Quantum Gates Between Mesoscopic Spin Ensembles

Introduction.– Large-scale quantum operations are difficult to develop, requiring isolation of a large number of individual qubits, precise control of their quantum states, error correction schemes and high-accuracy measurements. Existing qubits are implemented in environments requiring high vacuums or low temperatures. Several recent initiatives have been driving hardware development for the realization of new quantum technologies that are more scalable, robust, and less physically demanding. Qubits should have long coherence times or be able to retain quantum information for times much longer than the operation time of quantum gates. Arbitrary unitary transformations on the set of all qubits can be constructed up to desired precision by composing quantum gates chosen from a set of universal quantum gates. The latter typically consists of single-qubit gates (one per qubit) and one two-qubit gate (acting on pairs of qubits) [1]. Implementation of the two-qubit gate has proven to be the most challenging, and even the leading platforms for quantum computing, i.e., superconducting qubits [2], trapped ions [3], and Rydberg atoms [4] have error rates exceeding 0.1% per gate, which is an order of magnitude larger than the threshold required for fault-tolerant quantum computing [5].

Ensemble quantum computing has the advantage of many replicas, eliminating the need to repeat projective readouts thousands of times to obtain the statistical weights in the many-qubit wavefunction. Ensembles can have favorable properties such as ease of fabrication, built-in robustness, longer storage times, more sensitive detection, etc. Liquid-state NMR is of historical significance because it provided the platform for the first experimental demonstrations of a working quantum algorithm [6, 7]. Nuclear spins in the liquid-state feature long coherence times, typically in the order of seconds. They interact weakly, meaning that they behave independently and can be addressed individually. Qubits can be selectively addressed using frequency-selective (soft) pulses. This allows coherent control of quantum devices containing up to 12 qubits [8] and implementation of Shor’s quantum algorithm [9]. However, nuclear spin qubits are not scalable due to the difficulty of initializing pure quantum states (highly mixed pseudo-pure states above $\approx 1$ mK), and synthesizing molecules with a large number of individually addressable nuclear spins coupled to one another. Indeed, molecules can only accommodate the selective addressing of a small number of distinct nuclei due to the narrow range of chemical shifts. Therefore, obtaining entanglement in liquid-state qubits for ensemble NMR quantum computing is nearly impossible [10]. NMR ensemble computing involves performing collective quantum operations on molecules, followed by averaging the state of qubits over the ensemble. To address the important issues of scalability and ease-of-fabrication, we revisit the idea of ensemble computing and consider reversing the order of averaging: ensemble-average spins to obtain large qubits, then perform operations among the ensembles. In doing so, we use the rotational properties of large-$J$ angular momentum operators, which leads to a realization of qubit transformations similar to the single spin case [11]. In this paper we will assume the ability to fully polarize spins and work with spin-coherent states that we term E-qubits. Practical considerations are discussed.

For nuclear spins the notation $\mathbf{I}$ is used instead of $\mathbf{J}$ to denote the spin angular momentum. Suppose we have an ensemble of $N$ molecules, each with $n$ nuclear spins $\{\mathbf{I}^{1},\dots,\mathbf{I}^{n}\}$. The Hilbert space for the nuclear spin degrees of freedom in a molecule has dimension $\prod_{i=1}^{n}(2I^{i}+1)$. The state of each molecule’s nuclear spins is represented by a projector $P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$, $\omega=1,\dots,N$. Let $\ket{\varphi_{i}(\omega)}$ be the wavefunction describing the state of spin $i$ on molecule $\omega$. Each projector is initially a product state, i.e., $P_{\omega}=\ket{\psi_{\omega}}\bra{\psi_{\omega}}$, where $\ket{\psi_{\omega}}=\ket{\varphi_{1}(\omega)}\otimes\dots\otimes\ket{\varphi_{n}(\omega)}$, i.e.,

