Practical trapped-ion protocols for universal qudit-based quantum computing

The notion of universal quantum computation can be generalized to multi-level qudits, which offer advantages in resource usage and algorithmic efficiencies. However, qudits are likely to suffer from additional error sources compared to qubits, because of the experimental complexity of manipulating more quantum states. Trapped ions, which are pristine and well-controlled quantum systems, offer an ideal platform to develop qudit-based quantum information processing. Previous work has not fully explored the practicality of implementing trapped-ion qudits accounting for known experimental error sources. Here, we describe a universal set of protocols for state preparation, measurement, single-qudit gates, and two-qudit gates. We numerically simulate known sources of error from previous trapped ion experiments, and show that there are no fundamental limitations to achieving fidelities above $99\%$ for three-level qudits encoded in $^{137}\mathrm{Ba}^+$ ions. Our methods are extensible to higher-dimensional qudits, and our measurement and single-qudit gate protocols can achieve $99\%$ fidelities for five-level qudits. We identify avenues to further decrease errors in future work. Our results support the thesis that three-level trapped ion qudits can become a useful technology for quantum information processing.


I. INTRODUCTION
In current approaches to developing quantum computing hardware, each constituent building block -such as a trapped ion, superconducting resonator, etc. -is typically used to encode a two-level qubit.However, in contrast to classical computing hardware using binary transistors, it is less obviously an optimal choice to encode only two states within a unit of quantum information.Trapped ions, superconducting transmons, and many other quantum technologies typically feature many possible physical states, and must be artificially restricted to the two states used as a qubit.A natural question is whether we are optimizing the resources extracted from our quantum building blocks by choosing to use only two of these levels [1][2][3][4][5][6][7][8].Experimentalists have developed sufficient control to envision that a quantum processor could benefit from using more of the physical states afforded by the quantum system.Making use of multi-level quantum building blocks, or qudits, presents clear challenges.More controls will be needed to fully exploit the new degrees of freedom, while at the same time, more opportunities arise for errors during a computation.However, there could be substantial benefits if these challenges are overcome.
In this paper, we propose methods to perform quantum information processing with multi-level qudits, using trapped atomic ions as the hardware.
Trapped ions are one of the leading qubit technologies owing to their superb coherence and controllability [9][10][11][12], and as such are an attractive hardware choice for developing a qudit-based technology.
To determine whether qudit-based quantum processors could be more practical than qubit-based processors, several lines of inquiry are needed.First, we must determine whether idealized qudits offer advantages over idealized qubits.Second, we must show that the necessary qudit operations can be practically achieved in experiments.Third, we must investigate whether any advantages offered by idealized qudits are outweighed by tradeoffs with increased experimental complexity and more potential sources of error.The contributions in this article primarily address the second criterion.
The structure of the article is as follows.Section II provides a brief overview of theoretical literature supporting the thesis that qudits may offer practical advantages over qubits.Section III lays out the desiderata for qudit hardware (essentially, a generalization of the DiVincenzo criteria [13]), and discusses the encoding of the qudit basis states and their susceptibility to decoherence.Section IV describes a measurement procedure suited for discriminating among all qudit basis states.Section V briefly overviews the implementation of single-qudit gates.Section VI gives a detailed discussion of realistic twoqudit gate protocols.Section VII summarizes our protocols and the error estimates obtained.Within some of these sections, we deviate from a generalized arXiv:1907.08569v1[physics.atom-ph]19 Jul 2019 discussion to specific implementations using our chosen ion species, 137 Ba + .
Previous efforts have synthesized the motivations for pursuing qudit-based computation and described possible experimental toolkits [8,[14][15][16], while other efforts have implemented limited amounts of control over three-level trapped ion qudits (i.e.qutrits) [17][18][19].The novelty of our work is the demonstration that high-fidelity operations could be performed with existing technologies, even when accounting for realistic error sources.We hope this work is of particular interest to experts in qudit algorithms and error correction, for its demonstration of a clear path to experimental implementation, and to experts in trapped ion quantum computing, for its discussion of methods to increase the information capacity of their experiments.

II. MOTIVATION FOR PURSUING QUDIT EXPERIMENTS
Current understanding of practical quantum computing rests on several early theoretical discoveries.The notion of a universal gate set is typically assumed.Namely, there exists a finite set of operations that suffice to implement any "algorithm" (i.e.any unitary operation) with arbitrary precision, if these operations can be performed on any single qubit or two-qubit pair within a large enough collection of qubits [20][21][22].Furthermore, error mechanisms in physical hardware are pervasive enough to require fault-tolerant error correction protocols, in which logical qubits are encoded using multiple physical qubits.If physical error rates can be made sufficiently small, the structure of the error-correcting code guarantees that errors can be detected and corrected.Both of these notions are extensible to qudits [23][24][25][26].Therefore, by encoding qudits rather than qubits, a larger Hilbert space is accessible with the same physical information carriers, and there is no sacrifice of universality or of the potential for fault tolerant implementations.
It is not a priori clear that the larger Hilbert space accessed by using qudits instead of qubits should translate to a computational advantage.However, some algorithms can be shown to require fewer qudits of higher dimension to achieve comparable results to a qubit-based algorithm, suggesting that there will be computational advantages [27].In particular, the quantum phase estimation algorithm, which forms the basis of Shor's factoring algorithm [28] and of many quantum chemistry calculations [29], is known to benefit from an increase in the di-mension of the qudits used.For example, as seen in the Table from reference [30], making use of 5-level qudits roughly halves the number of atoms required to perform certain examples of quantum phase estimation as compared to qubits.There are also indications that simulations of higher-dimensional quantum systems, such as spins with S > 1/2, will be more efficient when performed on qudit-based processors [31].This suggests qudits could be useful for understanding questions from fundamental particle physics (where higher dimensions are necessary to simulate color charge) to exotic quantum material properties [32].
Qudits also may offer advantages for quantum error correction, which is a more important long-term concern compared to algorithmic advantages such as seen with quantum phase estimation.Current estimates for the number of qubits required to perform practical calculations, such as simulating reaction mechanisms for nitrogen fixation, or factoring numbers of the size used in RSA encryption, are in the range of 500-1000 logical qubits [33,34].However, millions to billions of physical qubits will be required to perform these calculations.The daunting overhead results from the extreme demands of fault-tolerant quantum error correction, which will not be substantially mitigated by halving the number of logical information units such as could be done by using 5-level qudits in lieu of qubits.Qudits can ameliorate the difficulty of resource overheads for quantum error correction in several contexts.In existing qubit codes, a frequently used construction called the Toffoli gate requires half as many physical operations when introducing a third level to the qubits [35,36].Qudit error correcting codes offer more favorable error thresholds than equivalent qubit codes [37], as well as improved efficiency in magic-state distillation [38], which in many cases is the most resource-intensive aspect of error correction [34].Work by Andrist et.al. [39] and Campbell et.al. [37,38] indicates that the error thresholds to successfully implement error correction increase with the number of qudit levels, indicating that fault-tolerant quantum computing for qudits may be able to sustain a higher error rate.For surface code quantum computing, this could mean that the fidelity needed for fault tolerance is lower than the 99.25% threshold [40] given for qubits.
There is one main class of qudits that we are interested in: small prime-dimensional qudits (d = 3, 5, 7).These are interesting because a set of single-qudit Pauli group gates and the generalized π/8 gate, along with a two-qudit gate, form a universal gate set [41] for the prime-dimensional Hilbert The requirements we consider for qudit-based information processing will be as follows: 1. Ability to encode multiple basis states.

