Triggering single-molecule qubit spin dynamics via non-Abelian geometric phase eﬀects †

We illustrate how macroscopic rotations can be utilised to trigger and control a spin dynamics within the ground doublet of both Kramers and non-Kramers-type molecular nanomagnets via the non-Abelian character of the time-evolution operator. For Kramers magnets, we show how this eﬀect can be harnessed to realise single-qubit quantum gates and give the explicit example of a recently reported CoCl 2 (tu) 4 single-molecule magnet (SMM). We demonstrate that gating operations could be performed on this magnet in as fast as 10 ps before the breakdown of adiabaticity, much faster than typical spin– lattice relaxation times. Based on this effect, we also suggest CoCl 2 (tu) 4 as a quantum gyroscope for sensing yaw-axis rotations. For integer spin nanomagnets where non-axial crystal field interactions often lift ground state degeneracy, we show how spin dynamics from the non-Abelian geometric propagator can be recovered using non-adiabatic macroscopic rotations not-necessarily resonant with the tunnel splitting gap. Using the well-known TbPc 2 single-ion magnet as a further example, we identify an experimentally plausible non-adiabatic rotation that induces a coherent superposition of tunnelling ground states, tantamount to preparing each member of a TbPc 2 ensemble in the maximal angular momentum state | m J = 6 i . The detection of an ensuing coherent oscillation of the macroscopic magnetisation polarised along the TbPc 2 principal magnetic axis after the completed rotation could then proceed via time-resolved magnetisation measurements.


Introduction
6][7][8][9] While the coherent and incoherent dynamics which result in 'under-thebarrier' fast magnetic relaxation has foiled current attempts at room temperature single-molecule magnet memory, [10][11][12][13][14] these dynamical processes can instead be harnessed for the manipulation of single-molecule magnet qubit states.6][17] The time-evolution of these linear combinations is then detectable as Rabi oscillations in time-resolved spin-echo 18,19 or non-equilibrium electrical current 20 measurements subject to decoherence from the local environment of the qubit.Two-qubit quantum logic gates (e.g.CNOT and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iSWAP p ) have also been implemented in coupled paramagnetic qubit architectures via g-factor engineering of lanthanide ions 21 and modular supramolecular linkers in paramagnetic rings. 22,23Addressing individual qubits in linked vanadyl-porphyrin dimers has also recently been reported by taking advantage of a weak intramolecular exchange coupling and small homogeneous linewidths intrinsic to the molecule. 24n 1984, Wilczek and Zee showed that the adiabatic transport of a quantum system around a closed path in Hamiltonian parameter space can induce a coherent dynamics between energy eigenstates of a degenerate manifold owing to an emergent non-Abelian gauge structure in the geometric component of the time evolution operator. 25This opened the door for holonomic quantum computation where gating operations correspond to the transport of a degenerate eigenspace (the holonomic qubit computational basis) along holonomies in parameter space. 26When the space of cyclically evolving parameters coincides with the coordinates of real space (e.g.macroscopic rotations), a connection between the Hilbert space of the quantum system and the geometry of the physical space which it occupies, is clearly established.The potential for rotationinduced non-Abelian dynamics in easy-plane Kramers-type spin This journal is © the Owner Societies 2023 systems (where time-reversal symmetry enforces an exact ground state degeneracy in the absence of external magnetic perturbations) was first exposed by Zee. 27In that work he pinpointed a mechanism for rotation-induced frequency splitting of nuclear quadrupole resonance spectra obtained for a I = 3/2 35 Cl nuclear spin. 283][34][35] As the electronic spin of the SMM adiabatically follows the rotating crystal field potential it can accumulate a non-Abelian geometric phase during the cyclic rotation.Similarly, the generation of a coherent spin dynamics via macroscopic rotations was suggested for ensembles of nitrogen-vacancy (NV) centres where a longitudinal magnetic field (rotating simultaneously with the NV centre quantisation axis) was proposed to enforce degeneracy. 36While largely successful as potential qubits and gyroscopes, NV centres lack the synthetic customisability of molecular nanomagnets.Since the rotation-induced spin dynamics of a SMM is strongly correlated with the microscopic parameters of its associated spin Hamiltonian (vide infra) this offers a bottom-up synthetic pathway to tune geometric effects in SMMs for applications such as holonomic quantum computation and inertial sensing.
In this manuscript we show how macroscopic rotations can trigger a spin dynamics in the degenerate or quasi-degenerate ground doublet of Kramers and non-Kramers molecular nanomagnets, respectively, via the non-Abelian character of their time-evolution operator.We first discuss the sensitivity of the non-Abelian Berry phase to the effective crystal field parameters and, with representative examples for each nanomagnet class, we construct single-qubit Hadamard quantum logic gates using specific rotation paths.In non-Kramers magnets, where nonaxial terms in the crystal field Hamiltonian are poised to remove ground state degeneracy through a tunnel splitting between the otherwise bistable ground states, we recover a non-Abelian time evolution operator for the quasi-degenerate ground doublet by relaxing the stringent requirement of adiabatic rotation.In this case, we derive an analytical relationship between the rotation time and rotation axis canting angle which leads to the realisation of all possible Hadamard gates with the tunnelling ground doublet acting as a computational basis.We finally discuss the detection of this effect via the resultant collective magnetic moment oscillation of the non-Kramers nanomagnet crystal.