Here, $\omega$ is standard notation in probability theory to denote the elementary outcome of a random variable. The unitary propagator corresponding to the quantum circuit is denoted $U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$. The final projector is $U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})P_{\omega}(\mathbf{I}^{1},\dots,\mathbf{I}^{n})U(\mathbf{I}^{1},\dots,\mathbf{I}^{n})^{\dagger}$. Averaging over $N$ molecules (denoted by overline):

where

is a statistical (density) operator and $\sum_{\omega}p_{\omega}=1$. The readout of observable operator $A$ is obtained as:

where $e_{1}$ distinguishes this form of ensemble averaging. On the other hand, suppose we start with $n$ ensembles, each described by a statistical operator, $\rho_{i}(\mathbf{I}^{i})$, describing this initial ensemble averaging. Each ensemble contains $N$ spins. The initial state is a product state $\rho_{1}(\mathbf{I}^{1})\otimes\dots\otimes\rho_{n}(\mathbf{I}^{n})$, where

The propagator for the quantum circuit is denoted $V(\mathbf{I}^{1},\dots,\mathbf{I}^{n})$. (The notation $V$ instead of $U$ indicates that the circuit implementations may be different, depending on the particular physical implementation of the qubits.) Evolution of this product state leads to:

The readout for this ensemble-averaging method, labeled $e_{2}$, leads to:

As a consequence of the different orders of averaging, we note that even if $U=V$, $A^{(e_{1})}$ and $A^{(e_{2})}$ do not generally give the same result. The averaging to obtain $A^{(e_{1})}$ only requires specifying the probability distribution $\{p_{\omega}\}_{\omega=1}^{N}$ whereas averaging for $A^{(e_{2})}$ is described by a set $i=1,\dots,n$ of distributions $\{p_{\omega,i}\}_{\omega=1}^{N}$, $\sum_{\omega}p_{\omega,i}=1$.

Averaging over molecules before any computation is expected to yield different outcomes. Qubits occupying larger physical volumes would significantly relax the manufacturing requirements, as the placement of single-atom qubits in a solid lattice (or in a host molecule, for that matter) at precise locations and spacings has been challenging due to the lack of methods to implant point defects with good precision [12]. A second advantage is the robustness of the quantum operations with respect to environmental noise. A single spin undergoing decoherence leads to the loss of its quantum state, whereas the quantum state stored in ensembles may survive longer due to the inherent robustness of ensembles (see Appendix I.3).

Suppose we have a large ensemble ($N\cdot n\gg 1$) of nuclear spins $\{\mathbf{I}^{i}\}$, where $N$ is the number of localized spin ensembles and $n$ is the number of spins per ensemble. We will call these mesoscopic spin ensembles E-qubits for reasons that will become clear shortly. To simplify the presentation we assume that $I=1/2$ for nuclear spins; we will also assume that all spins in each localized ensemble have the same gyromagnetic ratio. This choice only affects some constants and scaling factors below, and does not lead to a loss of generality. The magnetic dipole interaction between $N\cdot n$ spins cannot be used for quantum computing because of the impossibility of addressing each spin individually. Also, the exact couplings in atomic ensembles are not known, making it impossible to design precise gates. We consider: (1) two ($N=2$) localized ensembles $A$ and $B$ each containing $n$ spins; (2) assume full polarization of all spins in the ensembles; (3) spins within each ensemble are decoupled. Spin ensembles are rarely discussed as candidate qubits because of their multilevel energy structure and high degeneracies, which differs from the structure of a qubit. If each ensemble admits a $\mathfrak{su}(2)$-like algebra, the ensembles can be viewed as qubits, even though they are in reality multilevel qudits (Fig. 1). The magnetic interaction between all pairs of spins is

where $\vec{r}_{ij}$ is the vector connecting spin $i$ from $A$ to spin $j$ from $B$ (Fig. 1a). $\mu_{0}$ is the vacuum permeability and $\gamma$ is the gyromagnetic ratio. If there are $N$ such localized spin ensembles, each containing $n$ spin 1/2 particles the dimension of the spin algebra $\mathfrak{su}(2^{Nn})$ is $2^{2Nn}-1$. Clean quantum gates are still not possible due to the random distribution of $\vec{r}_{ij}$ values. Moreover, this type of interaction over macroscopic distances is rarely considered due to the rapid $1/r^{3}$ falloff of the dipolar interaction. However, in a specific cooperative geometry, clean gates are possible via cooperative couplings, as explained below.

