Investigation of Floquet engineered non-Abelian geometric phase for holonomic quantum computing

Holonomic quantum computing (HQC) functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging, and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold $^{87}$Rb atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in $SU(2)$, characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems.


I. INTRODUCTION
Quantum computing promises to solve some classically hard problems more efficiently than conventional (classical) methods, but both coherent and incoherent noise pose real barriers to practical deployment and use [1].Designing better qubits or error-correcting codes is a significant area of research [2], as is the search for new faulttolerant quantum control techniques.
Holonomic quantum computing (HQC) [3][4][5] is a promising approach that uses geometric phase in contrast to the more conventional gates that rely on dynamical phase.Geometric phases are independent of any details in the control Hamiltonian, instead depending only on the curvature in a state's Hilbert space as it varies with a set of control parameters.Geometric gates have long been thought to host intrinsic fault-tolerance when compared to dynamical gates, though recent work suggests that fault-tolerance does not depend on the type of phase, but rather on the details of the control Hamiltonian itself [6].
In HQC, computational states are encoded into a degenerate subspace of the control Hamiltonian.The Hamiltonian is varied adiabatically in a cyclic manner; states evolve and mix according to a non-Abelian connection in the parameter space.The evolution operator is referred to as a holonomy [7], due to its geometric interpretation [3][4][5]8].This protocol necessitates robust degeneracies, which are typically acquired through coupling to auxiliary levels [9].HQC protocols are also generally seen as slow due to the adiabatic criterion [9].As such, there are non-adiabatic generalizations [10] in which the role of degeneracy is relaxed, but the dynamical contributions to the phase that arise from this and the breaking of the adiabaticity impose strict conditions on the de-tails of each gate.There have been several experimental demonstrations of both conventional and non-adiabatic HQC in trapped ions [11,12], neutral atoms [13,14], liquid nuclear-magnetic resonance systems [15,16], Rydberg atoms [17], solid-state systems [18][19][20][21], photonics devices [22,23], and nitrogen-vacancy centers [24][25][26][27].
Topological quantum computing (TQC) also relies on non-Abelian holonomies, but differs from HQC.In TQC, quantum information is encoded in the phase of non-Abelian anyons [26,27]; these are particles that obtain a non-Abelian geometric phase under exchange, in contrast to fermions and bosons which only obtain Abelian phases of π and 0 under exchange, respectively.Qubits may be encoded onto a degenerate manifold of multianyon states, and gates are performed by moving anyons around each other, called braiding.In this case, the unitary transformation within this manifold depends only on the topological character of the path, that is, whether the anyon's path encapsulates another anyon or not.In this way the phase is still described by a non-Abelian holonomy, but is less sensitive to the detailed geometry of the adiabatic path than in HQC.
Recently, Floquet engineering was proposed as a path to produce robust degeneracies in systems which are otherwise non-degenerate [28,29].Through periodic modulation of a control Hamiltonian, degeneracies occur regardless of the underlying energetic structure.As such, this method may be readily applied as an HQC scheme without the need for any auxiliary levels or excited state couplings, with proposed implementations in ultracold neutral atoms [30] and Rydberg atoms [31].Additionally, for quantum simulation, this technique provides a path forwards to realizing interesting non-Abelian artificial gauge fields [32,33].
Here, we present a proof-of-concept experimental in-vestigation of Floquet engineered single-qubit holonomic gates in an optically trapped ensemble of 87 Rb.The holonomies investigated here may be readily implemented in other platforms.We perform several primitive gates and, through the tomographic reconstruction of the holonomies, we report their fidelities.Measurements are made in the presence of drifting background magnetic fields, the impact of which would normally be negligible over the time scales of each gate; however, due to the dynamics introduced by the Floquet driving this uncontrolled background has a substantial impact on the holonomy and its geometric nature.We quantify this impact and discuss the implications it has on the practicality of this protocol and its fault-tolerance, both in the cold-atom context, and more generally for generic Floquet-driven platforms.

