreover, this level of control would provide new routes for spin-to-charge conversion to achieve high-fidelity single-shot readout and extend both spin relaxation and coherence times in the solid state (Anderson et al. 2022). Opportunities to model, design, and synthesize electrically controllable molecular states with functional solid-state interfaces would provide a spatial level of control difficult to achieve through traditional inorganic fabrication techniques and may offer opportunities for multifunctional quantum control through optical and electronic properties. Chemically activated local gating has the potential to realize the atomistic control needed for activating quantum sensors, controlling quantum memories, and potentially gating entangled quantum registers between adjacent spin states (both electronic and nuclear). In particular, controllable atomic-scale doping through proximal molecular systems may activate coherent states as well as enable photonic processes where optical properties are dependent on the local ionization state.