Consider a geometry involving two spherical volumes $A$ and $B$ (Fig. 1a) whose centers are separated by a constant vector $\vec{r}_{0}$ and the distance of each spin from the center of each region is much smaller than $|\vec{r}_{0}|$, i.e., $\delta_{A}\ll r_{0}$, $\delta_{B}\ll r_{0}$. The internuclear vector $\vec{r}_{ij}$ is nearly equal to $\vec{r}_{0}$. Expanding $|\vec{r}_{ij}|^{-3}$ in the small parameter $\vec{\epsilon}=\vec{\delta}_{B}-\vec{\delta}_{A}$ about $\vec{r}_{0}$ gives

The zeroth order term (i.e., neglecting terms of order $|\epsilon/r_{0}|$ and higher) gives the effective Hamiltonian $\mathcal{H}_{D}$

where the coupling tensor is now constant. It is therefore independent of $i$ and $j$, the indices of summation. The Hamiltonian can be written as:

where $\mathbf{I}^{A}=\sum_{i=1}^{n}\mathbf{I}^{A,i}$ and $\mathbf{I}^{B}=\sum_{j=1}^{n}\mathbf{I}^{B,j}$ are total angular momentum operators. We thus have a bilinear coupling $\mathbf{I}^{A}\cdot\mathtensor{D}_{AB}\cdot\mathbf{I}^{B}$ involving two large spins $A$ and $B$.

Also, since $[I_{\alpha}^{A},I_{\beta}^{A}]=i\epsilon_{\alpha\beta\gamma}I_{\gamma}^{A}$, $[\mathbf{I}^{A},\mathbf{I}^{B}]=0$, the action of the total angular momentum operator on the composite $\prod_{i=1}^{n}(2I^{i}+1)$ dimensional Hilbert space of a localized ensemble constitutes a representation of the $\mathfrak{su}(2)$ Lie algebra that is reducible. Due to this rotational symmetry that maps to that of qubits, ensembles $A$ and $B$ are examples of E-qubits. (The $\mathfrak{su}(2)$ symmetry of large spins has been known since 1932 [13].) If we have $N$ localized spin ensembles, each with $n$ spins, the spin-Hamiltonian is the sum of Zeeman (static, control fields) and point dipolar interaction between the E-qubits,

with $\vec{h}^{Q}=(h_{x}^{Q},h_{y}^{Q},h_{z}^{Q})$ is an external field. It has $\otimes\mathfrak{su}(2)^{N}\simeq\mathfrak{su}(2^{N})$ symmetry. Even though E-qubits are multilevel systems, the Hamiltonian (4) of the $nN$-spins system is equivalent to an effective $N$-qubit Hamiltonian, which is controllable. Controllability would not be possible with the lower symmetric form (1). The spin operator algebra is also much larger, with dimension $(2I+1)^{2Nn}-1$ for  (1) vs $(2I+1)^{2N}-1$ for Eq. (4).

The microstates of an E-qubit can be viewed as a very large $d$-level system. For a $d$-level system a natural quantum computational basis consists of the top and bottom energy levels. Transitions between the top and bottom levels of a $d$-level system are multiple-quantum transitions requiring frequencies proportional to $d$. When $d$ is large, such energies become out of reach to the experimentalist. On the other hand, we know from decades of NMR and EPR experiments that very large $d$ (multi-quantum) transitions are never needed if the spins are uncoupled or weakly coupled (i.e., interaction strength is much smaller than the Zeeman energy). In the weak coupling limit, only single-quantum excitations are needed, as each spin can be viewed as behaving independently. In this case, unitary transformation rules of conventional quantum computing with qubits (spin 1/2) or those of qudits (see Appendices I.1 and I.2) for large spins apply, as far as the generation of single- and two-qubit gates. If the spins were coupled sufficiently strongly as to create a large spin, impractically high energy excitations may be needed to excite magnetic transitions between the lowest and highest energy states. To reduce the ion-ion couplings engineered materials may be required in which the sum of all interactions (e.g., dipolar, indirect spin-spin, or exchange between neighboring spins is weak. A notable approach has been magnon-cavity coupling where introduction of strong exchange coupling results in narrow-linewidth magnetostatic modes [15, 16].