II. THEORY OF FLOQUET-ENGINEERED DEGENERACY
In this work, we consider a system of N = 2F + 1 spin levels, coupled by an external field [30], which applies to any platform with quantized spin or pseudospin levels.Coupling is only between levels adjacent in energy, and thus the Hamiltonian is expressed in terms of the vector of N ×N spin matrices, F .The driving field's amplitude, frequency, and phase are periodically modulated, such that we obtain the Hamiltonian (in units where ℏ = 1), Equation 2 is the Hamiltonian for a spin in a magnetic field with magnitude Ω 0 , where the direction of the field is defined by the unit vector q = (sin Θ cos Φ, sin Θ sin Φ, cos Θ) ⊺ .This fictitious field is controlled through the phase of the driving field Φ(t) and control parameter Θ(t), which relates to the amplitude/frequency modulation envelope.Hence, the control parameter space is the unit sphere spanned by the angles {Θ(t), Φ(t)}.For a full derivation of this Hamiltonian, see Appendix A. The entire Hamiltonian V (t) is periodically driven with Floquet frequency ω.
Assuming that the timescales over which q(t) changes are very long compared to 2π/ω, we treat Ĥ(t) as nearly periodic and transform into the Floquet basis via the unitary operator Û = exp[i V (t) sin(ωt)/ω], usually called the micromotion operator.Following this we restrict our attention to the zeroth Floquet band [28][29][30], where the Hamiltonian becomes (see Appendix B), where g = 1−J 0 (Ω 0 /ω), and J 0 is the zeroth-order Bessel function of the first kind.This Hamiltonian only depends on changes to q; for any static choice of q (point  I), with detuning (Eq.7) ∆z = 0 (a), and ∆z = Ω0/20 (b).Colored lines show trajectories calculated in the rotating frame (Eq.2) with colors denoting progress through the loop, and points (circles) sampled stroboscopically, at each half-integer period of the Floquet driving frequency ω.The solid (red) line is simulated in the Floquet basis.The disagreement between the stroboscopic points and Floquet-basis simulations are due to non-adiabatic corrections, which for the parameters chosen here amounts to about a 3% error, in terms of the fidelity (Eq.10); as Ω/ω → 0 the evolution becomes more adiabatic.
in parameter space) the Hamiltonian is zero, hence the Floquet states are trivially degenerate.The Schrödinger equation for Eq. ( 3) can be written as where ∇ q is the gradient in the parameter space of q.Equation 5 is a purely geometric equation describing the parallel transport of a state |ψ⟩ in the parameter space of q with non-Abelian connection Â(q) [34].Changing q along some path in parameter space results in the accumulation of a fully geometric phase determined by the connection Â.As q changes in time, the Hamiltonian takes on instantaneous eigenvalues proportional to, but smaller than, the rate of change.As such, the system remains adiabatically degenerate, and its evolution will depend only on the geometry of the path traced out, and not on any dynamical details.
As is typical in HQC, we restrict our attention to the evolution of the system over loops in parameter space; we thus assume that q(t) varies over a cyclic path, ℓ in a period T = 2π/Ω.According to Eq. 5 the evolution operator over a path ℓ is the holonomy [3,4], where P is the path-ordering operator.Several loops are explored here, summarized in Table I.Regardless of the spin manifold F , these loops generate transformations in The rate at which loops are traversed, Ω, is constrained by the adiabatic condition [28], Ω ≪ ω.If we choose Ω to be a sub-harmonic of ω, then, at the end of a loop (and every half-integer period of ω), the Floquet basis and spin basis coincide [30].If the control fields are switched off precisely at this time, the projective measurements of the spins are equivalent to those of the Floquet basis states.For this reason the spins may be treated as the computational basis, despite the fact that the geometric phase is acquired by the Floquet states.For the experiments performed here, Ω 0 /ω = 1 setting the magnitude of the phase accumulated g, and the loop duration is T = 2π/Ω with Ω = Ω 0 /10.
The key feature of Eq. 1 is that the entire Hamiltonian (Eq.2) is modulated by a function with zero timeaverage, leading to the adiabatic degeneracy of the Floquet states.In any realistic attempt, this condition may be challenged; in our case the presence of stray fields either perturb the spin state energies or couple them, adding terms which do not average to zero over a Floquet period.A fairly general description of stray fields in this system requires only a minor modification of the Hamiltonian (Eq.1).We consider the case where stray fields are time-independent, since the fields affecting our experiment are stable over several tens of minutes.While the time-dependent case would follow a similar derivation the transformation to the Floquet basis may not be easily performed analytically depending on the specific functional dependence.Equation 1 is modified with an extra term, where we refer to ∆ as the detuning from resonance.The ∆ z component is equivalent to a mismatch between the driving field's carrier frequency and the level-splitting (see Appendix A).Other stray fields or a leaked control field would, in general, correspond to some combination of all three components.In our experiment there is sufficient extinction of the control field, and no other stray fields near resonance, so a ∆ z term was sufficient to describe the data.
Given that the detuning term in Eq. 7 is not modulated by the Floquet envelope, it will result in a new dynamical phase in the Floquet basis despite its lack of time-dependence; this is due to the fact that the Floquet basis transformation is itself time-dependent.If we transform the detuning term into the Floquet basis we obtain (see Appendix D), The extra detuning breaks the adiabatic degeneracy in the Floquet basis and produces dynamical coupling between states.As in the case of conventional HQC where small perturbations of the energetics break the degeneracy, the presence of any unmodulated term in the Hamiltonian produce similar effects here.While it may seem that the Floquet driving gives an easy and robust path to degeneracy, the challenges inherent with eliminating terms that do not average to zero are highly analogous to those with maintaining degeneracy in traditional systems [4].Furthermore, due to the Floquet basis being a time-dependent mixture of the bare spins, a miscalibration in resonance (which would normally result in a different phase accumulation rate and imperfect population transfer) now results in non-trivial coupling between states.Hence, any undriven term in the lab frame has non-trivial results in the computational basis.Due to this non-geometric term, we also consider the non-adiabatic generalization [35] of the holonomy, which is represented in time-ordered form, with T the time-ordering operator.
The terms ∂ t q • Â(t) and Ĥ∆ Floq.(t) represent the geometric and dynamical contributions to the phase, respectively.While we may write the two sources of phase separately in this way, their impact on the evolution is fundamentally inseparable due to the time-ordering [36].If we expand this operator, we would find an infinite sum of terms that depend on the nested commutators of the two terms.In certain similar circumstances, there are paths that result in net zero contribution from the dynamical term, which forms the foundation of non-adiabatic HQC [10].In our context, since ∆ is considered here to be outside of our control, the impact of the detuning given in Eq. 8 is unlikely to be avoidable.It may be possible through other means, such as dynamical decoupling [37][38][39][40][41][42], to suppress the effects of the detuning, but this requires further investigation.We will apply the detuned Hamiltonian (Eq.7) to the results that follow, demonstrating that even for relatively small detunings, the impact on state evolution is considerable (Fig. 1).

