Quantum logic operations and algorithms in a single 25-level atomic qudit

Crystal Senko; Gaurav A. Tathed; Nicholas C.F. Zutt; Pei Jiang Low

REVIEW 3 major objections 4 minor 3 cited by

Quantum logic operations and algorithms in a single 25-level atomic qudit

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single trapped barium ion stores 25 quantum levels and runs four-virtual-qubit gates.

desk verdict A genuine d=25 trapped-ion qudit demonstration with solid hardware results, but the 'no-free-parameter' error model has a real inconsistency and should be fixed before publication. read the letter →

arxiv 2507.15799 v2 pith:LMZBN2BZ submitted 2025-07-21 quant-ph

Pei Jiang Low , Nicholas C.F. Zutt , Gaurav A. Tathed , Crystal Senko This is my paper · ORCID

classification quant-ph

keywords trappedionquditbarium-13725-levelBernstein-VaziranialgorithmToffoligatenarrowbandopticalpumpingMonteCarloerrormodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a single trapped $^{137}$Ba$^+$ ion can act as a 25-level qudit—one ground-state level plus 24 metastable $D_{5/2}$ hyperfine-Zeeman levels—and that this register is good enough for real algorithms. The experimental evidence is a heralded state-preparation-and-measurement fidelity of $99.51(5)\%$ averaged over all 25 levels, coherent superpositions of up to $d=24$ states, a Bernstein-Vazirani key-finding run on two and three virtual qubits, and a four-qubit Toffoli gate on one ion. The paper also constructs a Monte Carlo error model with no fitted parameters, using independently measured noise sources, and shows it reproduces the measured contrasts and algorithm errors. It then uses the model to project that engineering improvements already demonstrated in other trapped-ion systems would push the dominant errors to the $10^{-3}$ level or below for $d \le 16$. A sympathetic reading is that this establishes large-dimensional qudit encoding as a viable scaling route for trapped-ion quantum computing.

What carries the argument

The load-bearing object is the hyperfine-Zeeman level structure of $^{137}$Ba$^+$: one $S_{1/2}$, $F=2$ level acts as the central node, and 24 $D_{5/2}$ levels provide the rest of the 25-state qudit. All coherent operations are electric-quadrupole transitions at 1762 nm, which connect the central state to every $D_{5/2}$ state; arbitrary unitaries are compiled into sequences of Givens rotations through this central node, with virtual-$Z$ phase shifts inserted for free. State preparation uses a narrow-band optical pumping extension that flushes unwanted $S_{1/2}$ levels and repeatedly shelves and repumps until the target sublevel is populated, while readout de-shelves each $D_{5/2}$ state in turn and checks for fluorescence, discarding shots whose herald fails. The error model is a shot-to-shot Monte Carlo simulation that samples independently measured magnetic-field noise, laser frequency noise, frequency miscalibration, and pulse-time errors, then propagates them through the actual pulse Hamiltonians. This combination—full connectivity through one central state, heralded SPAM, and a physically grounded noise simulation—is what lets the paper turn a 25-level atom into a working quantum processor.

What would settle it

Take the same $d=16$ qudit Ramsey sequence and the 3-virtual-qubit Bernstein-Vazirani circuit, and re-measure the magnetic-field noise with the D5/2 Ramsey protocol used in the supplement immediately before each run; then feed only those measured values into the Monte Carlo simulation. If the model's predicted success probabilities fall outside the quoted Wilson intervals, or if the simulation reproduces the data only when the field-noise width is changed from the independently measured $\sim 14.4\,\mu\mathrm{G}$ to the $\sim 24\,\mu\mathrm{G}$ used in the paper's Table I, then the no-free-parameter claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the 25 internal states of a single $^{137}$Ba$^+$ ion can be prepared, read out, and coherently manipulated with high fidelity, making it the largest digital trapped-ion qudit demonstrated to date. The authors encode 25 levels using the $S_{1/2}$ and $D_{5/2}$ manifolds, initialise any state on demand with narrowband optical pumping, measure all 25 levels by sequential shelving with fluorescence checks, and use 1762 nm electric-quadrupole transitions for coherent control. They report $99.51(5)\%$ heralded SPAM fidelity, Ramsey-type contrast for superpositions of up to 24 states, Bernstein-Vazirani success probabilities of $97.9(2)\%$ (2 virtual qubits) and $83.8(8)\%$ (3 virtual qubits), and a four-virtual-qubit Toffoli truth table with $99.5(2)\%$ average fidelity. A no-free-parameter Monte Carlo model based on independently measured noise sources reproduces the multi-level coherence and algorithm errors, and projects errors near $10^{-3}$ for $d \le 16$ with known engineering upgrades.