Stability of basis states against decoherence
processes.
3. Ability to prepare a fiducial initial state.

4.
A method to reliably measure in the physical basis.
6. Ability to apply an entangling operation between qudit pairs.
Each of these desiderata may be accomplished in an ion trap context, using straightforward generalizations of existing techniques.In this section, we discuss criteria 1-3.For specificity, we focus on the case of encoding qudit states in hyperfine sublevels of a ground S 1/2 electronic manifold of a hydrogenic atomic ion, e.g., using up to eight states for an ion with nuclear spin I = 3/2.A weak magnetic field is used to lift the degeneracy, and standard optical pumping techniques are used to prepare a fiducial initial state [43].

B. Encoding and coherence
Depending on the qudit dimension and the hyperfine structure of the candidate ion, there may be multiple ways to encode the qudit.We consider only encodings where the basis states form the nodes of a connected graph whose edges represent frequencyresolved transitions allowed under magnetic dipole selection rules (see Figure 1).The encodings which satisfy these requirements are the zig-zag and the bunched encodings in Figure 1.
Qudits will experience first-order sensitivity to magnetic field noise, which is the most common source of dephasing in ion trap experiments.Twolevel qubit states can be chosen to share the same magnetic field sensitivity, but that solution does not generalize to more than two states in a hyperfine manifold with J = 1/2.Technological solutions have been found to stabilize magnetic fields to the order of 1 pT by utilizing magnetic shielding and applying quantization fields with permanent magnet arrays, resulting in coherence times of order 1 second for magnetic-field-sensitive qubits [44,45].
Coherence times will suffer further as more qudit levels are added.In fact, a qudit cannot be generically characterized by a single characteristic dephasing time.Despite this difficulty, for sake of comparisons with qubits, we estimate a simplified version of a coherence time for anticipated qudit parameters.
A two-level qubit that responds to magnetic fields with H = µ 2 ∆B(t)σ z , in the limit where ∆B(t) is slowly varying compared to individual operations or algorithms (and the interaction time t is small compared to the magnetic field noise correlation decay time t c = 1/γ), will have a characteristic 1/ √ e dephasing time given by [46] In this expression, if the two most sensitive levels of the qudit are , where µ B ≈ h × 14 GHz/T is the Bohr magneton and g F 1 , g F 2 are the two levels' respective hyperfine g-factors.Here, = h/2π is the reduced Planck constant and σ z is the Pauli z operator.When making use of magnetic sublevels in the 6S 1/2 ground state, we have g F = ± 1 (I+1/2) , for F = I ± 1/2.Using the estimate of an achievable magnetic field noise from [44], where ∆B(t) 2 ≈ 2.7 pT, we may thus calculate a dephasing rate for any two hyperfine states.
The analytical form of the fidelity of a qudit in the arbitrary initial state can be obtained following a similar derivation as outlined in [46].We assume that ∆B(t) is Gaussian, and that the magnetic field noise process is stationary; for the case t 1/γ, the fidelity as a function Zig-Zag of time is We have a series of terms with different dephasing times τ l,l = To obtain a single parameter characterizing the qudit decoherence, we choose the pair of qudit states with the largest value of τ , which corresponds to the shortest dephasing time in the series of terms.For example, for the zigzag encoding in Figure 1 Coherence times will be maximized when the relative sensitivity of the entire set of states is minimized, as in the bunched configuration shown in Figure 1.Using this encoding scheme, the largest relative sensitivity for two states within a d-level qudit is µ = max( d−1 2 ) µ B I+1/2 ,where max(x) denotes the largest integer smaller than or equal to x.
In practice, encoding a qudit in the least sensitive levels is difficult to implement, because a great deal of polarization control is required.We choose instead to use the zigzag encoding exemplified in Figure 1(a), where each consecutive pair of states obey ∆F = 1, ∆m F = 1, for the d = 3 and d = 5 qudits.This encoding can be manipulated with Raman transitions that have straightforward laser polarization requirements, as discussed further in section VI.
In this case, the largest relative sensitivity for two states within the qudit is µ = dµ B I+1/2 .For comparison, the relative sensitivity of a Zeeman qubit used in reference [44], which uses the two states of a single electron spin, is 2µ B .This means that for any of the ions listed in Table I, the coherence time τ we estimate is greater than or equal to that in reference [44], depending on how many levels we are using; if we use all states, the lifetimes are equal.As pointed out in reference [47], this coherence time is already long enough to envision implementing error correcting codes with existing techniques.
C. Choice of Ion: 137 Ba + Table I compares many possible ion species options for encoding qudits.Atomic structure data for the selected ion species are presented, while the final columns show the coherence decay time in 3and 5-level qudits from magnetic field fluctuations.Having a metastable state is an important requirement in order to implement the shelving scheme described in the next section.The longer the lifetime of this metastable state, the more fiducious our shelving procedure becomes.We chose 137 Ba + to encode our qudits because it features the longest D 5/2 lifetime, and does not require an octupole transition for the shelving operation.With this species, we have enough hyperfine ground states to implement up to 8-level qudits -we just need to lift the degeneracy with a global magnetic field.For the calculations in this paper, we select a quantization field of 470 µT.Furthermore, as can be seen in Figure 2, most of the lasers we require are in the visible range, sim- state.Next, we list the relevant transitions in each ion species: the primary transition is used for Doppler cooling, optical pumping, and fluorescence measurement, and the metastable transition allows us to measure qudits using the shelving technique described in section IV.The last two columns are the dephasing/coherence times for a 3and 5-level qudit with 2.7 pT magnetic field fluctuations, calculated from Equation 1; we assume a zig-zag encoding centered at mF = 0.
plifying the optical technology required to build an experiment [43,54].
For the calculations in this paper, we assume that the metastable transition may be driven by a laser with 1 Hz linewidth [55].We furthermore assume that coherent operations may be performed using either microwave radiation or stimulated Raman transitions driven by a 532 nm pulsed laser [56].