III. EXPERIMENTAL RESULTS
We describe the details of an experiment in which we realize the spin Hamiltonian (Eq.2) on a specific platform: a non-interacting gas of ultracold 87 Rb atoms, which is similar to that in the proposal of Chen et al. [30].We explore both of the available stable hyperfine ground states, with total angular momentum quantum numbers F = 1 and F = 2; these manifolds have N = 3 and N = 5 magnetic sublevels, respectively, with magnetic quantum numbers m F = 0, ±1, ... ± F .We apply a background magnetic field to split the m F states by ω Z /2π = 1.25 MHz through the linear Zeeman effect; this field remains fixed for the duration of the experiment.
After initial laser cooling, forced radio-frequency (RF) evaporation in a magnetic trap, and further evaporation in an optical dipole trap (ODT), we obtain a 87 Rb Bose-Einstein condensate (BEC) of about 10 5 atoms in the |F = 2, m F = +2⟩ state.To prepare atoms in either of the F = 1 hyperfine states we use a microwave hornantenna to couple the F = 1, 2 manifolds through a magnetic dipole transition centered at ≈ 6.8 GHz, in addition to RF pulses.For the best population-transfer efficiency and long-term stability, we use 1-ms chirped microwave pulses to effect adiabatic rapid passage (ARP).In all state preparation sequences, state purity is ensured by intermediate resonant laser pulses that remove atoms remaining in the other hyperfine manifold following the microwave ARP pulse, at the expense of slightly reduced atom numbers.
To produce the spin Hamiltonian (Eq.2), we use an RF field that couples m F levels through a magnetic dipole transition on resonance with the ω Z splitting.The amplitude, frequency, and phase of this field are periodically modulated with an arbitrary waveform generator (AWG), (details in Appendix A).Each of the loops shown in Table I is implemented by the appropriate simultaneous variations of Θ(t) and Φ(t).
Following each gate operation, we perform state tomography.This is done using Stern-Gerlach (SG) imaging [43], in which atoms are spatially separated by a magnetic field gradient after the ODT is turned off, and subsequently absorption imaged to infer the relative spin populations.This constitutes a projective measurement in the spin basis.To gain information about the phase of the spin states, we precede SG measurements with resonant RF pulses of varied phase and pulse areas, which changes the measurement basis, permitting the full tomographic reconstruction of the prepared states.Combined with our ability to initially prepare atoms in each spin basis state, we measure the holonomies in full through a series of informationally complete projections and statepreparation pulses.Given that the absorption imaging process is destructive, each measurement represents the production of a new BEC, a process which takes about 25 seconds; as such, scans take upwards of 10 minutes depending on how many projections are being taken.

A. Results
Using the tomographic measurement techniques described above ( §III) we confirm several aspects of the expected holonomic evolution.The time evolution of the spins during each of the loops was verified by abruptly turning off the control fields, thus interrupting the loop.In changing the time of interruption we map the spin evolution and compare with theory calculations, as shown for ΓA (ℓ 1 ) in F = 1 and F = 2 [Fig.2(a-b)].We see that the evolution over loops results in coupling between states in a way that depends on the loop, as expected with non-Abelian geometric phases.
Additionally, we observe that the phase of the final state also depends on the path taken.We scan a readout RF π/2-pulse's phase, comparing two different holonomies, ΓA (ℓ 1 ) and ΓA (ℓ 2 ) with F = 1 and F = 2 [Fig.2(c-d)].These holonomies produce the same time evolution and final populations in the spin-basis, but the final states differ only in the relative phases between spin components.
Throughout the measurements presented here, background magnetic fields result in a detuning from resonance, as described by Eq. 7. Specifically, the system is susceptible to a ∆ z component to the detuning, while both other components were negligible.The typical detuning was ∆ z /2π 0.8 kHz, or when compared to the RF coupling strength, ∆ z /Ω 0 0.08.In the absence of magnetic shielding or feedback stabilization of magnetic fields, our ability to control or even detect a detuning of this magnitude is limited; hence, this effect represents a realistic complication in obtaining high-fidelity quantum control ( §III B).