Load-bearing premise

The central prediction depends on the assumption that the independently measured noise parameters inserted into the Monte Carlo model completely describe the dephasing that actually occurs during the multi-state pulses, so the agreement with the data is not achieved by tuning those inputs.

Editorial extensions

If this is right

A single $^{137}$Ba$^+$ ion can host up to four virtual qubits, so computational space per ion grows as $2^4 = 16$ without additional physical qubits.
The 4-qubit Toffoli gate, implemented as one transition swap between two basis states, reaches $99.5(2)\%$ truth-table fidelity, showing that complex multi-qubit gates can be cheaper in a qudit encoding than on multiple physical qubits.
SPAM at $99.51(5)\%$ with heralding positions the 25-level qudit on par with qubit SPAM on other platforms, and the error model identifies the specific sources (spontaneous decay, off-resonant driving, photon-discrimination errors) that must be reduced.
The Monte Carlo model attributes the dominant algorithm errors to A/C line-induced magnetic field changes and laser frequency noise, not to the qudit encoding itself, so hardware upgrades such as magnetic shielding and fibre noise cancellation should directly improve algorithm success.
With literature-reported low-noise parameters, the model projects contrast loss below $10^{-4}$ for $d\le10$ and about $10^{-3}$ at $d=16$, giving a concrete target for next-generation trap design.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the projected $10^{-3}$ or better error rates are reached, the extra levels of a 25-state qudit become useful as an in-situ error-correction resource, because a logical qubit can be encoded in a subspace while the remaining levels detect and correct errors without a destructive measurement.
Editorial inference: the star-topology decomposition routine is a general compiler primitive, not a barium-specific trick; any qudit platform with one strongly connected central level can reuse it, and the state-selection cost function in Methods 3 could serve as a hardware-aware heuristic for choosing which physical levels to encode.
Editorial inference: the Bernstein-Vazirani results imply a specific compilation trade-off for this architecture—minimise the number of rotations through the central state rather than the abstract gate count, because sequential Givens rotations cost time and accumulate noise, while virtual-$Z$ phases are free.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Quick take: the hardware results are real and worth taking seriously, but the paper oversells its error model. The d=25 demonstration, with 99.51(5)% heralded SPAM and coherence data up to d=24, appears solid. That is the largest digital trapped-ion qudit among the cited literature, and the NBOP extension plus the star-topology Givens decomposition with pulse compression are genuinely new. The public data and code on Zenodo are a plus. The 4-virtual-qubit Toffoli truth table at 99.5(2)% inclusive of SPAM is a clean demonstration.

Now the soft spots. The claim of a "full physical error model—each error source validated in independent experiments—which matches results to within 6.8% experimental uncertainty" does not hold up as stated. Table I lists the magnetic-field noise as a Gaussian width of 24 µG, while Supp. IV A reports the measured Gaussian standard deviation as 14.4 µG. Those are not equivalent. Second, the Monte Carlo BV predictions are 98.6% and 86.9% versus the measured 97.9(2)% and 83.8(8)%, which puts the simulations outside the quoted 1-sigma intervals. If the model truly has no free parameters, it should not miss the algorithm data by 3–4 sigma. Also, "within 6.8% experimental uncertainty" is never defined. None of this invalidates the direct measurements, but it does invalidate the "no-free-parameter" predictive-error-model claim, and by extension the projected 10^-3 error rates.