IV. QUDIT MEASUREMENTS
State measurement for trapped ion qubits is accomplished by exposing the ion to laser radiation, configured so that only one of the qubit states fluoresces, and the fluorescence is collected on a detector such as a CCD or photomultiplier tube.In generalizing the fluorescence technique to multiple levels, we must produce a signal that differs for each physical basis state.A straightforward way to accomplish this goal is to sequentially check each basis state separately: 1. Engineer a situation where only one of the basis states (e.g.|0 ) produces fluorescence when exposed to laser radiation 2. If no fluorescence is detected, engineer a situation where another state (e.g.|1 ) fluoresces 3. Etc, repeating this process until the presence of fluorescence has indicated which of the basis states is occupied.In case the ability to discriminate between all the basis states is required, the criterion that only a single basis state respond to the fluorescing lasers at any given step is crucial.If two or more states are both induced to fluoresce simultaneously, then typically the information about which of the states was occupied will be lost by the time fluorescence is detected, because the multiple spontaneous photon scatters will result in a random final state after fluorescence.
Many ions used for QIP feature metastable states, which can be exploited for this state readout.The metastable state chosen should not be part of the closed-cycle transition used for Doppler cooling.The "shelving" approach to measuring a qudit encoded in such an ion is illustrated in Figure 3 for 3levels.It consists of shelving all but one state in the metastable state, measuring the remaining state, then repeatedly de-shelving and measuring states until the state of the qudit is known.This approach assumes that the transitions between each qudit state and its corresponding metastable shelf state are resolved in frequency, so that each state can be checked independently during the fluorescence step.In order to circumvent the need for real-time changes to the measurement sequence (i.e.stopping the measurement process once fluorescence is detected), in practice the full pulse sequence for all d qudit states could be carried out for each measurement. 137Ba + energy structure [50].The 6S 1/2 ↔ 6P 1/2 optical transition is used for optical pumping, Doppler cooling, and fluorescence measurement.The 6S 1/2 ↔ 5D 5/2 transition is used to shelve qudit states.The 5D 3/2 ↔ 6P 1/2 transition is used to repump dark states back into the cooling/fluorescence cycle.The 5D 5/2 ↔ 6P 3/2 transition is used to empty the 5D 5/2 state.Because of its nuclear spin 3/2, each level is split into hyperfine levels: the frequency differences between these levels are shown [51][52][53] (2) ( The approach above implements a POVM with measurement operators corresponding to the projection operators for each basis state, An abbreviated version of the procedure described could be used to perform a POVM with measurement operators corresponding to the projection operators for certain subsets of the basis states: e.g., if only states other than |0 are mapped to a nonfluorescing state during the measurement, then the measurement operators are P 0 = |0 0| and P not0 = I − |0 0|.Note that our computational and measurement basis is, in a sense, preferred by the physical system.As with two-level qubits, measurements in any basis other than the computational basis defined by the hyperfine sublevels will require two steps to perform: first applying a unitary rotation that maps the desired measurement basis to the physical computation basis, then measuring in the physical basis.This is analogous to measuring in the σ x eigenbasis of a qubit by means of applying a σ y rotation before measuring in the σ z eigenbasis.
We additionally estimate fundamental limitations on the measurement speed and fidelity for our chosen ion, 137 Ba + .The finite lifetime of the metastable state D 5/2 imposes a fundamental limitation on the measurement fidelity as a function of the measurement duration.Furthermore, ensuring all the shelving transitions are frequency-resolved places limitations on the transition rate, imposing another tradeoff between measurement speed and fidelity.We propose to use rapid adiabatic passage to achieve these population transfers.A similar scheme was used in reference [57] for 138 Ba + shelving.
For a 2-level system, if we couple a field detuned from the transition by ∆ = ω − ω 0 (where ω is the laser frequency and ω 0 is the transition frequency) and use the rotating wave approximation (RWA), the Hamiltonian can be written as where Ω is the resulting Rabi frequency of the transition.The eigenstates of the system are called the adiabatic or dressed states.The important result is that if we sweep the frequency of the field adiabatically from some detuning ∆(0) across resonance, stopping at −∆(0), then the system will remain in whatever adiabatic state it was initialized in.While the adiabatic state doesn't change, the composition of this adiabatic state in terms of the diabatic (undressed) states changes so that we have near perfect fidelity population transfer between the two levels.
There are several sources for error during population transfer using adiabatic passage.The overall fidelity is given by where tan 2θ = Ω ∆ , i is the set of all unwanted transitions each transfer can couple to at the start and end of the transfer, C i is the transition strength of the i th unwanted transition, Ω i = ∆ 2 i + Ω 2 is the effective Rabi frequency of the laser coupling to some other transition detuned by ∆ i , ∆ν is the FWHM laser linewidth, and [58,59] (fidelity from the adiabadicity of the transfer).Finally, α = ∆ is the sweep rate of the laser frequency (we assume that the sweep is linear).The first term comes from imperfect adiabatic state preparation -ideally, we would have to start the sweep at detuning ∆(0) = ±∞ for the adiabatic states to correspond exactly to a diabatic state.This is not reasonable to do, so we instead start at some finite detuning.The second term comes from off-resonantly driving other transitions between the ground and the metastable shelving states.The third term is the dephasing and adiabacity error [60].Lastly, we have to consider decay of the shelving state from its finite lifetime (≈ 30 s for Ba + ).
For our simulations, we a quantization magnetic field of 470 µT.In 137 Ba + , the F = 3, 4 levels are separated by less than 1 MHz, so we use the F = 1, 2 levels, which are separated by ∼ 70 MHz, as our shelving states.We transfer |S 1/2 ; F, m F states to |D 5/2 ; F = F, m F = m F states in the shelving manifold.By orienting the 1762 nm laser wavevector and polarization in a useful geometry [61,62], we can completely suppress q = ±1 transitions (q = −(m F − m F )), and reduce the strength of q = ±2 transitions.The q = 0 transitions are, in the smallest case, ∼ 3.5 MHz apart in frequency.However, there are some q = ±2 transitions which are closer; the we encode using this level, we make sure to measure this state first, to avoid the need to shelve this state.
There are additional motional sidebands on this transition at the secular trap frequency.In a standard "blade" style trap, we expect this frequency to be ∼ 5 MHz.This can be tuned down to ∼ 2 MHz.In this case, for some shelving transitions, the second motional sideband is less than 1 MHz away from the transition we wish to drive.However, because the Lamb-Dicke parameter is less than 0.01, these couplings are negligible; this is because the strength of the second order motional sideband scales as η 2 ≈ 10 −4 .Therefore, we don't consider these couplings.With laser stabilization, we can get a laser linewidth of less than 1 Hz [55]; we assume 1 Hz laser linewidth for the following calculations.
With these properties, we calculated the fidelity of population transfer for different Rabi frequencies and overall passage times for the 3-level qudit F = 2, m F = 0 ↔ F = 2, m F = 0 shelving transition (see Figure 4(a)).We assume that the initial detuning ∆(0) = 1.6 MHz, which is 200 kHz detuned from the nearest motional frequency: the tilt mode at 1.8 MHz.If we pick optimal parameters (along the gray line in Figure 4), we can get much better than 99% fidelity for individual transfers.
For our overall measurement process, we also have to consider each fluorescence measurement.We assume that our imaging system has N A = 0.5 and a quantum efficiency of 80%.
A good estimate for our fluorescence Rabi frequency is (Ω f /2π) × 1/4 ≈ 5 MHz.Assuming we need around 10 bright-state photons to discriminate between a bright or dark reading, each fluorescence step takes 37 µs.
Figure 4(b) considers the entire shelving measurement process for different qudits up to 7-levels.During a measurement, the procedure is complete once fluorescence has been seen, so usually all of the transfers in the shelving procedure don't need to be performed.Here we assume the worst case where we end up having to do all of the transfers (for d levels, this is 2d − 3 transfers).Again, we assume that the initial detuning for each transfer is 200 kHz detuned from the nearest transition frequency.We plot the overall measurement fidelity using optimal parameters on each transfer.As can be seen in the Figure, it's possible to get better than 98.5% overall measurement fidelity for up to 7-level qudits.Both the 3-and 5-level qudits can be measured with better than 99% fidelity.Furthermore, because we can measure all of the states in the ground manifold with little error, state tomography for 3-and 5-level qudits is straightforward using this shelving technique.
Harty [63] was able to discriminate between a qubit state in the S 1/2 and the D 5/2 states with fluorescence in 43 Ca + ; their overall state preparation and measurement fidelity was better than 99.9%.An important distinction is that their transfer to the shelving state was not coherent like our proposed shelving operation.If we assume that their 99.9% error is mostly coming from measurement rather than state preparation, and that we see similar results, then our overall measurement error will increase by a factor of (99.9%) n , where n is the number of fluorescence measurements.
There are multiple avenues by which the error rates could be improved beyond the analysis in this paper.Another approach for the population transfer would be to use chirped pulses; this could give us better than 99% fidelities in a shorter measurement time [64].Alternatively, we could focus on improving our magnetic field stabilization, laser frequency, and intensity stabilization so that we can do normal Rabi transfers for an even shorter measurement time.Finally, when we do statistical measurements, we can use an adaptive algorithm to do state fluorescence on the state that the qudit is most likely in, based on the previous measurements.Such an adaptive measurement would make the number of adiabatic passages necessary approach d − 1, dramatically decreasing the measurement error.