B. Gate Fidelity
For a comprehensive analysis of the holonomic gates, we measured each holonomy in full, focusing on F = 1 for demonstration.This entailed preparing atoms in each of the spin basis states, applying a holonomy, and performing informationally complete projective measurements, as detailed above.This was done for each of the holonomies in Table I.
We scanned through a series of state-preparation pulses and measurement pulses for a given gate; the or- Γ ̂A( ℓ2 ) dering was randomized to prevent any bias coming from a predetermined measurement sequence.The resulting set of projections was then fit for residual detuning, and for the holonomy itself (Appendix G).We repeated these scans for each holonomy multiple times to account for the randomized detuning which was present in each scan.
After each scan we repeated the resonance calibration (Appendix E) in an attempt to detect any drift in the resonance during the measurement.
From holonomy measurements, we computed the gate fidelities.For a target holonomy ΓA (ℓ), we compute the pure-state fidelity of the measured holonomy LA (ℓ) from the inner product, If the two matrices are the same then F = 1.Our results are compiled in Fig. 3, where we compare our measurements to the target holonomy.Without considering detuning, the fidelities are low, with the average for each loop given in Table I.The detuning accurately captures the most significant source of error in our experiment, which we demonstrate by fitting each holonomy measurement scan for detuning ∆ z .Comparing our measured holonomy to a target which includes this detuning in the model (Eq.9) results in much higher fidelities (Fig. 3), which are more narrowly distributed, indicating the most significant error mechanism in our apparatus.The average measured fidelities with detuning are also summarized in Table I.Despite having higher fidelities when detuning is included, the results are still far from unity, with an average of 0.84 (7).This distribution of fidelities is consistent with our models of shot-to-shot fluctuations in detuning for each measurement of 0.2 kHz; therefore, we find that our infidelity is dominated by detuning fluctuations.The next largest source of error, on the order of about 2%, is due to extracting population data from TOF images, which is most significant when the number of atoms in a spin component is low, resulting in an inability to accurately fit for the atomic density distribution.

C. Wilson Loops
Another important consideration, in addition to the more practical one of gate fidelity, is to verify the non-Abelian nature of the connection Â.To do this, one needs to measure a gauge-invariant manifestation of the non-commutativity between the connection's vector components; in the absence of additional dynamical effects the Wilson loop is appropriate [23,33,44], which is defined as, The Wilson loop W is a gauge-invariant measure of the distortions experienced by an eigenbasis through a transformation, ΓA (ℓ).A familiar use of the Wilson loop is in the definition of gate fidelity, as implemented above ( §III B).In this framework, if the measured gate LA (ℓ) is the same as the target gate ΓA (ℓ), then the net distortion over the loop L † A ΓA is identity, that is, W = tr( 1).To show the non-Abelian character of Â, one must demonstrate that the Wilson loop is path-dependent, meaning it depends on the ordering of a series of holonomies [44].Due to the cyclic invariance of the trace, three distinct loops must be measured in two different orders, which are non-cyclic permutations of each other.More precisely, for the Wilson loop W ijk with loop order ℓ i , ℓ j , ℓ k , if W ijk − W jik ̸ = 0 then the transformations are path-dependent, and therefore the connection is non-Abelian.A detailed discussion of this may be found in Appendix C.
Given the significant dynamical contributions arising from detuning in our measured holonomies, as shown in §III B, we are unable to implement the Wilson loop in this way.The Wilson loop indicates whether the transformations being generated are Abelian or not, but says nothing about the geometric versus dynamical nature of them.Therefore, in a scenario such as this where the dynamical effects are too difficult to isolate, we would be unable to make any strong conclusions on the geometric phase alone.Furthermore, the detuning (Eq.7) drastically reduces the visibility of the non-Abelian signature W ijk −W jik , as its numerical value varies drastically with small values of detuning as shown in Fig. 4(a) , even converging to zero for certain detunings (which would otherwise indicate an Abelian transformation).Therefore, not only is it difficult to accurately measure the path dependence of the Wilson loop, but such a demonstration of path dependence can not be attributed to the geometric nature of the transformation due to the dynamical contributions.