One caveat: I do not think the authors necessarily fudged anything. The 14.4 µG measurement could be a different statistic or condition than the 24 µG used in Table I, but the paper does not say that, and in a no-free-parameter model the burden is on them to reconcile it. The fix is straightforward: report the exact definition of every distribution width, and either adjust the simulation inputs to the independently measured values or show that the mismatch does not affect the conclusions. The model section should be revised before publication.

Bottom line: the paper is a legitimate experimental advance, not a routine increment. Trapped-ion and qudit architecture people will get real value from it. Send it to review; the referee should focus on the error model, not the hardware.

Referee Report

3 major / 4 minor

Summary. The manuscript reports experiments on a single 137Ba+ ion encoded as a 25-level qudit in the S1/2 and D5/2 manifolds. It demonstrates narrow-band optical pumping initialization of any S1/2,F=2 sublevel, heralded single-shot SPAM with average fidelity 99.51(5)% for d=25 (with 2.82(5)% initialization and 1.51(3)% measurement data loss separately reported), Ramsey-type mutual-coherence measurements for superpositions of up to d=24 states, and virtual-qubit implementations of 2- and 3-qubit Bernstein-Vazirani key finding plus a 4-qubit Toffoli (CCCNOT) truth table. The paper attributes the observed errors to independently characterized noise sources, packages them in a shot-to-shot Monte Carlo simulation, and uses the model to project achievable error rates of order 10^-3 or below with literature-level noise suppression.

Significance. If the direct hardware results are taken at face value, this is the largest digital trapped-ion qudit demonstration among the cited literature and a useful data point for qudit-versus-qubit tradeoffs. The paper is commendable for reporting shot counts, Wilson-interval error bars, and public data and code; the SPAM, coherence, and algorithm results are presented with enough procedural detail to be checked. The forward-looking error-model projections are a valuable addition, but their current validation is the main weakness: the independent parameter basis for the model is compromised by an inconsistency in the magnetic-field noise input, and the model visibly misses the measured Bernstein-Vazirani success probabilities. The direct experimental claims are therefore plausible, while the 'no-free-parameter' predictive claim needs revision.

major comments (3)

[Methods 5, Table I; Supp. IV A] The Monte Carlo model uses a Gaussian magnetic-field noise width of 24 µG for the input labeled 'Magnetic Field' in Table I, whereas Supp. IV A reports a measured Gaussian standard deviation of 14.4 µG for the same noise source. These values are not equivalent under the stated Gaussian profile (a Gaussian FWHM would be roughly 34 µG, and twice the standard deviation would be roughly 29 µG), so the paper contains two inconsistent values for one independently measured input. Because the abstract and Methods 5 claim that each error source is validated in independent experiments, this discrepancy must be resolved before the model can be called a no-free-parameter, independently validated description.
[Fig. 4(c)-(d); Methods 5] The simulated Bernstein-Vazirani success probabilities of 98.6% for n=2 and 86.9% for n=3 lie outside the quoted 1σ Wilson intervals of the measured values of 97.9(2)% and 83.8(8)% by roughly 3–4σ. The abstract's claim that the model 'matches results to within 6.8% experimental uncertainty' is therefore not supported by the algorithm data unless that uncertainty metric is defined and shown to be distinct from the per-point 1σ intervals. Please provide the definition, state whether any parameter or hidden loss term had to be adjusted to bring the simulations into agreement, and if so, revise the forward-looking 10^-3 error-rate claims to be conditional on that adjustment.
[Abstract; Methods 5; Supp. V] The phrase 'within 6.8% experimental uncertainty' is used in the abstract but is never defined in the main text or Methods. The reader cannot verify whether it refers to a single aggregate contrast, all data points, or a particular normalization, and the BV disagreement described above shows that the relevant metric matters. The projected 10^-3 to 10^-4 error rates in Supp. V inherit this issue because they use the same Monte Carlo model, so the manuscript should state explicitly which model inputs are measured in this work, which are literature values, and which, if any, are adjusted to match the experimental datasets.

minor comments (4)