V. SINGLE QUDIT GATES
Single qudit gates can be decomposed into sequences of two-level operations as outlined in [6,65].Physically, these operations are implemented using sequences of microwave or laser pulses, each implementing an evolution operator of the form G(j, k; θ, φ) = exp iθ e iφ |j k| + e −iφ |k j| .(6) Here |j and |k are two of the qudit basis states, θ represents a pulse angle (which physically depends on the Rabi frequency for the transition |j ↔ |k and the pulse duration), and φ is a phase that can be controlled by manipulating the phase of the microwave or optical radiation.These constituent operations G(j, k; θ, φ) are referred to as Givens rotations, in keeping with prior nomenclature.
Single-qudit gate decompositions can be made provided that the allowed transitions form a connected graph.For a qudit of dimension d, at most d(d − 1)/2 Givens rotations are required to synthesize an arbitrary single-qudit unitary, up to a set of phase factors on the qudit basis states.Essentially, the desired gate can be written as a sequence of population transfers between two levels from the qudit space: where Vi are unitaries generated by individual pulses applied to a transition between states and Θ is a set of phase factors in a diagonal matrix.If necessary, these phase factors can further be eliminated by at most 2(d − 1) additional rotations.Further details of this decomposition are given in Appendix A 1.
We describe a library of pertinent qudit operations in Appendix A 2, in which we map operations of interest for qudit circuits and error correcting codes into a corresponding set of Givens rotations.We focus on prime-dimensional qudits, formulating the elements of the generalized single-qudit Pauli group and the generalized π/8 gate, which, along with a two-qudit gate, complete a universal gate set [41].We also decompose the Quantum Fourier Transforms (QFT).
In qubit-based quantum processors, the error rates for single qubit operations are often significantly lower than those for state preparation and measurement or for two-qubit gates, and this statement will likely hold for single-qudit gates too.To drive a single qudit gate, we consider two possible methods.The first method is a direct transition with a microwave source while the other is via a Raman transition with laser beams in the visible range.For microwaves, an unpolarized source is assumed.Using microwave pulses makes it more difficult for us to do individual addressing of ions.While there are techniques to do it [66][67][68], Raman transitions have much better individual addressing control.
In order to assess the practicality of a trapped ion qudit system, the expected errors for the constructed gate set should be evaluated.From previous qubit experiments, sources of error for a trapped ion system are well-understood [69].Thus, focus is given to known errors that are fundamental to our trapped ion system.Other error sources pertaining to hardware control imperfections are excluded as they can arguably be improved as technology advances, thus not posing a fundamental limit to the fidelities of  the single qudit gates.For single qudit gates, the first error we consider is magnetic field noise.Fluctuations in the overall magnetic field result in magnetic sub-level energy fluctuations, decohering the qudits while transitions are being driven.Brown and Brown [47] found that a single-qubit hyperfine gate can be error corrected if the magnetic field fluctuations are smaller than 100 nT.Another source of error will be off-resonant coupling.This error is relevant for an unpolarized driving source as an unpolarized source is unable to make use of selection rules for state transitions to mitigate off-resonant coupling.We performed numerical simulations to estimate the errors for both magnetic field noise and off-resonant coupling for different qudit gates with magnetic field fluctuations of 2.7 pT (achievable using shielding and feedback [44]).Details of these simulations can be found in Appendix A 3, and the results for a Rabi frequency of 10 kHz are shown in Figure 5.For the more intensive gates such as the QFT and the π/8 gate, the errors are at worst, which are ∼ 10 −4 for 3-level qudits and ∼ 10 −3 for 5-level qudits.For all of these gates, the fidelities are better than 99.8%.To isolate the contribution of each effect, simulations are also run for each error by itself, and the average results for the QFT gate with a Rabi frequency of 10 kHz are shown in Table II.With such a well controlled magnetic field, off-resonant coupling error is more than 5 orders of magnitude larger than the magnetic field noise error.To see how much we can relax our magnetic field control, we use equation 3 as an estimate for the QFT gate error, assuming a coherent superposition of all qudit states.For the magnetic field noise error to be of the same order as the off-resonant coupling error, the magnetic field noise ∼ 1 nT.For higher Rabi frequencies, the gate times decrease, and thus the magnetic field fluctuation errors are reduced; however off-resonant coupling error becomes worse.With a Rabi frequency of 100 kHz, the QFT error increases to around 1% and 10% for 3-and 5-level qudits respectively.These results with microwave gates indicate that Raman transitions are the better option for single qudit gates.Using Raman transitions with polarized laser beams avoids errors from off-resonant coupling.However, there is another significant error stemming from photon scattering.To obtain a characteristic value for this error (using Equation C5), we pick the error from transition that has the largest photon scattering error among all the |l to |l + 1 transitions.For simplicity, we do not differentiate between Raman and Rayleigh scattering from the total scattering rate.We assume that any scattering decoheres the state completely, which overestimates the true error because Rayleigh scattering will not cause errors for single qudit gates [70].For d = 3 and using the zig-zag encoding shown in figure 7(a), the largest scattering rate is where R sc,0 is the scattering rate from state |0 , Ω 0 is the Rabi frequency for the transition |0 ↔ |1 , γ 1/2 = 9.53 × 10 7 s −1 is the decay rate from the 6P 1/2 state of the Ba + ion, γ 3/2 = 1.11 × 10 8 s −1 is the decay rate from the 6P 3/2 state [71], ∆ 1/2 is the detuning of our laser frequency for Raman transition from the 6P 1/2 state, and ∆ 3/2 is the laser detuning from the 6P 3/2 state.For d = 5 and the zig-zag encoding scheme as shown in Figure 7(b), the largest scattering rate is where R sc,1 is the scattering rate from state |1 and Ω 1 is the Rabi frequency for the transition |1 ↔ |2 .The scattering probability, P sc , which is treated to be equivalent to the errors, is then where t g is the gate time.The QFT gate, which has the longest duration, is used to obtain an upper bound for the error.The QFT gate has a gate time t g = 280.4µs for d = 3 and t g = 678.5 µs for d = 5 with a Rabi frequency of 10 kHz.This very conservative upper bound for the error is displayed in table II.We expect the actual scattering error to be much smaller than what is presented, and that the overall Raman single-qudit gates will be more fiducious than microwave gates, since the off-resonant coupling error can be mitigated by polarization control.In addition, using Raman transitions allow us to more easily do individual addressing.