Quantum gates.– To have universal quantum gates we need arbitrary single qubit gates complemented by one entangling two-qubit gate. Suppose the initial state of an E-qubit is fully polarized, e.g., $\ket{I^{A},\pm I^{A}}$. This fully polarized state corresponds to a linear combination of fully polarized high-rank axial tensor operators $\mathscr{Y}^{(k)0}(\mathbf{I}^{A})$, $0\leq k\leq 2I^{A}$, such that the single-qubit rotations are described by $\sum_{q}\mathscr{D}^{(k)}_{q0}(\Omega)\mathscr{Y}^{(k)0}(\mathbf{I}^{A})$, where $\mathscr{D}^{(k)}_{qq^{\prime}}(\Omega)$ are elements of Wigner $D$ matrices representing the rotation group in the subspace of E-qubit $A$ (see Appendices I.1 and I.2). These collective rotations can be implemented using unitary operators of the form $\text{e}^{-i\theta\hat{n}\cdot\mathbf{I}^{A}}\otimes\mathbb{1}^{B}$, where $\hat{n}$ is a rotation axis and $\theta$ is the angle of rotation. Transformations in the full rotation group generate coherent rotations needed for single-qubit gates. This will work as long as the spins within each ensemble are decoupled from each other. The fidelity of quantum gates depends on linewidth. The latter is influenced by couplings between spins in each E-qubit. The fidelity of a quantum gate, to the first order, depends linearly on relaxation rate and inverse of gate time [17]. We discuss decoupling schemes below. Figure 2(a) demonstrates such rotations of spin-coherent states describing the E-qubit with $n=5$ and $N=2$. Initially polarized states such as $\ket{I^{A},I^{A}}$ in the coupled spin basis can easily be rotated into (say) $\ket{I^{A},-I^{A}}$ using standard operations.

The CNOT gate where the first E-qubit is the control and the second is the target can be implemented with the following sequence of unitaries:

where $\|\mathtensor{D}_{AB}\|$ is the norm of $\mathtensor{D}_{AB}$, $U_{zz}(\frac{1}{2\|\mathtensor{D}_{AB}\|})=\exp(i\pi I_{z}^{A}\tiny{\otimes}I_{z}^{B})$ describes evolution under an Ising ($zz$)-type interaction derived from Eq. (4). Figure 2(c) indicates simulated time-evolution under the full $U_{zz}$ dynamics for two qudits $N=2$, with $n=1,2,3,4,5$ spins, showing that the behavior is similar to the evolution of qubits, except that the evolution frequency scales with the number of spins in each ensemble. Figure 2 proves that the coupling between E-qubits scales linearly with $n$. Although cases $n>5$ ($>$ 10 spins total) are difficult to simulate on an average classical computer, this linear relationship holds for any $n$. We discuss below specific examples for much larger $n$. With spin ensembles, storage of quantum information is inherently robust to noise if the spin-flips affect only a small fraction of the total moment. This is an advantage over single-atom qubits. We now discuss decoupling schemes.

Solid-state nuclear spins.– In solids, the magnetic dipole interaction between nearest neighbors is strong. In contrast to the liquid state interactions, the spatial part of dipolar coupling, $(1-3\cos^{2}\theta_{ij})r_{ij}^{-3}$, is not averaged to zero as the result of molecular motion. Spin evolution under homonuclear dipolar interaction $\sum_{i<j}D_{ij}(2I_{z}^{i}I_{z}^{j}-\frac{1}{2}(I_{+}^{i}I_{-}^{j}+I_{-}^{i}I_{+}^{j}))$ transforms uncorrelated spin terms such as $I_{x}^{i}\otimes\mathds{1}^{j}$ into correlated terms such as $I_{y}^{i}\otimes I_{z}^{j}$. By connecting multiple spin pairs, a network of multispin correlated terms is generated that grows in size exponentially fast. These correlated terms are not directly observable and result in short $T_{2}$ times, on the order of microseconds [18, 19, 20, 21].