[Supp. VI, Fig. 14 caption] The caption of Fig. 14 says 'Three virtual qubit Hadamard (H⊗2) gate'; this should read H⊗3 to match the three-virtual-qubit circuit it describes.
[Methods 5, Fig. 3(e)] The text specifies 1024 Monte Carlo shots for the simulations in Fig. 3d, but the main contrast curve in Fig. 3e and the BV simulations would benefit from an explicit statement of the number of shots and whether error bars are included for the simulated curve.
[Section II A, Fig. 2(c)] The statement 'Off-diagonal populations above 0.1% are indicated with text' is difficult to verify in the rendered figure; please ensure that all such annotations are legible in the final version or provide the population matrix in a table.
[Supp. IV A] The text says the Gaussian magnetic-field noise profile was 'also verified with an independent direct measurement of the magnetic field,' but the magnetometer data shown in Supp. Fig. 7 were taken outside the vacuum chamber; please clarify whether the in-vacuum noise was measured directly or only inferred from Ramsey decay.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the qudit demonstrations and algorithm results are measured, the Monte Carlo error model uses independently characterized noise parameters, and the Table I versus Supp. IV A magnetic-field discrepancy is a soundness concern rather than a circular one.

full rationale

The paper's central results—25-level SPAM fidelity, d <= 24 Ramsey contrasts, Bernstein-Vazirani success probabilities, and the CCCNOT truth table—are directly measured experimental observables, not quantities derived from their own inputs. The Monte Carlo model in Methods 5 takes noise profiles (magnetic-field noise, laser frequency noise, frequency and pulse-time miscalibration, A/C line signal) from independent Ramsey, Rabi, and magnetometer characterizations described in Supp. I and IV, then propagates them through the known Hamiltonians; the contrasts and BV rates are outputs, not fit parameters. The projected 10^-3-level error rates use literature noise values from Refs. [53-55] with stated calibration assumptions, so they are extrapolations rather than restatements of the inputs. The self-citations [41,42] describe prior apparatus and level-structure control and are not invoked to prove the present results. The discrepancy between Table I's 24 µG magnetic-field width and Supp. IV A's measured 14.4 µG standard deviation, and the BV simulations falling outside the measured 1-sigma intervals, undermine the 'no free parameters / within 6.8%' claim, but these are correctness and parameter-selection concerns: nothing in the text reduces a prediction to a fit of the same dataset. Hence no circular step is identifiable.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central hardware claim rests on standard atomic physics and measured calibrations. The error model is not circular in structure because its noise inputs are measured independently, but its completeness is uncertain due to the unresolved magnetic-field-noise discrepancy and the mismatch between simulated and measured BV success rates. No invented entities are introduced.

free parameters (7)

Magnetic field magnitude B = 4.209 G
Fit to two reference transition frequencies to calibrate the hyperfine/Zeeman Hamiltonian (Methods 1, Supp. I B). This is a calibration input, not a parameter fit to the target results.
Magnetic field noise sigma_B = 14.4 uG in Supp. IV A, 24 uG in Table I
Used as the Gaussian noise width in the Monte Carlo model. The two values are inconsistent in the preprint, which is a flagged issue because the model is claimed to have no free parameters.
Laser frequency noise width = Voigt FWHM 287 Hz (from Gaussian 81.6 Hz and Lorentzian 77.1 Hz components)
Measured from a magnetically insensitive Ramsey transition (Supp. IV B). Input to the error model.
Frequency miscalibration sigma = Gaussian FWHM 296 Hz (sigma = 126.3 Hz)
Measured from deviations of calibrated transition frequencies from fitted lines (Supp. I B). Input to the Monte Carlo model.
Pulse-time miscalibration and drift = 1.77% and 2.61%
Measured from transition-strength calibration errors and long-term drift (Supp. I C). Input to the error model.
A/C line magnetic field amplitudes = 128 uG at 60 Hz, 40 uG at 180 Hz
Measured line-synchronized field variation (Methods 5, Supp. IV C). Used in the Monte Carlo model.
Projected noise parameters = 0.04 uG field noise, 70 uG line amplitude, 0.5 Hz laser linewidth, 10 Hz calibration error, 0.1% pulse-angle error
Literature and assumed engineering values used for the projected 10^-3 error estimates. These are projections, not measurements in this work.