IV. DISCUSSION
Our results demonstrate that Floquet-engineering may be used to produce non-Abelian geometric phases in systems which are otherwise non-degenerate.Furthermore, in the context of HQC, the computational basis is conveniently the same as the spin basis due to the stroboscopic nature of the Hamiltonian, without the need for any auxiliary levels or intermediate-state couplings.This technique therefore holds significant potential as an alternative approach to performing holonomic gates, as well as in the generation of interesting artificial gauge fields [32,45].
Despite these successes, the Floquet-engineering approach to HQC is not without its limitations.The magnetic field instability present in a typical setup like ours demonstrates a realistic complication in achieving highfidelity quantum gates, in the form of small miscalibrations in the RF resonances.While Floquet engineering provides a fully degenerate computational basis regardless of the underlying Hamiltonian's energetics, we find that one simply inherits many of the same issues in working with degenerate quantum systems.In fact, it is possible that the degeneracies obtained here are less robust than those in other schemes, such as the dressed-state basis of a tripod scheme [9].
Our analysis of the effect of detuning also reveals a problem that is more general to all HQC.In the presence of a degeneracy-breaking term like in Eq. 8, the traditional form of the holonomy in terms of the Wilczek-Zee non-Abelian phase [34,46] must be replaced by that in Eq. 9 [10,35].In this representation, modelling or isolating the effects of dynamical noise sources  I) on a z-component of the detuning, ∆z.The fidelities approach zero for relatively small detuning values, but also exhibit periodic revivals.
quickly becomes non-trivial, especially in the case of time-dependent noise; this is not just an issue with the Floquet-engineering approach discussed here.
To further characterize the effects of detuning we simulate the gate fidelity for a spin-1 system as it varies with detuning ∆ z [Fig.4(b)].The fidelities fall off quickly with ∆ z , but also exhibit periodic revivals.For the loops ℓ 1 , ℓ 2 , ℓ 4 , and ℓ 6 the revivals never reach unity for finite ∆ z due to the dynamical coupling the detuning produces in the Floquet basis.For ℓ 3 , the detuning term is ∝ Fz in the Floquet basis (Eq.8), and therefore commutes with the geometric portion of the phase.The detuning therefore only changes the relative phase between states, but at certain detunings this net relative phase is 2π so the fidelity is one.Lastly, for ℓ 5 , the fidelity also approaches one at a particular detuning; however, due the geometric phase not commuting with itself at different times in this loop, a closed-form solution of the holonomy Γ∆ A (ℓ 5 ) (Eq. 9) is unknown.As such, this was verified numeri-cally.
Given the effects of the detuning it is essential in future implementations to reduce the relative detunings.It should be noted, however, that simply increasing the Rabi-frequency Ω 0 would be insufficient on its own: while this would permit faster gate operations (scaling the Floquet drive ω and gate frequency Ω = 2π/T accordingly), the ability to detect detuning as outlined in Appendix E decreases with Ω 0 .Therefore, calibrating the gates in this way would result in similar detunings relative to the Rabi frequency.To avoid this issue, one also needs to adopt a different detection technique.There are many magnetometry techniques that could be readily implemented [47], but they may require additional hardware outside of the gate control-scheme itself (such as an external laser for Faraday magnetometry).Therefore, for a more scalable quantum computing setup one should calibrate the resonance with the gate-control scheme itself, as in our experiment.For instance, measuring populations after an RF-pulse with pulse area Ω 0 t = nπ with n an odd integer.For large values of n the detuning sensitivity increases.Ideally, one would apply pulses with much longer durations than their gates.This change, coupled with the more obvious additions of magnetic shielding and/or stabilization should be sufficient to realize Floquet-engineered gates with competitive fidelities, in addition to a proper characterization of the Wilson loop as outlined in §III C. For instance, with the magnetic field stability reported in similar ultracold atom systems [33], we expect the fidelities of the gates shown here would exceed 0.99.The gate fidelity expected in other quantum computing platforms would require detailed analysis, but would follow from the detuning calculations provided here (Eq.9).
It is also important in principle to consider the effects of the quadratic Zeeman shift, which is a second-order correction to the spin-state energies proportional to the linear Zeeman splitting, ω Z (Appendix F).Like the detuning, this term is unmodulated by the Floquet envelope and therefore leads to dynamical degeneracy-breaking terms in the Floquet Hamiltonian; however, unlike the detuning, the magnitude of this additional shift is approximately constant across all measurements, and can therefore be treated as a systematic effect rather than a transient miscalibration (as in the case of detuning).This effect was included in the analyses presented here, however, since this term only appears in higher-spin systems and thus does not generalize to qubits, in the generic context of HQC it is irrelevant.The generalized detuning described in Eq. 7 and App.D are sufficient to describe time-independent calibration errors that may occur in traditional qubit implementations.
Despite these limitations, Floquet-engineered gates may still have an important place in the quantum-control engineer's toolbox.There are many examples of experimental systems with excellent isolation of background magnetic fields and qubit-resonances [33,48,49].This is typically a necessary step in precise quantum control regardless of what gate architecture is used.It is unlikely that any gates are tolerant to all faults (even in the case of topological gates [50,51]), and so eliminating their shortcomings will always be required.Gates that exhibit some level of fault-tolerance are therefore welcome in reducing the complexities of quantum computers.In the case of the Floquet-engineered gate demonstrated here, more work must be done to overcome coherent and decoherent noise sources [52][53][54][55][56][57], in order to achieve proposed performance that is tolerant to high-frequency fluctuations, as is typical for geometric gates [30,31].
Compared to alternate approaches to HQC [9], Floquet engineering offers the significant advantage of choosing a convenient computational basis without the need for auxiliary levels or intermediate-state couplings.While the degeneracies and phases are best represented in the Floquet basis, its stroboscopic coincidence with the bare states (spin basis in this case) provides experimental simplicity in state preparation and measurement.This scheme has already been generalized to multi-qubit control in Rydberg atoms [31], and could be similarly extended to other popular quantum computing platforms which host sufficient control mechanisms.