Nanomagnet effective spin Hamiltonian
The microscopic origin of magnetic anisotropy in SMMs results from the interplay between electrostatic crystal field effects and spin-orbit coupling mainly affecting the electronic states of the paramagnetic ions' 3d or 4f unpaired electrons.In polynuclear nanomagnets, intramolecular spin-spin interactions can too influence the overall magnetic anisotropy.For most nonlanthanide SMMs, the effective spin Hamiltonian H S = DS z 2 + E(S x 2 À S y 2 ) is often a good starting point to describe the zero-field splitting of orbital angular momentum quenched magnetic levels within a given ground manifold 2S+1 G 0 . 37The first term of H S accounts for an axial magnetic anisotropy of the electronic spin S = (S x , S y , S z ) and is always negative for singlemolecule magnets.The second term accounts for rhombic magnetic anisotropy and, when S z is quantised along the principal magnetic axis, we have that 0 r E o D/3.We utilise the aforementioned effective spin Hamiltonian throughout this manuscript except in Section 3.5 where the example of a rotating lanthanide single-ion magnet necessitates the inclusion of higher-order irreducible tensor operators into the effective spin Hamiltonian.
In the laboratory frame, the Hamiltonian for the rotating system takes the form H(a, b, g) = R(a, b, g)H S R † (a, b, g) where the rotation operator is parametrised by the Euler angles a, b and g. 38 The instantaneous energy eigenstates |Z a (a, b, g)i of the rotated Hamiltonian H(a, b, g) are constructed from rotations of the eigenstates |f a i of H S using the operator R (a, b, g). 27It is thus simple to show that H(a, b, g) |Z a (a, b, g)i = e a |Z a (a, b, g)i where the eigenvalues e a defined by H S |f a i = e a |f a i remain constant throughout the rotation.
For single-molecule magnets with half-odd integer spin length, Kramers theorem guarantees that diagonalisation of H S will always yield a ground state doublet |f AE i whose instantaneous energy e + = e À = e 0 can be harmlessly shifted to zero. 30As a consequence, ground state degeneracy is maintained throughout any adiabatic rotation and thus triggers the onset of a non-Abelian Berry phase, a unitary transformation producing a linear superposition of the two initial degenerate states.In general this is not the case for integer spin SMMs where the rhombic anisotropy term introduces a tunnel splitting gap between the ground doublet states.We will return to this interesting case in Sections 3.4 and 3.5 but for now, we assume a perfect ground state degeneracy.

The non-Abelian geometric propagator
Let M 0 represent an N-fold dimensional manifold composed of the subset of instantaneous ground states |Z a (k)i of a Hamiltonian H(k) which remain degenerate throughout a cyclic and adiabatic variation of d classical parameters k = (l 1 , . .., l d ).For N Z 2, an initial wavefunction prepared within this subspace evolves into linear combinations of states from M 0 according to the Berry phase (see Supplementary Note 1 for a careful derivation, ESI †) where P denotes path ordering, G is the cyclic path traced out in d-dimensional parameter space and A m dl m is the associated non-Abelian gauge potential (here we adopt the Einstein summation convention).The A m 's are N Â N matrices with matrix elements between all states of M 0 .
Paper PCCP In this work, we study the spin dynamics triggered within the degenerate ground manifold, M 0 , of nanomagnets undergoing rotations in physical space parametrised by the Euler angles a, b and g.The non-Abelian gauge potential (derived explicitly in Supplementary Note 2, ESI †) is defined by the infinitesimal line element A m dl m = A a da + A b db + A g dg where The S 0 i s are matrix representations of the angular momentum operators S i on the basis of the ground spin states of H S .Unlike in previous studies of rotation-induced spin dynamics, 27,36 we note that the loss of cylindrical symmetry in H S (owing to the inclusion of rhombic anisotropy) necessitates that the gauge potential A m be defined in the three-dimensional space spanned by all three Euler angles, rather than just a and b alone.