assumptions (7)

domain assumption The hyperfine plus Zeeman Hamiltonian for 137Ba+ energy levels (Supp. I A, Eq. 1), including the octupole term, is correct with higher-order terms neglected.
All transition frequencies and magnetic-field sensitivities are computed from this Hamiltonian. Its validity is standard atomic physics but is an unproved modeling input.
domain assumption Quadrupole transition strengths are computed using the method of Ref. [1] and calibrated against five reference Rabi frequencies (Supp. I C).
The control pulses assume these strengths. The measured mean absolute error of 1.77% partially validates the assumption.
domain assumption The noise distributions (Gaussian field, Voigt laser, Gaussian calibration and pulse errors) fully describe shot-to-shot variability, with parameters constant within a shot (Methods 5, Table I).
The Monte Carlo model's predictive claim depends on this completeness assumption. The 24 vs 14.4 uG discrepancy shows it is not fully pinned down.
domain assumption The A/C line field variation is modelled as two harmonics (Eq. A.15) with measured amplitudes and phases.
Used to compute detunings and phases in the error model. Validated by fits to line-phase scans.
standard math Ideal Givens rotations with calibrated Rabi frequencies implement the unitaries in the qudit Ramsey and Bernstein-Vazirani decompositions, with off-resonant leakage neglected in the derived ideal expressions.
The analytic contrast expressions in Methods 4 assume ideal rotations. The Monte Carlo model adds noise but still assumes effective two-level Hamiltonians.
standard math The star-topology unitary decomposition (Methods 6) is exact given arbitrary Givens rotations, and the L-BFGS-B compression reaches the stated tolerance.
This is a mathematical algorithm; no physical evidence is required beyond the matrix identities shown.
domain assumption Literature noise values (Ruster et al., Alnis et al.) are transferable to 137Ba+ and the assumed calibration improvements are achievable (Supp. V, Table I).
The projections to 10^-4 or 10^-3 error depend on this transferability. It is not demonstrated in this work.

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Cite this review

Pith. "Pith review of Quantum logic operations and algorithms in a single 25-level atomic qudit." pith.science (2026). https://pith.science/paper/LMZBN2BZ

@misc{pith2026250715799,
  author       = {Pith},
  title        = {Pith review of: Quantum logic operations and algorithms in a single 25-level atomic qudit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMZBN2BZ}},
  note         = {Machine review of arXiv:2507.15799}
}

abstract

Scaling quantum computers remains a substantial scientific and technological challenge. Leveraging the full range of intrinsic degrees of freedom in quantum systems offers a promising route towards enhanced algorithmic performance and hardware efficiency. We experimentally study the use of $^{137}$Ba$^+$ ions for quantum information processing, achieving high-fidelity state preparation and readout of up to 25 internal levels, thus forming a 25-dimensional qudit. By probing superpositions of up to 24 states, we investigate how errors scale with qudit dimension $d$ and identify the primary error sources affecting quantum coherence. Additionally, we demonstrate high-dimensional qudit operations by implementing a 3-qubit Bernstein-Vazirani algorithm and a 4-qubit Toffoli gate with a single ion. Our findings suggest that quantum computing architectures based on large-dimensional qudits hold significant promise.

Figures

Figures reproduced from arXiv: 2507.15799 by the authors.

**Figure 1.** Narrow-band optical pumping (NBOP) initialisation. (a) The ion starts out in one of the 8 states in S1/2. The NBOP sequence, consisting of 1. F = 1 state flushing, 2. 1762 nm shelving pulses, and 3. repumping with 614 nm light, is applied repeatedly, increasing the probability that the initial state falls into the one S1/2, F = 2 state not being actively pumped out (in this case |m = 1⟩). (b) Pulse sequence for NBOP… view at source ↗

**Figure 2.** 25-level SPAM infidelities and data rates. Data loss rates per qudit basis state |i⟩ due to initialisation errors (a) and measurement errors (b). Open circles are data, bars are theoretical predictions based on transition strengths and magnetic field sensitivities as well as S1/2 initialisation error rates. (c) SPAM measurement results showing an average fidelity of 99.51(5) % over all states. Off-diagonal populatio… view at source ↗