V. CONCLUSION
We demonstrated and characterized an approach to HQC in an ultracold ensemble of 87 Rb.Through periodic modulations of a control Hamiltonian, the resulting Floquet-engineered system behaves as though it is fully degenerate regardless of the underlying level structure.Using adiabatic evolution of control parameters, we generated non-Abelian geometric phases for the purpose of universal single-qubit quantum gates.While our demonstration used the entire F = 1 or F = 2 ground state manifolds containing three and five spin states respectively, the control Hamiltonian generates transformations in SU (2), and is therefore readily applicable to any two-level system.Our demonstration was limited to the context of single-qubit gates, but the approach, including the detuning analysis, may be readily generalized to two-qubit gates in a similar manner [31].
Further study could also illuminate how the scheme might generalize to arbitrary SU (2F + 1) state transformations: additional coupling fields and a non-linear splitting between levels, such as through the quadratic Zeeman effect, could yield connections with such a symmetry.Such schemes are of interest to the control of qudits, or in the generation of new artificial gauge fields.
In our experimental demonstration, the existence of ambient magnetic field fluctuations revealed the limits of the approach.While Floquet engineering quite easily produces degeneracies in systems which would otherwise not support them, the technique inherits many of the same difficulties that come with maintaining them.This resulted in limited gate fidelity.Despite this, these gates offer the freedom to encode information on any readily applicable basis, without the need for auxiliary levels, removing a level of difficulty in state readout and preparation.Furthermore, the degree of fault-tolerance from sources other than static-detuning has not been investigated in great detail.Clearly, the Floquet-engineering approach deserves ongoing attention.

FIG. A1
. Spin manifold F with N = 2F + 1 magnetic sublevels, denoted mF , that are separated in energy by ωZ through the Zeeman effect.Adjacent levels are coupled by an RF magnetic field (Eq.A1) with Rabi frequency Ω(t) and phase φ(t), both of which are modulated in time to trace out loops in parameter space (Eq.A9 and Eq.A13).The frequency of the driving RF field ωRF is modulated close to resonance with the level splitting, δ ≈ ωZ − ωRF (Eq.A8).γ(t), which will be determined by the frequency modulations computed later.We may compute the transformation (Eq.A3) by rewriting in terms of the raising and Defining |ψ⟩ = Û † |ϕ⟩ and applying the rule to the ∂ t term, such that the Hamiltonian in the basis of the micromotion operator Eq.B1 becomes Where we have used the relationship that Ĥ ∝ V , such that [ Ĥ, Û ] = 0. We can then use a version of the Baker-Hausdorff lemma for differential operators and define c = sin(ωt)/ω to write where we have used that [ V , V ∂ t c] = 0 to only consider the derivatives with respect to V for the terms involving commutators.We now notice that the iterated commutators between V and ∂ t V form a recurrence relation Now, we expand ĤU (t) in a quasistatic Fourier series in ωt, essentially assuming that V (t) is approximately constant over a 2π/ω period: so that the Schrödinger equation the Floquet Hamiltonian, where the fast oscillations in the original Hamiltonian have been factored out and replaced with a new set of quantum numbers (n) which are analogous to band indices from materials with periodic spatial structure.
To perform the quasistatic Fourier transform on Ĥ(n) Floq , we use the identities where F is the spin quantum number corresponding to F .In the slow limit where F |∂ t q| ≪ ω, we may then neglect couplings between different Floquet bands, since the energetic gap between different bands will be very large relative to the off-diagonal terms.With transitions between states in different Floquet bands suppressed, we may focus on the evolution of a state solely within a given band.Therefore, we choose to restrict all attention to the n = 0 Floquet band, where the Floquet Hamiltonian becomes: and we have defined the non-Abelian connection.This agrees with the derivation in Ref. [28] for the special case where q is of constant unit length.
In the Floquet basis, the time-ordered evolution operator generated by ĤFloq. is given by: Now, we may perform a change of variables to replace the explicit time-ordering and an integral over dt ′ to an integral over dq itself with path-ordering, where ℓ is the path traced out by q from times t 0 to t.
In this way, due to the special structure of ĤFloq., the evolution of the system may be interpreted fully geometrically, depending only on the path traced out by q so long as the system is in the adiabatic limit.For a closed path ℓ, U (ℓ) is known as a holonomy which encodes geometric and topological information about the Hilbert space and the group generated by Â.We typically denote holonomies with the symbol Γ A , The trace of a holonomy is a gauge and basis invariant quantity known as a Wilson loop: As demonstrated in Ref. [44], showing that the evolution operators for two distinct loops do not commute is insufficient to prove that a system has truly pathdependent evolution.The authors describe situations where basis-dependent effects can cause two evolution operators from an Abelian gauge theory to fail to commute.They then argue that to properly determine that an atomic gas hosts a non-Abelian geometric phase, one should measure basis-independent quantities, such as the difference between two Wilson loops with different orders: for three independent loops p 1 , p 2 , p 3 .If I ̸ = 0, then the connection ΓA is truly non-Abelian regardless of any gauge or basis dependent effects.One choice of the paths p i which are useful to work with are the set of unit-radius great circles: which are chosen such that for q along these paths, F ×q is constant along the q direction.Note that this set of loops produce nearly equivalent transformations the loops considered in the experiment (Table I), with ℓ 1 identical to p 2 , ℓ 2 being the reverse of p 1 , and ℓ 3 identical to p 3 .The path-ordered evolution operator in Floquet basis can be written as: but if F × q is constant along p i , then the path ordering becomes unnecessary and the evolution operator becomes: Via Stokes' theorem, this can be interpreted as a non-Abelian flux of 2π g Fi through the loop p i .
Noticing that the operators ΓA (p i ) are SU (2) rotation operators, we decompose them in terms of Euler angles in the Z − Y − Z convention: ΓA (p 3 ) = e −i 2π g F3 e −i0 F2 e −i0 F3 = R (2π g , 0, 0) .(C10) The matrix elements of a rotation operator can be found in terms of Wigner's d-matrix [58] as: where F is the total spin quantum number corresponding to F , and: and k min = max(0, m − m ′ ), k max = min(F + m, F − m).These matrices have the properties and are manifestly real in this basis.Thus, using the d-matrices, we can write where the sums of m and m ′ run from −F to F .Similarly so that Now, we can observe that and that (if it exists) I 0 is purely imaginary, and so must vanish.Hence, we have: a manifestly real quantity which can be experimentally measured.In the absence of dynamical effects, this quantity being non-zero would demonstrate a non-Abelian geometric phase, but more generally signals path-dependent evolution.