To average out the dipolar interactions inside each E-qubit while preserving the inter-qubit interactions, well-known techniques of dynamic decoupling can be applied to average out unwanted dipolar interactions and obtain the desired effective Hamiltonian. The idea is to either suppress unwanted interactions or shift them to a different frequency band. One may leverage solid-echo cycles such as MREV-8 [22], or sequences based on magic echo [23], or use magic angle spin-locking such as Lee-Goldberg (LG) [24]. Consider the polarization-inversion spin-exchange at the magic angle (PISEMA) experiment which is a variant of LG decoupling sequence. The homonuclear dipolar interaction is averaged to zero by locking the spins to an effective field $\mathbf{B}_{\text{eff}}$ along the magic angle, $\theta_{m}=\arccos(\sqrt{1/3})$. The spatial part of the dipolar interaction averages to zero in each E-qubit, as long as $\|\mathbf{B}_{\text{eff}}\|\gg\|\omega_{D}\|$ (Fig. 3a). To understand the influence on the inter-qubit interaction, consider the magnetic moment of spins in each qubit. As their magnetic moment vectors rotate around $\mathbf{B}_{\text{eff}}$, the perpendicular component gains a time dependence and averages out over the cycle, while the parallel component survives. For all the spins with initial magnetic moment along the $z$ axis, a portion of their initial magnetic moment remains after the application of the locking field $\tilde{I}_{z}^{\|,i}=I_{z}^{i}\sin{\theta_{m}}=0.82\times I_{z}^{i}$  [25]. This is a large scaling factor in comparison to rival decoupling sequences. Figure 3b shows the pulse sequence that combines these ideas for quantum control of two E-qubits. By applying the PISEMA sequence on both E-qubits simultaneously, we ensure that the intra-qubit interactions are averaged to zero. Selectivity in applying $\mathbf{B}_{\text{eff}}$ (direction and amplitude) means that the effective inter-qubit interactions can be engineered. In particular, the effective magnetic moment in each qubit can be oriented at a different direction and with different precession frequencies resulting in the inter-qubit interaction of the form $\sin{(\theta_{m})}^{2}\sum_{i}I_{z}^{A,i}\otimes\sum_{j}I_{z}^{B,j}$.

To get an idea of the effective coupling strength, consider the example of solid hydrogen ($T<14.01$ K, density $0.086$ g/cm³). The gyromagnetic ratio of ¹H is $26.75\times 10^{7}$ rad/T/s. Consider an arrangement of spin ensembles each confined to spherical volumes with radius of $r=30$ nm. The centers of the ensembles are placed $r_{0}=100$ nm away from the centers of their nearest neighbors. There are $5.9\times 10^{6}$ spins in a sphere of solid hydrogen, leading to $D_{AB}=7.51\times 10^{-4}$ rad/s. The magnetic field seen by a spin in sphere $A$ due to the field of spins in sphere $B$ is $104~{}\mu$T. This corresponds to a precession frequency of 4.4 kHz. Entangling gates between the two ensembles can therefore be executed in 227 $\mu$s. This proposed implementation is limited by the efficiency of the control sequence in removing intra-qubit interactions. The fidelity of two-qubit gate is mainly determined by the robustness of two-qubit unitary interactions such as $zz$. Figure 3d shows the effect of residual intra-qubit interaction in the implementation of a macroscopic Ising $zz$ interaction.

Liquid state or gas phase.– In the gas or liquid state Eq. (4) automatically takes the form of a mesoscopic Ising ($zz$)-type interaction. In high magnetic fields, dipolar interactions between nearby molecules vanish due to diffusion, i.e., $\braket{3\cos^{2}\theta-1}_{S^{2}}=0$, leading to intra-qubit decoupling. However, beyond the diffusion length, the magnetic dipole interaction does not vanish, leading to a nonzero mesoscopic inter-qubit dipolar interaction. The end result is an Ising $zz$ form $\mathcal{H}_{AB}\propto I^{A}_{z}I^{B}_{z}$ [26, 27, 28, 29], where the proportionality constant depends only on the time-averaged internuclear vector connecting pairs of spins between each ensemble. In spite of molecular motions this effective mesoscopic long-range interaction is static in the geometry described here. For spherical volumes ($r\sim 30$ nm) filled with liquid hydrogen, $r$ is less than the diffusion length of H₂ molecules during the timescale of a NMR experiment, the dipolar interaction is averaged out within each sphere. The effective coupling strength for inter-qubit interactions when $r_{0}=100$ nm is 3.6 kHz. This is to be contrasted with the long relaxation times ($>$ 1 s) of nuclear spins in liquids.