Rotations about loops enclosing simply and non-simply connected domains
Although the most general rotation is defined by all three Euler angles discussed at the end of last section, we choose here special closed paths in parameter space that are defined by constant, but generally non-zero, values of g.In particular, we investigate rotations about two topologically distinct closed loops traced out by rotations on the surface of the unit sphere: (i) square paths G square which do not enclose the poles of the sphere and (ii) circular paths G circle which do.Examples of each of these paths are depicted schematically in Fig. 1 as green and purple line segments, respectively.Note that the third Euler angle g could also be allowed to vary along these paths provided that initial and final coordinate axes vectors e x , e y , e z coincide on completion of the rotation.This ensures that the macroscopic rotation forms a true closed loop in parameter space.
In the first instance, the geometric propagator U square which takes the system around a closed square path on the unit sphere can, due to path-ordering, be decomposed into the ordered product of geometric propagators U (4) square U (3) square U (2) square U (1) square along each of the four legs (enumerated in Fig. 1) in which the system is rotated along constant lines of either latitude or longitude.Explicitly, these four propagators are Provided that 0 r a 1 r a 2 o 2p and 0 o b 1 r b 2 o p, the area enclosed by the square path is a simply connected domain on the unit sphere.Thus, it is simple to show that in the limits where the square loop is shrunk to an infinitesimally small point, U square approaches the N Â N unit matrix I N as intuition dictates.
The circular loop G circle does not enclose a simply connected domain of the unit sphere.Consequently, the gauge potential defined in eqn ( 4) is ill-suited for the calculation of the geometric propagator around these kinds of loops for spins with non-integer ground state magnetic moments. 27In order to eliminate this pathology it is necessary to make a change of basis using the unitary operator r = exp(i(a + g)S z ), yielding a gauge transformed potential which is now well behaved everywhere on the unit sphere except for the south pole.In general, the transformed potential A 0 n describes precisely the same physics as A m (i.e. the potential is gauge invariant) provided that the integration path G in eqn ( 2) is closed. 27While the new potential could in principle make the evaluation of the path-ordered intergal eqn (2) more involved, in practice a slightly modified version of the simpler gauge potential can be used, due to the equivalence where r i and r f are the unitary transformations evaluated at the initial and final points of the loop respectively.The geometric propagator along the circular path thus reads 3 Results

The influence of magnetic anisotropy on non-Abelian dynamics
As a consequence of the anisotropic energy landscape of the SMM spin states caused by the interplay between crystal field interactions and spin-orbit coupling, different paths in parameter space will lead to different dynamical properties.It is therefore intuitive that varying the crystal field parameters D and E (which encapsulate this interplay in our model Hamiltonian H S ) will influence the non-Abelian spin dynamics discussed here.This simple fact could be used either as a design principle for SMMs optimised for desired quantum dynamical applications, or to indirectly probe crystal field Hamiltonians via quantum dynamics experiments.
Given that the most easily measurable effect of the quantum dynamical behaviour in SMMs triggered by a non-Abelian geometric phase is a change in the magnetic moment brought about by the modulation of the degenerate ground state amplitudes, in this section we set out to study the effect of D and E on the achievable population transfer dynamics between the degenerate ground states of Kramers-type SMMs.To achieve this we consider rotations about the square path G square with a 1 = 0, a 2 = 901, b 1 = 601 and b 2 = 1201, and about the circular path G circle with b = 901.For simplicity, we initially set g = 01 for both rotations.When b = 901 and g = 01, the propagator for the great circle (eqn ( 8)) is where M x = hf À |S x |f + i and M z = hf + |S z |f + i.The propagator for the square path is similarly a 2 Â 2 matrix depending on the matrix elements hf À |S x |f + i, hf À |S y |f + i and hf + |S z |f + i, though its analytical presentation is rather cumbersome and unenlightening.We assume that prior to rotation, the system is initialised in the |f + i ground state of H S and, in Fig. 2, we report the squared amplitude of this state after rotation around the square loop (top) and circular loop (bottom).
For both rotation paths shown in Fig. 2, the final composition of the ground doublet wavefunction differs the most from the initialised state when S is small and E is near critical.This trend is intuitive since the non-Abelian dynamics here is triggered by the S x and S y matrices via eqn ( 5) and (8).For these matrices to be non-zero, the matrix elements hf + |S AE |f À i (with S AE = S x AE iS y ) must be non-vanishing and hence a portion ) must be recovered in the ground doublet via the rhombic anisotropy term in H S .This condition is most effectively met for small S and large |E|.Interestingly, for the rotation paths chosen here, critical rhombicity does not necessarily lead to maximal population transfer to |f À i.Instead, maximal transfer occurs for S = 3/2 nanomagnets when |E/D| B 0.2.It turns out that this value of |E/D| leads to a magnetic moment which maximises the offdiagonal matrix elements of eqn (9) and minimises its diagonal elements.An analogous situation is observed for the propagator of the square loop at around |E/D| B 0.25.
It is worthwhile noting that we also investigated the role of the g rotation in determining the final ground state composition of the magnet after rotation about G square and G circle .While the final amplitudes of all spin systems rotated about G square were only imperceptibly changed from those reported in Fig. 2, the choice of g had some impact on the final wavefunction compositions of the S = 3/2, 5/2 and 7/2 nanomagnets rotated about the great circle G circle .The largest change was observed for g = 901 where population transfer to the |f À i state was maximsed for S = 5/2 and S = 7/2 at critical rhombicity, and for S = 3/2 at |E/D| B 0.15.