Appendix D: Generalized Detuning
As described in the main text, uncontrolled environmental factors may contribute to unknown shifts in the experimental parameters, such as the detuning.In our case, for example, unknown ambient magnetic fields that change between calibrations may result in shifts to this detuning, ∆ z = ω RF − ω Z .
Given a general, static detuning term in the lab frame we can transform Ĥ∆ to the Floquet basis with the micromotion operator: using the Baker-Hausdorff lemma: The commutators between Ĥ∆ and the terms in the power series representation of Û form a recurrence relation: where we have asssumed that q • q = 1, so that we may write As in Refs.[28,29], we restrict our attention to the zeroth Floquet band: where we take q and ∆ to be approximately static over the course of the integral.Noting that we find that in our approximation scheme, the Floquettransformed detuning term becomes: where g = 1 − J 0 (Ω 0 /ω).

Appendix E: Experimental Methods
Our experiment uses an ensemble of ultracold neutral 87 Rb atoms in a Bose-Einstein condensate (BEC), where we prepare and manipulate a new BEC for each measurement, treating each BEC as a single quantum object (with all atoms acting in unison).To create these ultracold ensembles, we use standard laser cooling in a magneto-optical trap (MOT), and then perform forced RF-evaporative cooling in a magnetic trap, leaving atoms in the |F = 2, m F = +2⟩ ground state.Atoms are then loaded into a crossed optical dipole trap (ODT) and evaporated further, until we obtain a nearly pure BEC of about 10 5 atoms.Atoms are held in the ODT for the remaining duration of the experiment, and only released in time of flight (TOF) prior to projective measurement.
State preparation used magnetic-dipole transitions, with microwave-frequency fields to couple the F = 1 and F = 2 hyperfine manifolds, and/or RF fields to couple m F levels within a manifold.A static magnetic bias field is applied to control the Zeeman splitting ω Z between m F levels, and was adjusted so that the RF driving field was resonant when ω RF /2π = 1.25 MHz.The microwave field is produced by mixing the output of a microwave function generator, which is detuned from the hyperfine clock transition by about 100 MHz, and an RF signal from an arbitrary waveform generator (AWG); these signals are amplified together, and transmitted through a horn antenna towards the BEC.Each of the microwave source's polarization components address different |F, m F ⟩ → |F ′ , m ′ F ⟩ transitions, which are frequency dependent in the presence of Zeeman splitting.By adjusting the microwave carrier frequency, the mixed RF signal can be tuned to independently address the desired transitions [43].
Through the combination of these fields, we prepare atoms in any of the three m F levels in the F = 1 manifold with near purity (See §III).Preparing atoms in the |F = 1, m F = ±1⟩ states can be achieved by a single microwave pulse, followed by a resonant RF π-pulse, in the case of the m F = −1 state.
For In addition to the RF channel that mixes with the microwave carrier pulses, another AWG channel is used to generate the RF pulses for state preparation, generating holonomies, and state readout.The output from this AWG channel is amplified and sent through a pair of coils located near the BEC vacuum chamber; the oscillating current in the coils produces an RF oscillating magnetic field (Eq.A1), which couples internal m F states.To achieve stronger fields after amplification, the signal is sent through a home-built circuit to match the impedance of the source to the transmission line and coils.
Following state preparation and the application of a holonomy (Eq.6), we perform state readout.The ODT beams are turned off, allowing atoms to fall in TOF.During this time, a small magnetic field gradient is applied that, through the Stern-Gerlach effect, spatially separates atoms according to their m F levels.This is a projective measurement in the Fz basis, where we obtain ensemble statistics by looking at the relative populations in each spin component through absorption imaging.To measure in other bases, we precede the TOF measurement with a short RF "readout" pulse with varied pulse area and phase.From a set of measurements with varied RF readout pulses we can tomographically reconstruct the prepared state.
Prior to each set of measurements, and intermittently throughout data collection, we calibrate the RFresonance against the background detuning by applying an RF π-pulse to a pure initial state.By measuring the final state, we can detect large scale detunings, which would result in imperfect population transfer.The bias static magnetic field was adjusted to match the ω RF /2π = 1.25 MHz RF carrier frequency of the driving field.This technique is limited by our ability to image the small number of atoms remaining in the initial m F state, and the short duration of a single πpulse.To further zero the detuning we would then use the holonomy ΓA (ℓ 1 ) in a similar way, looking for the expected state populations.This gate is more sensitive to detuning than an RF π-pulse of the same duration due to its multi-spectral decomposition.In the absence of other more sophisticated magnetometry techniques, this provided an excellent resonance calibration.