Nuclear-spin polarization.– In the initialization step all spins must be fully polarized, which is a non-trivial task for nuclear spins. Depending on the physical implementation of the E-qubits, hyperpolarization techniques have been developed which may be applicable including dynamic nuclear polarization [30], spin exchange optical pumping [31] and parahydrogen induced polarization [32]. We also note that nuclear ferromagnetic ordering has been observed at nanokelvin temperatures [33, 34, 35, 36, 37].

Molecular magnets.– Moving on from nuclear spins to discussing electronic spins we switch our notation from $\mathbf{I}$ to $\mathbf{S}$. Molecular magnets are high-spin clusters in a generally anisotropic crystalline structure [38]. They are modeled as giant spins [39]; Mn₂₅ has $S=\tfrac{51}{2}$ [40] whereas Chen et al. [41] reported a giant spin ground state of magnitude $S=91$. A $T_{2}$ time of 31 ms was recently demonstrated for ¹⁷¹Yb³⁺ at 1.2 K [42]. They are possible building blocks for the physical implementation of quantum processors. A realization of Grover’s quantum search algorithm was achieved by coherent manipulation of a TbPc₂ molecular magnet [43]. Unlike nuclear spins, nearly full thermal polarization of electron spins, $\langle S_{z}\rangle=\mbox{Tr}(S_{z}^{\dagger}e^{-\beta\mathcal{H}})/\mbox{Tr}(e^{-\beta\mathcal{H}})\approx|S|$, $\mathcal{H}=D[S_{z}^{2}-\tfrac{1}{3}S(S+1)]+E[S_{x}^{2}-S_{y}^{2}]-g\mu_{B}\mathbf{S\cdot B}$, can be asymptotically approached at high magnetic fields and low temperatures. As an example, Gd³⁺ ions in GdW₃₀ complexes [44] ($D=1281$ MHz, $E=294$ MHz, $S=\tfrac{7}{2}$, $g=2$) can reach 95% polarization at 3 T and 2 K. The crystal-field (CF) term is sufficiently weak (for GdW₃₀, 3% of the Zeeman term) and does not interfere significantly with the spin dynamics. Weak CF interactions are a general feature of rare-earth ions, stemming from the isolated nature of the $4f$ electrons, making it possible to treat them like nearly free paramagnetic ions (unlike transition-metal ions whose CF splittings are much larger [45]). Other methods could be considered, such as chirality-induced spin selectivity (CISS), whose helical electrons are fully polarized at high temperatures [46].

The methods of the previous section can be extended to ensembles of large spins ($S>1/2$) provided that the algebra of large spins is used. Molecular magnets are normally treated as multipole tensors, $\mathscr{Y}^{(k)q}(\mathbf{S})$ where $0\leq k\leq 2S$, $-k\leq q\leq k$, in the EPR literature [45]. A full quantum treatment of multipoles (e.g., Ref. [11]) is presented in Appendices I.1 and I.2. There, we demonstrate the creation of one- and two-qubit gates involving arbitrary spins. In this context, it is worth reviewing the well-known relationship from angular momentum theory between coupled and uncoupled representations. It is the basis that gives rise to large composite electronic spins. A molecular magnet is made up of $n$ magnetic atoms each with spin $\{\mathbf{s}^{i}\}$ ($i=1,\dots,n$). The total spin is $\mathbf{S}=\sum_{i=1}^{n}\mathbf{s}^{i}$. The Zeeman interaction with an applied magnetic field $\mathbf{B}(t)$ is $\mu_{B}\mathbf{B}(t)\cdot\sum_{i=1}^{n}\mathtensor{g}_{i}\cdot\mathbf{s}^{i}$, where $\mu_{B}$ is the Bohr magneton and $\{\mathtensor{g}_{i}\}$ are $g$-tensors. Using the Wigner-Eckart theorem