**Figure 3.** Multi-level coherence probes and dimensional contrast scaling. (a) Pulse sequence circuit diagram - each rotation addresses a transition between state |0⟩ and |j⟩, written as R(θ, ϕ) for pulse angle θ and phase ϕ. θj = 2 arcsin 1/ √ d − j + 1 , and ϕj = ϕ · j for ϕ ranging from 0 to 2π over the course of a full phase scan. Multi-level superpositions and coherence probing for (b) d = 3, (c) d = 5, and (d) d = 9, as … view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: Bernstein-Vazirani circuit diagram, state selection, 2 and 3 virtual qubit results, and CCCNOT truth table. (a) Bernstein-Vazirani algorithm circuit diagram. (b) State choices for 2 virtual qubit (blue) and 3 virtual qubit (orange) implementations of the key finding al…

**Figure 1.** Figure 1: Quadrupole transition frequencies and magnetic field sensitivities. Solving the full Hamiltonian in Equation 1 allows for the prediction of transition frequencies and the calculation of transition sensitivities to changes in the magnetic field, shown here for a magneti…

**Figure 2.** Figure 2: Frequency calibrations overview. (a) Diagram explaining the Ramsey calibration scheme using just two measurements at a revival pulse phase of π/2 and 3π/2, with pulse sequence inset. (b) Linear relation on the same transition as in (a). Each point plotted here represen…

**Figure 3.** Figure 3: Calculated relative transition strengths, comparison with measured transitions. Vertical lines in the top panel correspond to transitions (with frequencies on the x-axis calculated as in Section I B) with heights denoting the relative strengths found from theory using …

**Figure 4.** Figure 4: NBOP transition choices and repumping pathways probability map. (a) NBOP shelving transitions chosen to initialise S1/2 , F = 2 states with high efficiency. (b) Re-pumping pathways probabilities found by calculating the transition matrix elements for D5/2 → P3/2 states…

**Figure 5.** Figure 5: Contributors to 25-level SPAM error. (a) Calculated SPAM fidelities as a function of x− and y−secular frequencies of the trap. Calculations made with η = 0.014, n = 140 (previously measured on this system [7]), axial-secular frequency of 215 kHz, and magnetic field val…

**Figure 6.** Figure 6: Magnetic field and laser noise characterisation. (a) Decays in Ramsey contrast measurements as a function of wait times in superpositions for pairs of states in D5/2 . (b) Two-level Ramsey contrast decay times plotted against the relative magnetic field sensitivities …

**Figure 7.** Figure 7: Magnetic field noise, outside vacuum chamber. Magnetic field measurements as taken by a magnetometer outside the vacuum chamber, with the y-axis aligned with the quantisation axis of the system. We find that the noise profile in all directions is well described by a st…

**Figure 8.** Figure 8: A/C line signal measurement. Measured change in magnetic field as a function of the line A/C mains line signal phase. The two main components to this signal are at 60 Hz (amplitude 0.128 mG, phase −0.636 rad) and 180 Hz (amplitude 0.04 mG, phase −1.55 rad). Blue and or…

**Figure 9.** Figure 9: Noise levels necessary for 10−4 level errors. Turning on each noise source alone, and for each dimension from d = 2 to 18, we estimate how the noise sources must scale as a function of qudit dimension to maintain less than 10−4 contrast loss. Values are normalised to t…

**Figure 10.** Figure 10: Qudit Ramsey contrast measurements versus pulse times. Contrasts of the |0⟩ state populations as a function of qudit dimension d, along with the prediction based on known noise sources affecting the ion. The sub-plots represent the same data on a linear scale (top) an…

**Figure 11.** Figure 11: Full encoding schemes for all dimensions of qudit Ramsey measurements. State encoding schemes for all qudit Ramsey experiments, from d = 2 to 24. The colours for each encoded state indicate the transition used to reach that encoded state from S1/2 , see the legend in …

**Figure 12.** Figure 12: Two virtual qubit Hadamard (H⊗2 ) gate. All transitions involve the |00⟩ encoded state, as necessitated by the star-type topology of the state set chosen in Fig. 4b. θi = 2 arcsinp i/3 . [10] C. Huang, C. Wang, H. Zhang, H. Hu, Z. Wang, Z. Mao, S. Li, P. Hou, Y. Wu…