VI. TWO QUDIT(ENTANGLING) GATES
The two-qudit gates can be performed using generalizations of a common technique often referred to as the Mølmer-Sørensen(MS) gate.Lasers applying optical dipole forces to the ion crystal can be used to implement a state-dependent force; combined with the Coulomb repulsion between ions, this force can mediate an entangling interaction.
In this section, we give a detailed derivation of a MS like entangling protocol for qudits (part A).In addition, we investigate the effects of a variety of possible error sources (part B).

A. Ideal generalized MS gate
We describe a generic approach to implementing two-qudit gates by addressing an appropriate combination of motional sideband transitions.We assume that the qudit levels are chosen such that there are dipole-allowed transitions between each pair of levels |l ↔ |l + 1 , and that the energies are chosen in a zig-zag configuration, where even-numbered states are higher in energy than odd-numbered states.
In the case of a qubit MS gate, we desire an interaction Hamiltonian of the form [72] where η is the Lamb-Dicke parameter, Ω is the resonant Rabi frequency between the two levels, â and â † are the lowering and raising operators of a vibrational mode in an ion chain respectively, ω M is the target vibrational mode frequency, µ is the detuning of laser frequencies from the resonant frequency, t is time, and σx is the Pauli x operator.n is the index for each ion in a chain of N ions.
In analogy to the qubit case, we generalize the MS gate to a qudit system with the following Hamilto- with d = 2s + 1 being the total qudit levels.This is analogous to generalizing a spin half system in the qubit case to an arbitrary spin system in the qudit case.
The desired time evolution unitary operator generated by the Hamiltonian in Equation 12is where θ 0 is the qudit entangling gate phase.
In order to arrive at the desired Hamiltonian, we start with a chain of N ion qudits.The static Hamiltonian is where Ĥ0,M describes the Hamiltonian of the motional state, Ĥ0,S describes the Hamiltonian of the internal energy states, the subscript m denote the m th vibrational normal mode, E l is the energy of state |l , and l = l + s.We assume that for each transition level between |l and |l + 1 , we apply a laser perturbation with frequency ω L,l , close to the transition energy between the two target levels and far off-resonant to (or forbidden by selection rules for) transitions to the other levels.The interaction Hamiltonian is then approximately where Ω l,n is the Rabi frequency for the target transition from |l to |l + 1 for the n th ion, x is the position of an ion along the motion of the phonon mode being used for entanglement, k is the wavevector of the laser perturbation along x, and φ is the initial laser phase.The total Hamiltonian is then Ĥ = Ĥ0,M + Ĥ0,S + Ĥint .
Assigning odd qudit levels to lower energy levels and even qudit levels to higher ones in a zig-zag pattern, we define In the interaction picture with respect to Ĥ0 , the effective Hamiltonian is then where x = e i Ĥ0,M t xe − i Ĥ0,M t is the position operator in the interaction picture.
The above description assumes one laser frequency per transition.In analogy to the MS scheme, let us have two laser perturbations tuned close to resonant for each |l to |l + 1 transition.One set of laser fields is blue-detuned while the other is red-detuned, with frequencies For small µ, we can apply a RWA to obtain the effective Hamiltonians for blue and red-detuned laser perturbations respectively: Defining and adding Ĥb and Ĥr gives the resulting effective Hamiltonian of the form × e −i(−1) l φ S,l e −i(−1) l kx n |l + 1 l| n +e i(−1) l φ S,l e i(−1) l kx n |l l + 1| n . ( For small kx , we can apply the Lamb-Dicke approximation (LDA), which gives Expressing kx in terms of raising and lowering operators of ion chain vibrational modes, where η m,n is the Lamb-Dicke parameter for the m th motional mode and the n th ion, ω m is the motional frequency of the m th motional mode.We arrive at Ĥtotal = Without loss of generality, we let the mode m = M to be close to the laser frequency detuning µ ≈ ω M , and far-detuned from the other vibrational mode frequencies.With the condition Ω l µ, another RWA can be applied, which gives the resultant effective Hamiltonian of Ĥtotal = For simplicity, we specialize to the case where the near-resonant vibrational mode is the centre-of-mass mode, η M,n = η C and we rewrite ω M = ω C (we will from here on use the subscript C to denote centreof-mass mode).To arrive at Equation 12, we let For the case where the spin phase φ S = 0, and the motional phase φ M = 0, with Equations 27 and 28, we get the exact form in Equation 12.This dictates the phases of the blue and red-detuned laser perturbations [73] The time-evolution operator generated by the Hamiltonian of Equation 12 is obtained by solving Schrödinger's equation We evaluate the time-evolution operator with the Magnus expansion where M k (t) is the k th term in the Magnus expansion.For the Hamiltonian at hand, the generated Magnus expansions are Ŝx,n where Here, we have neglected terms in M 2 (t) which are bounded with t.
To minimize coupling to the phonon states (which is equivalent to minimizing M 1 (t) and closing the loop in the phase space picture in Figure 6(a)) and obtain the desired entangling gate, we require where K is a positive integer.The resultant unitary of the qudit entangling gate is then  .
(34) The ion qudits in eigenstates of Ŝx after the gate in Equation 34 gain phases of where λ n is the eigenvalue of the n th ion with respect to Ŝx .For a two-qudit gate, N = 2, and the output is an entangled 2-qudit state in general.
In the phase space picture as shown in Figure 6(a), this operation corresponds to displacing the system in the phase space with a radius proportional to S x,1 + S x,2 .The geometric phase gained after closing the loop is proportional to the area enclosed by the trajectory, which is proportional to (S x,1 + S x,2 ) 2 .