Towards a holonomic implementation of Grover's algorithm
We now show how this effect can be utilised to construct singlequbit gates, with a mind towards the holonomic implementation of Grover's search algorithm. 39We follow here an approach to Grover's algorithm proposed by Fahri and Gutman in 1998 40 which does not require the use of quantum oracles.This version of Grover's algorithm has subsequently been employed using resonant microwave pulses delivered to the I = 3/2 nuclear qudit of a TbPc 2 single-molecule magnet. 16In that work, two types of quantum gates were demonstrated (i) a single qubit Hadamard gate which prepared the qudit states in a coherent linear superposition, and (ii) logic gates which selectively populated just one of the four qubit states at a time.
We endeavour to show how these gates could be implemented via macroscopic rotations of a recently reported CoCl 2 (tu) 4 single-molecule magnet 41 using the ground doublet of the S = 3/2 spin manifold as a computational basis.A rather good agreement was found between experimental magnetic measurements of CoCl 2 (tu) 4 and theoretical simulations using the same effective spin Hamiltonian H S as reported here, with the parameters D = À63 cm À1 and |E/D| = 0.22.We adopt the same values in the following sections.
3.2.1 Single-qubit Hadamard gate.To begin, we assume it possible to initialise the qubit in the |f + i state via the application of a small magnetic field for a sufficiently long time; the field is removed before any rotation takes place. 5Then, a coherent linear superposition of the qubit states (up to a global phase y) is achieved by applying the Hadamard gate which, in matrix notation is To implement the Hadamard gate on the initialised qubit, we rotate its magnetic anisotropy axis around the square loop defined in eqn ( 5 To investigate in what limit adiabaticity is satisfied for the rotation of this particular system, we numerically integrated the time-reduced Schro ¨dinger equation q s |c(s)i = (ÀiT/h)H(a, b, g) |c(s)i where the Euler angles describing the aforementioned square path have been parameterised by the dimensionless quantity s = t/T (0 r s r 1) with T the total time taken to complete the rotation.In Fig. 3 we show the final amplitudes and relative phases of the ground doublet states as function of rotation duration T. For loops which are completed in T 4 10 ps the single-qubit Hadamard gate is implemented with little error.This can be attributed to the rather large axial splitting of the system which separates the ground doublet from the first excited doublet by DE = 135 cm À1 .In this case, h /DE E 5 Â 10 À14 s, so for T = 10 À11 s, adiabaticity is satisfied and the formalism presented in Section 2.2 is valid.We again endeavoured to verify the adiabatic limit implicitly assumed in the above calculations by direct numerical integration of the Schro ¨dinger equation.In Fig. 4, we plot the final amplitudes of the ground doublet states after rotations around the two square paths specified in the previous paragraph.Again, after initialisation with the Hadamard gate from the previous section, with T Z 10 ps, each member of the ground doublet can be selectively populated to near unity, depending on the choice of rotation along either one of the quite different paths.

Single-ion magnets as quantum gyroscopes
While the non-Abelian geometric propagator formalism developed in Section 2.2 is gauge-invariant only for closed paths in parameter space, macroscopic rotations of nanomagnets along open paths can still lead to a measurable effect provided that the initial state of the system is known a priori, i.e. the gauge has been fixed by the initial measurement.Monitoring a change in amplitudes of the nanomagnet degenerate ground states can hence provide useful information regarding the orientation of the SMM principal magnetic axis in physical space with respect to a known initial direction.This journal is © the Owner Societies 2023 In Fig. 5 we plot the squared amplitudes of the ground |f AE i and first excited |c AE i doublet states of CoCl 2 (tu) 4 as it undergoes a 3601 rotation in the xy-plane obtained from numerical integration of the time-dependent Schro ¨dinger equation with a = 2pt/T and T = 10 ns.Note that the squared amplitudes of the ground state doublet are essentially single-valued functions of the angle a.Thus, a measurement of the ground state populations at a given time provides information on how far the molecular magnet has rotated from its initial orientation about the global z or ''yaw'' axis in the lab frame.We repeated this calculation for shorter timescales and found identical results as shown in Fig. 5 up until T r 10 ps where, from the previous section, adiabaticity begins to breaks down for the particular example of CoCl 2 (tu) 4 .For rotations which occur slower than T = 10 ps, the nanomagnet operates as a quantum gyroscope in the xy-plane.