**Figure 13.** Figure 13: Fast superposition pulse using 7 Givens rotations. All transitions involve the |000⟩ encoded state, as necessitated by the star-type topology of the state set chosen in Fig. 4b. The total gate time for these 7 transitions is 170 µs. Here, θi = 2 arcsinp 1/(d − j + 1)…

**Figure 14.** Figure 14: Three virtual qubit Hadamard (H⊗2 ) gate. All transitions involve the |000⟩ encoded state, as necessitated by the star-type topology of the state set chosen in Fig. 4b. θ1 = 2 arcsinp 1/3 . 29 [PITH_FULL_IMAGE:figures/full_fig_p044_14.png]

**Figure 15.** Figure 15: State encoding scheme for CCCNOT gate. States chosen to encode 4 virtual qubits into the ion, using transitions that are all connected via the [PITH_FULL_IMAGE:figures/full_fig_p045_15.png]

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Experimental setup We use the same experimental setup as outlined in our previous work [ 42], with some changes to the ex- perimental parameters. We employ a two-step, isotope selective ablation loading scheme as detailed in [ 41]. In this work, the laser intensities are 13 mW...
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Narrow-band optical pumping The NBOP approach introduced in Sec. II A in- volves several steps which each have opportunity for tuning/optimisation. Firstly, the S1/2, F = 1 states ﬂushing step is sensitive to the power and pulse time of the 493 nm light used. A lower power, sh...
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State choices for encoding SP AM, qudits, and virtual qubits State preparation and measurement For the d = 25 level SPAM result, all states in D5/2 are encoded, along with a single state in S1/2, F = 2 . The |0⟩ state in S1/2 was determined to be the state with the shortest tr...
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Qudit Ramsey-type experiment - analytic expressions In intermediate magnetic ﬁeld strengths of order 1 G, in the context of maximally encoding the D5/2 states in 137Ba+, it is more practical to utilise the 1762 nm for coherent manipulation and connecting the qudit states as co...
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To ﬁnd the detuning f∆ correctly then, we ﬁt the measurement results to a unitary evolution given by |ψf ⟩ = ˆU3 ˆU2 ˆU1 |ψ0⟩ = ˆU3 ˆU2 ˆU1 |0⟩ for ˆUi = exp ( −i ˆHit )

Finding detunings from Ramsey accumulated phase It is worthwhile to note that the phase diﬀerence ∆ϕ accumulated during the (detuned) Ramsey sequence is not equal to f∆ ·τ in practice (they are only equal in the limit where the pulse times for the rotations go to zero). To ﬁnd...
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For each shelving pulse intended to move population to a desired D5/2 initial state in the SPAM measurement, calculate the probability for oﬀ-resonantly driving population to an undesired D5/2 state using the well-known Rabi fringe expression Pi(f,t ) = Ω2 i Ω2 i + (f −fi)2 si...
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preparation

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The 60 Hz signal has an amplitude of 0.128 mG and phase of −0.636 rad, and the 180 Hz component is ﬁt to an amplitude of 0.04 mG, with a phase of 19 Figure 8

Each component is plotted using dashed grey lines, with the total ﬁt in black. The 60 Hz signal has an amplitude of 0.128 mG and phase of −0.636 rad, and the 180 Hz component is ﬁt to an amplitude of 0.04 mG, with a phase of 19 Figure 8. A/C line signal measurement. Measured c...
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The current bright/dark discrimination error in our system is 5 × 10−5, which yields an overall ϵRO = 6.25 × 10−4 for d = 25

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Oﬀ-resonant driving: The error source fundamentally stems from the density of states in the D5/2 manifold, all of which are within around 150 MHz of one another. However, many other parameters also inﬂuence the degree of oﬀ-resonant driving (mostly through motional sideband tr...
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the encoded states for every qudit Ramsey measure- ment from d = 2 to 24, with colour-coded state labels that indicate which transition (i.e. from whichS1/2 state the transition is driven) is used. In Supp. ﬁgs. 12 and 14, we show the full pulse sequences used for the implemen...
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