B. Error estimates
In order to estimate the expected error of the twoqudit entangling gate, we consider sources of error that are intrinsic to the formulation as well as experimental imperfections that are difficult to overcome with current technology.The intrinsic sources of error are: 1. Inaccuracy from LDA.

Photon scattering.
The experimental imperfections deemed difficult to overcome are: 5. Imperfect cooling of ions.

Magnetic field noise.
In order to obtain the output state and thus the fidelity without making the above-mentioned approximations, the time evolution of an input state is simulated by numerical integration according to the differential equation in Equation 30 using the Hamiltonian in Equation 23.In order to obtain the error due to imperfect cooling of ions, the input phonon state is modelled to be in a thermal state with phonon Fock state population distribution of [74] where n is the average phonon number.In order to obtain a crude (over)estimation of the error due to motional heating of ions during the gate, the phonon state of the motional mode is increased by one when the phase space displacement is maximal, from which we compute F heat .The overall fidelity is then computed by where P heat is the probability that a phonon hop happens due to motional heating from the environment, F 0 is the fidelity when no phonon hop happens, and F heat is the fidelity when a phonon hop has happened.
To compute the scattering probability during the gate time, the calculation is done based on reference [70].For the case of a 3-level qudit system, the photon scattering probability is calculated with the states |1 , |2 and |3 in Figure 1(a) being the 3 computational qudit states.The photon scattering in this case is derived to be With the 5-level qudit in the zig-zag encoding, the total photon scattering probability is derived to be We acknowledge that the derived formula is a factor of 2 larger than that given in reference [70], which may be due to a different definitions employed in the formulations.The details for these derivations are available in Appendix C.
To estimate the lower bound of our gate fidelity due to photon scattering, we assume that the gate has zero fidelity once a scattering event has occurred.The probability of zero scattering event for both ions is The fidelity is then computed by Since it is more computationally intensive to implement a time-varying magnetic field noise for the simulation of an entangling gate, we obtain an estimate of the error introduced by magnetic field noise by setting a constant magnetic field offset error of 2.7 pT.This modifies the original Hamiltonian to where Ĥorig is the original Hamiltonian and ∆E l is the energy shift of state |l due to magnetic field shift.
To model a realistic ion trap, the parameters used are η C = 5.07 × 10 −2 , ω C = 2π × 2 MHz, ω T = 2π × 1.8 MHz (the frequency of spectator mode, subscript T denotes tilt mode), µ = 2π × 2.01 MHz, K = 1 and thus a gate time of t g = 2π |ω C −µ| = 100 µs.We set θ 0 = − π 4 as an example.This value of θ 0 is chosen as it results in a non-trivial entanglement result that is not replicable by a single qubit MS gate for a 3-level qudit system.For example, θ 0 = − π 2 acting on the state |2, 2 of a 3-level qudit system can be shown to give the same output as a qubit MS gate acting on the appropriate transition levels (see Figure 6(b) at t = 200 µs).We kept θ 0 = − π 4 for the 5-level qudit for simplicity.The motional heating rate is  assumed to be 100 s −1 , which is a realistic estimate [75].The initial two-qudit state is chosen to be |d − 1 |d − 1 .Variations of the parameters and Hamiltonian were used to pinpoint the magnitude of error contributed by a certain error source.The details of the simulations are available in Appendix B. To compute the error from photon scattering, Raman beams with wavelengths of 532 nm are assumed, which gives ∆ 1/2 = −44.08THz and ∆ 3/2 = −94.78THz.
The fidelity obtained with all the error sources taken into consideration for d = 3 is 0.9932.For d = 5, off-resonant transition frequencies distorts the Hamiltonian significantly from the encoding scheme in Figure 1(c), and results in a fidelity much smaller than 1, which is F = 0.0296 (see Appendix B 5).We note that this error is present due to symmetry of the zig-zag encoding scheme that we have chosen, and may be overcome with other encoding schemes.Neglecting this error, an overall fidelity of 0.9791 is obtained for d = 5 with these parameters.From Table III, the spectator phonon mode and motional heating of ions are the major sources of error.To reduce the error due to a spectator phonon mode, a direct way is to tune the trap parameters such that the spectator mode is detuned farther from the desired phonon mode frequency.This would reduce the contribution to the state evolution from the spectator modes.To eliminate the spectator mode contribution without the need to tune the trap parameters, clever pulse shaping could be performed which removes spin-phonon coupling of spectator modes, which is shown for the qubit case [76].Assuming that spectator mode error can be eliminated FIG. 7. Schematics of laser perturbations applied to 137 Ba + for (a) 3-level qudit and (b) 5-level qudit.ωR,n denotes the n th laser frequency applied for the entangling gate.
by clever pulse shaping techniques, the fidelity for this 3-level qudit entangling gate can be increased to 0.9959.Neglecting the error due to off-resonant frequencies again, the fidelity for the 5-level qudit entangling gate is 0.9901 if the error from spectator mode can be overcome.
Overall, it is possible to achieve more than 99% for 3-level qudits with this generalized entangling gate.For d ≥ 5, this gate is not applicable for our specific encoding scheme using 137 Ba + due to error from off-resonant frequencies.