The case of tunnel-split Ising ground states
For a single-molecule magnet with an integer spin S, small but non-negligible non-axial terms in the crystal field Hamiltonian that can connect the bistable |AESi ground states in high-order perturbation theory, lead to a quasi-degenerate tunnelling ground with tunnel splitting D. Assuming that the system is initialised in this quasi-degenerate ground manifold and the rotation is not so fast as to activate a dynamics between excited crystal field states, it is possible to write the timeevolution operator of just the quasi-degenerate SMM ground doublet as 42 Uðt; t 0 Þ ¼ Te Within this quasi-degenerate subspace, the S i matrices take the simple form S x ¼ S y ¼ 0 and S z ¼ Ss x (where s x corresponds to the usual Pauli matrix).Accordingly, the propagator in eqn ( 12) can be expressed analytically as the 2 Â 2 matrix where z ¼ h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8p 2 S 2 ð1 þ cos 2bÞ þ ðTD=hÞ 2 p .Note that owing to the simplicity of the gauge potential for this rotation path, eqn ( 13) is invariant to g.We now show how eqn (13) can act as a single-qubit Hadamard gate, or in other words can create a coherent linear superposition of the qubit computational basis (here the |f AE i quasi-degenerate ground doublet states).Supposing that the nanomagnet is initialised in the ground state |f À i, then a macroscopic rotation specified by eqn (13) creates a coherent linear superposition of |f À i and |f + i when b and T are chosen such that Since the RHS of eqn ( 14) is purely imaginary, for such a relation to hold we must have that cos z/2h = 0 or equivalently that z/2h = (2n + 1)p/2 for n 2 N [ f0g.Using this condition in eqn ( 14) yields T = (4pSh /D)cos b, so we find that S and consequently Interestingly, given the spin quantum number S there are several allowed values of canting angle b, and corresponding rotation times T, that will result in a coherent linear superposition of the non-Kramers SMM ground doublet tunnelling states with the condition that n is a whole number and 0 n o ð4 ffiffi ffi 2 p S À 1Þ=2.To maximise the possible choices of rotation speed and canting angle for practical implementations of this effect, molecular magnets with large S spin projections in the quasi-degenerate tunnelling ground states should be favoured.
The above rotation acting on the tunnelling ground state is tantamount to the preparation of the SMM in the |Si quantum state.Since we have assumed that S z does not commute with the nanomagnet Hamiltonian, once the nanomagnet has been prepared in the |Si state by the appropriate macroscopic rotation, a periodic tunnelling of the SMM magnetic moment triggered by the non-axial terms of the crystal field potential, will ensue along its principal magnetic axis taking the maximal values m z = AEgm B S. In a crystal of nanomagnets, the collective in-phase oscillation of each molecular magnetic moment will give rise to an oscillating macroscopic magnetisation that can be readily detected.We note en passant that, following a similar line of reasoning as above, the |ÀSi state can also be prepared from |f À i with precisely the same macroscopic rotation but now performed clockwise instead of anticlockwise, or equivalently with bp À b.

TbPc 2 as a rotating qubit candidate
Results from the previous section suggest that macroscopic rotations of integer spin single-molecule magnets could be employed to realise single-qubit gating operations for a holonomic implementation of Grover's algorithm where the quasi-degenerate ground doublet acts as the two-dimensional computational basis.In parallel with a recent implementation of Grover's algorithm on the nuclear qudit states of the well-known (bis)-phthalocyanine terbium SMM (TbPc 2 ), 16 we demonstrate here how non-adiabatic macroscopic rotations of the same molecule can be utilised to construct logic gates which act instead on the quasi-degenerate 4f tunnelling ground states.In particular, we will demonstrate (i) a single qubit Hadamard gate which prepares the quasidegenerate ground doublet states into a coherent linear superposition, and subsequently (ii) logic gates which selectively amplify each member of the ground quasi-degenerate doublet beginning from the initialised coherent linear superposition.
The [Xe]4f 8 electronic configuration of Tb 3+ results in an energetically well isolated 7 F 6 spin-orbit ground multiplet characterised by the J = 6 (S = 3, L = 3) total angular momentum quantum number.The splitting of this multiplet due to the crystal field of the two phthalocyanine ligands which sandwich the Tb 3+ ion is captured particularly well with the Hamiltonian 43,44 where y 2 , y 4 , y 6 and O q k (J) are the well documented Stevens parameters and Stevens operators for Tb 3+ , respectively. 45The A q k hr k i terms are experimentally determined ligand field parameters which take the values A 0 2 hr 2 i = 414 cm À1 , A 0 4 hr 4 i = À228 cm À1 , A 4 4 hr 4 i = 10 cm À1 and A 0 6 hr 6 i = 33 cm À1 . 46,47The axial terms in eqn ( 16) lead to a degenerate |m J = AE6i ground doublet which is thermally well isolated (by 4400 cm À1 ) from all other |m J i states.The non-axial y 4 A 4 4 hr 4 iO 4 4 (J) term mixes these states in third order perturbation theory introducing a tunnel splitting D = 7.29 Â 10 À7 cm À1 and results in the quasi-degenerate tunnelling ground . Using the result from the previous section (with S understood as J = 6), n can be chosen as any whole number between 0 r n r 16 resulting in 17 different combinations of b and T that will implement a Hadamard gate on the quasi-degenerate TbPc 2 ground doublet.We show each possible path in Fig. 6a but highlight especially the n = 10 path which requires a canting angle b = 521 and a rotation time of T = 339 ms.In principle, one ought to perform the rotation as fast as possible (i.e.choose the path corresponding to minimal n) to ensure that decoherent processes such as spin-lattice relaxation do not interfere with the coherent dynamics of the SMM undergoing rotation.We choose to focus on the n = 10 path here since the required rotation time is in line with similar experimental setups which have already exposed Berry phase splittings in a rotating NaClO 3 crystal 28 as well as measurements of a rapidly rotating diamond qubit. 48,49o investigate the coherent dynamics of the quasidegenerate tunnelling states during rotation as well as to verify the assumptions leading to eqn (13), we numerically integrated the time-dependent Schro ¨dinger equation for the rotating TbPc 2 molecule using the Hamiltonian from eqn (16).The resultant amplitudes |c AE (t)| of the tunnelling states |f AE i are plotted as functions of time t in Fig. 6b in the eigenframe of the rotating TbPc 2 quantisation axis as well as the relative phase difference between the |f AE i states in Fig. 6c.Indeed we find that once a single rotation has been completed, a linear superposition of the tunnelled states is prepared from the initial conditions c À (0) = 1 and c + (0) = 0. Curiously, Fig. 6b and c show that throughout the rotation, the system oscillates eleven times between the |f À i tunnelling eigenstate and the linear