FIG. 1 .
FIG. 1. Spin-1/2 simulations of state trajectories on the Bloch sphere (see Appendix G for details), with axes indicted by the −1, −1, and +1 eigenstates of σx, σy, and σz respectively.In both (a) and (b), Ω0/ω = 1, and Ω = Ω0/10.Trajectories are shown for loop ℓ1 (see TableI), with detuning (Eq.7) ∆z = 0 (a), and ∆z = Ω0/20 (b).Colored lines show trajectories calculated in the rotating frame (Eq.2) with colors denoting progress through the loop, and points (circles) sampled stroboscopically, at each half-integer period of the Floquet driving frequency ω.The solid (red) line is simulated in the Floquet basis.The disagreement between the stroboscopic points and Floquet-basis simulations are due to non-adiabatic corrections, which for the parameters chosen here amounts to about a 3% error, in terms of the fidelity (Eq.10); as Ω/ω → 0 the evolution becomes more adiabatic.

2 .
FIG.2.Measured spin populations (points) superimposed onto theory calculations (lines), with F = 1 (a, c) and F = 2 (b, d).For all measurements, Ω0/ω = 1, and Ω = Ω0/10.(a-b) Time evolution throughout the ΓA(ℓ1) gate, where Ω0/2π = 14.27 kHz.(c-d) Comparison of RF-phase scans between ΓA(ℓ1) (left) and ΓA(ℓ2) (right), which have identical evolution in the Fz basis but clearly differ in the final phase of the state.The field amplitude was Ω0/2π = 10.64 kHz and Ω0/2π = 14.27 kHz for (c) and (d), respectively.The shaded bands in (a-d) show the effect of the extra detuning term (Eq.7); for detunings sampled from a Gaussian distribution with mean ∆ fit (the result of a numerical fit for the detuning ∆z) and standard deviation 2π × 0.2 kHz, they display the interquartile range of resulting populations.The quadratic Zeeman shift, as described in App.F, was included in the numerics (see Sec. IV for discussion).

FidelityFIG. 3 .
FIG.3.Measured gate fidelities (Eq.10) for each of the loops in TableI, plotted as density distributions with individual measurements shown scattered beneath.Black vertical bars represent the mean fidelities.A fidelity of one indicates that two transformations are equal.Purple (dark) color show fidelities when detuning is not considered.Yellow (light) color show results when detunings fit to each data set are considered.The quadratic Zeeman shift (App.F) was also considered in both measured and predicted holonomies.

FIG. 4 .
FIG. 4. (a)Numerical calculation of difference between Wilson loops for two different path orderings (App.C), which are non-cyclic permutations of each other, as they vary with the detuning ∆z.W ijk corresponds to consecutive application of loops ℓi, ℓj, and ℓ k , respectively.The results vary with the spin manifold F in which the gates are performed.If the Wilson loops depend on the path order, their difference is non-zero, indicating a non-Abelian generator of the transformation.With detuning, there is reduced visibility, and one is unable to separate the impact of non-Abelian dynamical and geometric contributions.(b) Numerical integration of Eq. 9 with F = 1 showing dependence of the fidelity for each loop (TableI) on a z-component of the detuning, ∆z.The fidelities approach zero for relatively small detuning values, but also exhibit periodic revivals.

TABLE I .
The various loops considered here parameterized by Θ and Φ, and the corresponding holonomies.Evolution occurs over a single period of Ω = Ω0/10.The appearance of all three spin matrices in the phase demonstrates how these loops generate transformations in SU (2).