VII. SUMMARY
In this paper, we have described a suitable operation set to implement qudit-based quantum computing.We discussed how to satisfy all of DiVencenzo's requirements for quantum information processing with qudits.Using the hyperfine sublevels of a trapped ion, we are able to encode the different qudit states such that we can arbitrarily create any superposition state for the system.Standard optical pumping can be used to initialize our qudits reliably.An optical shelving method using a metastable state was discussed in detail, which allows us to measure the state of the qudit with low error.Finally, we presented a Mølmer-Sørensen-like entangling gate which, along with single qudit gates, allows us to create any arbitrary entangled quantum state in our qudits.With these conditions satisfied, our proposed trapped ion system can be considered a universal quantum computer.As a comparison for our operations' fidelities, we use the 99.25% fault-tolerant error threshold found for qubits [40] on the grounds that there is significant evidence that qudit-based codes will have more relaxed thresholds [37].Table IV lays out all of the error sources considered for our qudit platform.The simulation codes for the error estimations presented in this paper can be found in the repository [78].We acknowledge as well that this is not an exhaustive study; more details could be included such as noise from Rabi rate fluctuations and differentiating types of photon scattering.However, our main goal is to show that there are no fundamental roadblocks towards qudit implementations, and we have taken measures to ensure the errors considered are upper bounds for this study.For qutrits, we find no fundamental obstacles to achieving this error threshold.For 5-level qudits, more work needs to be done to improve the entangling gate, but if we succeed in overcoming the parasitic coupling, these gates could be done with a fidelity of at least 97.91%.microwave E(t) = E 0 cos ωt + φ (ω = ω e − ω g ), the pulse hamiltonian in the atomic hamiltonian interaction picture looks like ĤI = (A10) Note that some transitions have the same frequency; we could align our electromagnetic polarization to the magnetic field to select a single transition.Each decomposed pulse R[θ, ρ] ij corresponds to the physical parameters θ ↔ | Ωmi−mj |(t − t 0 )/2, ρ ↔ φ − arg( Ωmi−mj ).

Gate Library
Unitary Pulse Transition Pulse Angle, C Phase, φ  Here, we give a list of useful qudit gates decomposed into these pulses.We assume that all qudits are in the zig-zag configuration, so that |l is con-nected to |l ± 1 and we only perform pulses on these transitions.Different pulses are notated R[θ, ρ] ij , where θ is the angle from the z-axis(pulse angle), ρ is the angle from the y-axis(phase) in the bloch sphere picture, and ij denotes the transition |i ↔ |j .
The These gates are useful in quantum information theory as supplements to the Pauli and Clifford gates [41].We used the generalized 3-and 5-level π/8 gates presented in reference [41].All of these gates are listed out for 3-and 5-level qudits in tables V and VI.
where g F is the hyperfine g-factor, µ B is the Bohr Magneton, and ∆B(t) is the random fluctuation of the magnetic field from the set magnetic field.The subscripts g and e denote the lower and higher energy state in the hyperfine splitting respectively.The resultant Hamiltonian is then where Ĥideal is the ideal Hamiltonian for a single qudit gate.The output state under this Hamiltonian is obtained by numerically solving Schrödinger's equation.
To account for off-resonant coupling the Hamiltonian has to be modified to where ĤOR is the component of the Hamiltonian due to off-resonant coupling.It has the form where |l and |l + 1 are the states where resonant transition is desired, Ω l is the Rabi frequency for the desired transition, ω l is the transition frequency between |l and |l + 1 , Ω k,k is the Rabi frequency for the transition between the states |k and |k , ω k is the energy for the |k state, sgn (x) is the sign function sgn For the simulation, we find that it is too computationally intensive to simulate both off-resonant and magnetic field noise error simultaneously with a time-varying noise.Thus, the deviation in magnetic field is set at a constant offset at the standard deviation of 2.7 pT as an estimation for simulations with both errors taken into account.We found no discernible difference in the average fidelity obtained whether a magnetic field offset is present as the error is dominated by off-resonant coupling.For the simulations with only magnetic field noise present, we generate random magnetic field noise using a Ornstein-Uhlenbeck function with a mean of 0, inverse correlation time γ = 0.5 ms −1 , and volatility σ = 2γ ∆B 2 ; we assume that the magnetic field noise is Gaussian and stationary [46].For simulation of an entire gate, pulses are applied immediately after one another; we calculate the fidelity by comparing the final state from the evolution of Equation A12 to the desired state from applying the gate.
(LDA).This gives us a more realistic fidelity that we expect to get when we carry out the experiment.Thus, the Hamiltonian used for the simulation is Equation 23.
First, we choose some convenient motional and spin phases Then, the Hamiltonian is Ĥ Due to computational limitations, we can only simulate 2 phonon modes (center-of-mass and tilt modes).Thus, the position operator for each ion is where the subscripts C and T denote centre-of-mass mode and tilt mode respectively.The Hamiltonian is then Since the phonon operators for different modes commute, Ĥ = In the presence of a magnetic field mismatch, the Hamiltonian is modified to With Equation B5, the time evolution operator can be solved numerically according to Equation 30.We are only concerned about the output spin state and not the phonon states at the output.Thus, |ψ ideal is an ideal spin state and fidelity is computed after tracing out the phonon states of the output density operator.The fidelity at the end of the gate is where ρ 0 is the initial density operator before applying MS gate.

Simulating Mixed Initial State
We are taking into account error due to having an initial state that is not absolutely in the ground state, which is a realistic assumption.The initial motional state is assumed to be in a mixed state with the density operator ρ 0 = m n P C (m)P T (n)|ψ 0 , m, n ψ 0 , m, n|, (B8) where P is the phonon Fock state population.The C and T subscripts again refer to the centre-of-mass and tilt modes respectively, and |ψ 0 is the initial qudit state.For a thermal state, we have [74] where n is the average phonon number.
The best strategy for a faster simulation is to evaluate the time evolution operator, then apply it to the initial density operator to get the output and compute the fidelity.However, this can be too memoryintensive, which is the case for us.
As an alternative approach, we compute the evolution of pure phonon Fock states, |ψ 0 , m, n , then weigh each fidelity by the phonon populations, P C (m)P T (n): The total fidelity is then which is equivalent to Equation B7.Since we are numerically solving the problem, the summation over the Fock state population n P (n) cannot be an infinite series.Thus, the number of allowed Fock states for the centre-of-mass, m max and tilt modes, n max have to be chosen such that they are large enough for accurate results.For our simulations, where nC = 0.1 and nT = 0, we have determined that m max = 20 and n max = 2 is accurate enough such that further increment of allowed Fock states do not increase the accuracy of the fidelity at the fourth decimal place.
To further speed up the process, phonon states where P C (m)P T (n) < 10 −5 are ignored.