PCCP Paper
This journal is © the Owner Societies 2023 (including some global phase w).We also checked all other combinations of b and T within the range 0 r n r 16 using the same numerical integration procedure and found consistently that the TbPc 2 system undergoes n + 1 oscillations between |f À i and during each rotation.In Fig. 6d we plot the time-evolution of the TbPc 2 magnetic moment as the SMM undergoes the cyclic macroscopic rotation corresponding to n = 10 (0 ms r t r 339 ms) and is then left stationary (339 ms o t r 600 ms).During rotation, Fig. 6b shows clearly that the TbPc 2 wavefunction oscillates between |f À i and the coherent linear superposition corresponding to |m J = 6i.Naturally, the TbPc 2 magnetic moment oscillates between m z = 0 and its maximal value m z = 6g J m B (g J = 4/3) during this time as well.After the rotation is completed (i.e. the Hadamard gate has been executed) the TbPc 2 molecule is prepared in the |m J = 6 i state which is not an energy eigenstate of Hamiltonian ( 16) due to the y 4 A 4 4 hr 4 iO 4 4 (J) non-axial term.As a result, the nanomagnet tunnels between |m J = 6i and |m J = À6i angular momentum states and thus the magnetic moment oscillates from m z = 6g J m B to m z = À6g J m B accordingly.Notably, from our proof-of-concept model above we have assumed a perfect D 4d symmetry of the TbPc 2 crystal field and have neglected hyperfine coupling between the TbPc 2 electronic spin and its I = 3/2 nuclear spin.With some minor alterations to the experimental protocol, we show in Supplementary Note 4 (ESI †) that this effect persists regardless of crystal field symmetry lowering and hyperfine coupling.
Finally we note that for a given b, T pair specified by n, the same rotation can be performed to amplify the |f À i energy eigenstate to unity from the linear combination f . The |f + i state can also be amplified to unity by rotating the quantisation axis along the same path but now clockwise instead of anticlockwise (i.e. with the substitution bp À b).

Rotation time and angle deviations
We have so far considered idealised rotation paths and times without concern for errors introduced by experimental/instrumental uncertainties.We use the n = 10 rotation path of the TbPc 2 single-molecule magnet described in the last section as a test case to investigate the effect of rotation time and path fluctuation errors on the holonomic implementation of a Hadamard gate on a single-molecule magnet.
For the perfect canting angle b and rotation time T, the action of the Hadamard gate should result in equal populations and phases of the tunnelled TbPc 2 ground doublet states.In Fig. 7a

Discussion
While we have focused on the specific examples of CoCl 2 (tu) 4 and TbPc 2 single-molecule magnets, there are a myriad of alternative candidate molecules reported in the literature which may also show these effects.For example, the tunnel split |J = 8, m J = AE8i ground doublet of a recently reported Ho 3 + singlemolecule magnet synthesised and characterised by Wu et al. 50n that study, a tunnel splitting of 10 À5 cm À1 was obtained from ab initio calculations suggesting that rotations should be carried out on ms timescales to observe the above non-adiabatic rotationinduced spin dynamics of the tunnelling ground states.A further example is a recently reported Ni 2+ field-induced single-molecule magnet synthesised and characterised by Miklovic et al. 51 In that work, excellent agreement between experimental results and simulations was afforded using the S = 1 effective spin Hamiltonian H = DS z 2 + E(S x 2 À S y

2
) with D = À13.7 cm À1 and E = 0.07 cm À1 .The rhombic magnetic anisotropy term resulted in a zero-field tunnelling ground doublet with splitting D = 0.14 cm À1 .To observe a rotation-induced coherent dynamics between the tunnel split ground doublet states for such a large D, would require rotations to be performed with T o 50 ns which is several orders of magnitude faster than conventional set-ups so far reported in the literature. 28,48,49Nevertheless, more sophisticated molecular rotor systems may in the future be leveraged to achieve such rapid, constrained rotations. 52e note in addition, that our theoretical model has not accounted for population transfer between the energy eigenstates of H S mediated by incoherent transitions from the local environment.These could result from lattice phonon-mediated spin transitions in the nanomagnet, 37 stray fields from a bath of nuclear dipoles 53 or potentially from continuous electrical measurement if the SMM can be realised in a spintronics device. 54,55While we haven't treated these effects directly, we note that for both explicit examples, the quantum logic gating and gyroscopic applications were based on macroscopic rotations that could be performed on timescales much faster than typical incoherent processes (for example the spin relaxation between TbPc 2 doublet states has been measured T 1 B 10 s), 54 thus reducing the risk for spurious, incoherent population transfer.While the prototypical results presented in this manuscript represent a first step towards the holonomic implementation of Grover's algorithm using macroscopically rotating singlemolecule magnets, since the rotation-induced coherent dynamics is confined to the SMM ground doublet, one is restricted to only ever searching a database of two objects.The database might be expanded however, by the extension of the computational basis to include hyperfine states of the SMM.In addition, the macroscopic rotation of different species of SMM confined to the same unit cell could also offer a way to grow the computational basis for a useful holonomic implementation of Grover's algorithm.The explicit discussion of these more complicated scenarios is beyond the scope of this work and will be considered in a future publication.