applied to vectors:

where $\mathscr{Y}^{(k)q}$ are components of an irreducible tensor $\mathscr{Y}^{(k)}$, $\braket{jk;mq}{jk;j^{\prime}m^{\prime}}$ is a Clebsch-Gordan coefficient and $\braket{\alpha^{\prime}j^{\prime}}{\mathscr{Y}^{(k)}}{\alpha j}$ is a reduced matrix element, it is customary to write this interaction as $\mu_{B}\mathbf{B}(t)\cdot\mathtensor{g}\cdot\mathbf{S}$. Here, $\mathbf{S}$ is the total spin of the molecular magnet and $\mathtensor{g}=\sum_{i}\mathtensor{g}_{i}c_{i}$, with $c_{i}=\braket{\alpha^{\prime}S}{\mathbf{S\cdot s}^{i}}{\alpha S}/\hbar^{2}S(S+1)$. This is certainly the case when only the multiplet with maximum spin $S$ is populated. Inside this multiplet, application of this theorem for $\mathbf{V}=\mathbf{s}^{i}$, $\mathbf{J}=\mathbf{S}$, $\ket{jm}\equiv\ket{SM}$, enables replacing $\mathbf{s}^{i}$ by $c_{i}\mathbf{S}$, where $c_{i}$ is a numerical constant, i.e., $\pi\mathbf{s}^{i}\pi=c_{i}\pi\mathbf{S}\pi$, where $\pi$ is the projection operator onto the multiplet of highest spin allowed in the ground state of the molecular magnet, i.e. $\pi=\sum_{M=-S}^{S}\ket{SM}\bra{SM}$. The numerical constants $c_{i}$ are determined by the nature of the exchange interactions (e.g., ferro- vs antiferromagnetic) between the constituent spins.

Solid State Electronic Spins.– Solid state systems such as rare-earth ion or transition-metal-doped semiconductors could be a potential platform for E-qubits. In the dilute limit, spins are weakly interacting and residual intra-qubit couplings can be addressed using decoupling schemes as discussed earlier. In the dense limit, however, ions interact strongly. Fortunately, uncoupling may be possible with the use of co-dopants or materials engineering. Long decoherence times of Er³⁺ ions in the presence of oxygen co-dopant have been observed in the system Er-Si [47, 48, 49]; we note that the system Er:YSO is also of interest. For Er³⁺ the ZFS is smaller than the Zeeman splitting from Tesla fields, making it possible to polarize electron spins above $>$ 90%. Weak ZFS are a general feature of rare-earth ions.

Conclusion.– The design requirements of quantum processors based on large collective spins (ensembles, molecules) may be less stringent that those based on individual atomic spins. A scalable implementation of the latter would require precise placement of atomic defects in the lattice with atomic resolution, something that is not currently feasible at this time using foundry techniques. Also, once the single atom decays, the entire quantum state is lost. That is not the case for ensembles (E-qubits), where survival of the fittest prevails, so-to-speak. Moreover, E-qubits can be selectively addressed through spatial degrees of freedom, rather than frequency (spectral) selectivity, providing a sensible path towards scalability. To date, a prescription of gate implementation between collective spins has been lacking. Herein we examined the feasibility of using direct macroscopic magnetic dipolar interactions for the implementation of universal gates between E-qubits. We have shown how to create coherent evolution where each E-qubit behaves isomorphically to $\mathfrak{su}(2)$. Under conditions of intra-qubit decoupling and effective mesoscopic Ising $zz$ interactions, possible implementations could include nuclear-spin ensembles in the solid, liquid and gas phases, as well as giant electronic spins from molecular magnets (including possibly, ensembles of molecular magnets or rare-earth ion dopants). Previous implementations of ensemble qubits such as neutral atoms rely on interactions with an external field to apply quantum gates, where the coupling scales with $\sqrt{n}$ [50, 51] as compared to $n$, a clear advantage in our proposal. High-fidelity control and readout methods are of course, critical to the success of such a proposal. To this end, we note the possibility of electric-field control [52], nanomagnet control [14], nanowire control [53] and nanosquid readout [54].