Optimum Rabi Frequency
The Rabi frequency-geometric phase relation in Equation 35 is derived with LDA and RWA.Without the RWA applied to Equation 26 to arrive at Equation 27, the Hamiltonian we arrive at is Ŝx,n , (B12) and θ 0 is modified to Without LDA, the geometric phase from the entangling gate for a certain phonon Fock state is [72] θ where where L α n (x) are the generalized Laguerre polynomials Thus, to obtain the desired geometric phase with optimal fidelity for input qudits in phonon Fock state n, the laser amplitude should be tuned to the corresponding Rabi frequency of where Ω LDA is the optimal Rabi frequency with LDA.
For input states with a superposition of or mixed phonon states, the fidelity with errors only from the shifted geometric phase from the LDA case can be written as C l e −i(−1) l (kx n −µj t− π 2 ) |l + 1 l| n + e i(−1) l (kx −µj t− π 2 ) |l l + 1| n C l e −i(−1) l (kx n +µj t− π 2 ) |l + 1 l| n +e i(−1) l (kx +µj t− π 2 ) |l l + 1| n , (B27) where 3 , and the quantity ∆ z is the energy of the Zeeman splitting in frequency.We further simplify the problem by letting ∆kx → 0 in the off-resonant component of the Hamiltonian.The fidelity of 0.0296 in the main text is obtained from simulations with the Hamiltonian in Equation B27, which only has the error from off-resonant frequencies to verify that this error alone causes failure for the 5-level entangling gate.We employ the Magnus expansion again to evaluate the time-evolution operator generated by this Hamiltonian.
The first term in the Magnus expansion is (B28) By changing the laser frequencies or Zeeman splitting, such that (1 − e ±i(−1) l µj t ) = 0 when t = K 2π |ω M −µ| , it is still possible to minimize the contribution of the off-resonant in the first Magnus expansion.The second order Magnus expansion is  (C7) Similarly, the Rabi frequency for each of the Raman transitions in Figure 7(b) can be derived to be

FIG. 1 .
FIG. 1. Examples of qudit encodings.(a) We prefer a zigzag configuration because it simplifies laser manipulations.(b) A bunched configuration minimizes decoherence due to magnetic field fluctuations but requires more polarization control of lasers.(c) Disconnected configurations are not preferred due to the experimental complexity of transferring population among all possible states.
FIG. 4. (a) Equation 5 plotted for various applied Rabi frequencies and passage times t = 2∆ α .The horizontal axis is a log-scale.(b) The measurement time and fidelity for different prime-dimensional qudits.Fluorescence time is included in the measurement duration, and we assume that the amount of adiabatic passages needed is 2d − 3 (the maximum amount of transfers needed for an arbitrary measurement).

FIG. 5 .
FIG.5.Simulation histograms of gate errors for all Pauli gates (X, Y, Z), the Quantum Fourier Transform (QFT), and π/8 gates (T)[41] for (a) 3-and (b) 5-level qudits.Each gate has a sample size of 500 where the initial qudit state is a randomized superposition of the encoded states.A magnetic field offset is set at the standard deviation of ∆B(t) 2 ≈ 2.7 pT (see Appendix A 3) and off-resonant coupling to other states are incorporated into the simulations.The Rabi frequency is set at 10 kHz for the data sets in this figure.

1 √ d p l=0 e−i π 8 0 0 e i π 8 .
first set of useful gates are the generalized Pauli gates.The simplest way to generalize the Pauli gates to d-dimensions is by the following prescription: X|j = |j + 1 mod d , Ẑ|j = ω j |j , and Ŷ |j = i X Ẑ, where ω = e 2πi/d [41].The next set of gates are the Quantum Fourier Transforms(QFT), defined as QF T |j = 2πijl/d |l .The final set of gates we present are called π/8(pi-over-eight) gates; in qubit form, they look like T = e

3 .
Single Qudit Gate Error Simulations Magnetic field noise can be modeled by a perturbative Hamiltonian Ĥnoise = Fe me=−Fe g F µ B m e ∆B|F e , m e F e , m e | − Fg mg=−Fg

∞ n=0 PĤ
n | ψ 0 |e i(θn−θ ideal )( N n=1 Ŝx,n) 2 |ψ 0 | 2 .(B19)5.Off-Resonant Error for 5-level QuditEntangling GateWith the encoding scheme as shown in Figure8for the 5-level qudits, we apply laser perturbations with frequencies as shown in the Figure to implement the entangling gate.However, there are some (unwanted) frequencies in each transition that are allowed by selection rules.For example, the required right and left-circularly polarized light acting on state |3 for the entangling gate acts on state |1 too, but at unwanted frequencies for |1 state.For the transitions |0 → |1 and |1 → |2 , two additional blue-detuned off-resonant frequencies are introduced to each of the transition, whereas two additional reddetuned off-resonant frequencies are introduced to each of the transitions |2 → |3 and |3 → |4 .From Equation21, the off resonant frequencies modify the ideal Hamiltonian in Equation12to

TABLE II .
Error budget for single qudit QFT gates with a 10 kHz Rabi frequency, and a magnetic field noise of 2.7 pT. Off-resonant coupling simulations were run 500 times each and magnetic field noise simulations were run 300 times each, varying the initial state randomly; the average error from these are shown.Scattering comes from a conservative upper bound using Equation 8, 9, and 10. *Only present for microwave gates.**Only present for Raman gates.

TABLE III .
Error estimate from error sources for the qudit entangling gate.Each error estimate except for photon scattering is obtained by the increase in fidelity when the error source is removed from the simulation.*Error for photon scattering listed here is 2Psc, where Psc is the photon scattering probability as defined in text.**The error estimates for d = 5 listed here are obtained for the case without the large error from offresonant frequencies (see text).
|F g , m g F e , m e | +e −iφ Ωmg−me |F e , m e F g , m g | , g , m g |F e , m e ; 1, q . F

TABLE V .
Unitary decomposition for various three dimensional unitary gates of interest.

TABLE VI .
Unitary decomposition for various five dimensional unitary gates of interest.