Conclusions
With a theoretical model, we have demonstrated that a spin dynamics can be triggered in the ground doublet of both halfodd integer and integer spin SMMs by macroscopic rotations of its principal magnetic axis.
For half-odd integer spin SMMs, we explored how the crystal field parameters of an effective spin Hamiltonian influence the spin dynamics of the ground Kramers doublet via an emergent non-Abelian gauge potential that arises due to the electronic spin adiabatically following the rotating crystal field.As well as an unconventional probe of the spin Hamiltonian parameters, we suggested how this might be utilised to realise holonomic single-qubit quantum logic gates in a CoCl 2 (tu) 4 single-molecule magnet.Direct numerical integration of Schro ¨dinger's equation revealed that the holonomic gating operations required for such an implementation could be executed on picosecond timescales, orders of magnitude faster than typical spin-lattice relaxation mechanisms.In addition, we demonstrated how this effect could be utilised for a quantum gyroscope inertial sensor based on CoCl 2 (tu) 4 .Again, provided that the rotation speed does not exceed 10 ps, the ground doublet squared amplitudes remain single-valued for rotations in the xy-plane and thus their measurement provides direct information about the rotation of the SMM principal magnetic axes.
By lifting the stringent condition that rotations proceed adiabatically, we recovered a tunable non-Abelian spin dynamics in the tunnelling ground (pseudo) doublet of an otherwise general integer spin SMM.We showed that when the quantisation axis is canted by an angle b ¼ arccos½ð2n þ 1Þ=4 ffiffi ffi 2 p S from the lab-frame z axis and rotated along a circular path on the unit sphere for a duration T ¼ ð2n þ 1Þp h= ffiffi ffi 2 p D (with n a whole number between 0 and ð4 ffiffi ffi 2 p S À 1Þ=2), the rotation prepares the SMM in a coherent linear superposition of the quasi-degenerate ground doublet states, effectively acting as a single-qubit Hadamard gate.We made a specific example of the well-known single-molecule magnet TbPc 2 and showed how this effect could be observed with state-of-the-art set-ups from the current literature.Via numerical integration of Schro ¨dinger's equation, we vindicated the approximations implicit in our non-adiabatic geometric propagator approach and exposed a curious n + 1 oscillation effect in the coherent dynamics of the rotating TbPc 2 magnetic moment.We used this prototypical example to explore the effects of rotation time and path fluctuations on the fidelity of a Hadamard gate induced by an SMM macroscopic rotation.

Fig. 1
Fig.1Rotation paths on the unit sphere.We illustrate the two closed loops of interest, G square in green and G circle in purple, generated on the unit sphere by rotating the coupled electronic and nuclear spin by the Euler angles a and b.The square path is broken into four segments (labelled in the diagram) in which the coupled system is rotated along either constant lines of longitude or latitude.The circular path G circle can be seen to enclose a non-simply connected domain of the sphere, containing either the north or south pole.For both paths, the third Euler angle g remains constant but, in general, non-zero.

Fig. 2
Fig. 2 Squared amplitude |c + | 2 of |f + i as a function of the ratio |E/D| for a range of half-integer spin lengths S, on completion of rotation of the SMM around the square (top) and circular paths (bottom) described in the main text.The system is initialised in the |f + i ground doublet energy eigenstate before rotation.
) with the Euler angles a 1 , a 2 , b 1 , b 2 and g obtained from numerical solution of (U square ) 11 = (U square ) 21 .The solution space of this equation can be reduced by taking a 1 = 0 and fixing the gap b 1 À b 2 .We find that rotation about the square loop defined by a 1 = 0, a 2 = 721, b 1 = 601, b 2 = 1201 and g = 871, drives the system into the desired coherent linear

3 . 2 . 2
State specific amplification.Beginning from the coherent linear superposition e iy f þ þ f section, we now demonstrate the selective amplification of |f + i and |f À i via rotations along the closed square loops G AE .In general this is achieved by rotating the qubit along different square paths where the rotation parameters are solutions to the relations (U square ) 11 = (U square ) 12 and (U square ) 21 = (U square ) 22 , respectively.We find for our CoCl 2 (tu) 4 example that rotation about the square paths G + with a 1 = 0, a 2 = 701, b 1 = 601, b 2 = 1201 and g = 541 and the path G À with a 1 = 0, a 2 = 1631, b 1 = 601, b 2 = 1201 and g = 1241 serve to amplify to near unity, the |f + i and |f À i states, respectively.

Fig. 3
Fig. 3 Final amplitudes (top) and relative phases (bottom) of the ground doublet states of CoCl 2 (tu) 4 after rotation shown as a function of the rotation duration T. The rotation path was that of a closed square loop with a 1 = 0, a 2 = 721, b 1 = 601, b 2 = 1201 and g = 871.The blue and orange curves represent the amplitude/phase of the |f + i and |f À i state respectively.The final wavefunction was obtained from direct numerical integration of Schro ¨diner's equation with the system initially prepared in the |f + i eigenstate.
A(t) has elements A ab ðtÞ ¼ Z a ðtÞ h j_ Z b ðtÞi and the matrix Y has elements Y ab = hZ a (t)|H(t)|Z b (t)i/h = e a d ab /h which are time-independent when expressed on the basis of instantaneous eigenstates |Z a (t)i of H(t).The integral over time can be converted to a path-ordered integral (see Supplementary Note 1, ESI †) involving the same matrix gauge potential A as in eqn (3) where l m correspond to the Euler angles a, b and g.After the system has undergone a complete rotation (when t = T) around the circle path G circle specified above, the non-adiabatic time evolution operator is U non-ad circle Ài2p Sx sin b cos gÀSy sin b sin gÀSz cos b ð Þ ÀiYT :

Fig. 4
Fig. 4 Final amplitudes of the ground doublet states of CoCl 2 (tu) 4 after rotation shown as a function of the rotation duration T. The two closed square loops are obtained from eqn (5) with G + : a 1 = 0, a 2 = 701, b 1 = 601, b 2 = 1201 and g = 541 and G À : a 1 = 0, a 2 = 1631, b 1 = 601, b 2 = 1201 and g = 1241.The blue and orange curves represent the final populations of the |f + i and |f À i states respectively.The final wavefunction was obtained from direct numerical integration of Schro ¨diner's equation with the system initially prepared in the coherent superposition e iy ð f þ þ f À j iÞ= ffiffi ffi 2 p from the previous section.

Fig. 5
Fig. 5 Squared amplitudes of the ground |f AE i and first excited |c AE i Kramers doublets of CoCl 2 (tu) 4 as it undergoes a 3601 rotation in the xy-plane.The squared amplitudes are obtained via numerical integration of the Schro ¨dinger equation after initialisation in the |f + i energy eigenstate and then rotation by a = 2pt/T for T = 10 ns.Here the squared amplitudes correspond to a basis quantised along the rotating anisotropy axis.

Fig. 6
Fig. 6 Hadamard gate implementation through the non-adiabatic macroscopic rotation of a TbPc 2 single-molecule magnet.(a) Schematic depiction of the rotation path (dark green loop) of the TbPc 2 quantisation axis (blue arrow) about the lab-frame z axis (black dashed arrow) when n = 10.Other possible paths are shown as translucent green loops and a select few are labelled.(b) Populations of the quasi-degenerate ground doublet states of TbPc 2 (c À in blue and c + in yellow) undergoing the rotation specified in (a).(c) Phase difference between the quasi-degenerate ground doublet states of TbPc 2 as it undergoes the rotation specified in (a).(d) Time evolution of the TbPc 2 magnetic moment as it undergoes the rotation specified in (a) up until T = 339 ms (dashed line).For t 4 339 ms the SMM quantisation axis remains stationary and the magnetic moment tunnels between À9m B and 9m B solely due to the non-axial crystal field.The amplitudes, phase difference and magnetic moment plotted in (b)-(d) were obtained from numerical integration of the Scho ¨dinger equation over the full Hilbert space of the Tb 3+ ground 7 F 6 ground multiplet and are plotted in the eigenframe of the rotating SMM.
we show the difference in these populations Dc(T + DT) = |c + (T + DT)| À |c À (T + DT)| and relative phases Dy(T + DT) = y + (T + DT) Ày À (T + DT) (mod 2p) at the end of the rotation, due to under/over rotation by a time DT.The error in the populations Dc(T + DT), grows approximately quadratically with the over/under rotation time DT, leading to almost a complete annulment of the Hadamard gate for |DT/T| B 10%.The phase error does not grow as rapidly but reaches a maximum of p/2 when |DT/T| B 10%.To approximate the effect of stochastic fluctuations in the rotation path we obtain the mean values of Dc(T) and Dy(T) from 500 simulations of the TbPc 2 rotation described above whereby the Schro ¨dinger equation is numerically integrated up until T = 339 ms.For each of these rotations we pseudo-randomly vary the canting angle b throughout the rotation path by drawing from a normal distribution with mean value 521 and standard deviation s.Typical fluctuations in the b Euler angle for fixed s during the TbPc 2 rotation are plotted in Note S5 (ESI †).For reference, standard deviations of s = 0.5 lead to fluctuations in the b canting angle of B 21.In Fig. 7b we show the mean values of Dc(T) and Dy(T) as function of the standard deviation s.Stochastic path fluctuations begin to reduce the fidelity of the Hadamard gate when the b Euler angle fluctuates 421.

Fig. 7
Fig. 7 Quantifying the Hadamard gate errors associated with rotation time and path fluctuations.(a) Difference between the populations (blue) and relative phases (yellow) of the TbPc 2 tunnelled ground doublet states after the cyclic rotation described in Section 3.5 with total rotation time T + DT.(b) Average difference between the populations (blue) and relative phases (yellow) from 500 simulations of the TbPc 2 rotation with a stochastically fluctuating canting Euler angle b.The fluctuations are drawn from a normal distribution centred at 521 with standard deviation s.