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Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions

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

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Pith's one-line read This paper shows that the five-qudit perfect code, when protected by a flag qudit and decoded with belief matching, sustains circuit-level error thresholds around $10^{-4}$ for qudits of dimension 2, 3, and 5, placing qutrits and ququints…

desk verdict Useful qudit decoder engineering, but the headline thresholds are fixed points of a power-law fit, not measured concatenated thresholds. read the letter →

arxiv 2502.05992 v2 pith:ABAGBW3Q submitted 2025-02-09 quant-ph

James Keppens , Quinten Eggerickx , Vukan Levajac , George Simion , Bart Sorée This is my paper · ORCID
classification quant-ph MSC 81P7381P68 PACS 03.67.Pp
keywords quditquantumerrorcorrectionperfectcodeflagcircuit-levelnoisethresholdbeliefmatchingminimum-weight
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

The paper asks whether qudits—quantum systems with more than two levels—can be used for error correction without paying a large noise penalty. It constructs and simulates the smallest perfect quantum error correction code, which uses five data qudits, for prime dimensions $q=2$, $3$, and $5$, under depolarizing and circuit-level noise. After adding a flag qudit that catches hook errors propagating from noisy ancillas, the simulated logical error rates yield comparable thresholds of $4.95\times10^{-4}$ (qubit), $3.24\times10^{-4}$ (qutrit), and $2.32\times10^{-4}$ (ququint). The paper thus argues that higher-dimensional qudits remain viable for fault-tolerant computation despite a noise model whose Pauli error set grows like $q^2$.

What carries the argument

The load-bearing object is the five-qudit perfect code, the smallest stabilizer code that corrects an arbitrary single qudit error, generalized to every prime dimension $q$. Three mechanisms carry the argument: a general encoding circuit derived from the code's parity-check matrix; detector matching graphs in which each ancilla is replaced by $q-1$ nodes, one per nonzero eigenvalue, so that minimum-weight perfect matching and belief matching decoders can process qudit syndromes; and a flag qudit coupled to each ancilla that flags hook errors, with corrections stored in a lookup table. The threshold estimate itself rests on the power-law model $P_L(p_s)=a p_s^b$ for the circuit-level logical error rate, recursively applied to approximate concatenation levels $l=2$ and $l=3$ and intersected to read off the threshold.

What would settle it

Simulate the two-level concatenated 5-qudit code under circuit-level noise for $q=3$ (and ideally $q=5$) at physical error rates around the reported thresholds, then compare the observed logical error rate with the prediction obtained by composing the fitted power law; a significant mismatch would falsify the threshold claim.

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Extended reading notes

Core claim

The central claim is that the circuit-level error threshold of the 5-qudit perfect code does not degrade sharply with dimension once hook errors are suppressed. With a flag qudit and the adapted Belief Matching decoder, the estimated thresholds are $4.95\times10^{-4}$ for $q=2$, $3.24\times10^{-4}$ for $q=3$, and $2.32\times10^{-4}$ for $q=5$. Hook errors—correlated errors spread from a noisy ancilla back onto data qudits—were the main obstacle; the flag qudit detects their occurrence and a brute-force lookup table supplies the correction. Without the flag qudit the thresholds are much worse, particularly for $q=5$ where the estimate drops to $4.36\times10^{-11}$. The threshold estimates are obtained by fitting a power law $P_L(p_s)=a p_s^b$ to one-level circuit-level simulations and composing it to represent concatenation levels two and three, rather than by directly simulating concatenated codes.

Load-bearing premise

The headline thresholds rest on the assumption that the logical error probability of one concatenation level can be modeled by composing the same fitted power law $P_L(p_s)=a p_s^b$ with itself, rather than by actually simulating concatenated codes; if first-level logical errors are correlated or the power law fails outside the fitted range, the threshold estimates are not established.

Editorial extensions

If this is right

  • A physical implementation of the 5-qudit code with $q=3$ or $q=5$ would need a per-step error probability near $3\times10^{-4}$ or $2\times10^{-4}$ to sit below threshold, a target within roughly a factor of two of the qubit requirement.
  • Two-qudit gates acting on qutrits or ququints can be noisier per step than their qubit counterparts by up to about a factor of two and still yield comparable logical error rates for this code.
  • Without a flag qudit, the $q=5$ code is effectively unusable (threshold near $10^{-11}$), so hook-error correction is not an optional optimization for higher-dimensional qudit codes.
  • The decoder construction using $q-1$ nodes per ancilla applies to any prime-dimensional stabilizer code, giving a general route to simulate and decode larger qudit codes with existing matching-based decoders.
  • Smaller threshold gaps between dimensions imply that the overhead advantage promised by qudits is not erased by circuit-level noise, so comparisons at fixed logical error rate and fixed encoded information become the relevant next benchmark.

Reading between the lines

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

  • Editorial inference: if the flag-qudit thresholds survive direct concatenation checks, qudit codes could encode more logical information per physical system without a proportional drop in threshold; a concrete test is to compare logical error rate per encoded qubit-equivalent for the $q=5$ code against the $q=2$ code at the same physical error rate.
  • Editorial inference: the disconnected structure of the $q=5$ matching graph suggests the cheaper minimum-weight perfect matching decoder should become more competitive at higher dimensions, which could be checked by simulating MWPM with the flag qudit at $q=5$ and $q=7$.
  • Editorial inference: the level-by-level concatenation extrapolation can be validated at modest cost for $q=3$ by simulating the two-level concatenated code at physical error rates near $3\times10^{-4}$; agreement would confirm the power-law recursion, while disagreement would indicate correlated logical errors.
  • Editorial inference: the flag qudit is assumed to be measured and reinitialized fast enough not to add idle time on data qudits; hardware with slow mid-circuit measurement would need the threshold re-derived with a finite flag-reset duration included.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper studies the smallest perfect quantum error-correcting code (five qudits) for q = 2, 3, and 5 under both standard depolarizing and circuit-level noise. The authors construct encoding and syndrome-extraction circuits valid for any prime qudit dimension, adapt the MWPM and belief-matching decoders to higher-dimensional codes, and introduce a flag qudit to handle hook errors. Their main quantitative claim is that, after the flag qudit is added, the error thresholds for q = 3 and q = 5 are comparable to the qubit threshold, on the order of 10^-4, with Table I reporting 4.95e-4, 3.24e-4, and 2.32e-4. These thresholds are not obtained from direct simulation of concatenated codes; instead, the paper fits the level-1 logical error probability to the power law P_L(p_s) = a p_s^b (Eq. 16) and recursively composes this fitted curve to represent higher concatenation levels, identifying the threshold as the intersection point of the resulting curves.

Significance. If the threshold comparison were firmly established, the paper would provide concrete evidence that qudit codes can achieve circuit-level thresholds comparable to qubits despite a noise model whose per-qudit error probability grows with dimension, which is of real practical interest for qudit-based platforms. The paper's strengths include a publicly available simulation extension on GitHub, a systematic generalization of matching-graph construction to prime qudit dimensions, and a careful level-1 circuit-level simulation with flag-qudit handling. The clean observation that the BM decoder outperforms MWPM on hyperedge errors at higher dimensions is also a useful contribution. The headline threshold claim, however, rests on a model-based extrapolation and is not directly simulated; this is the main weakness and the reason the paper requires revision.

major comments (3)
  1. [Sec. 5.4, Eq. (16), Table I] The reported thresholds are not measured concatenated thresholds. The text fits P_L(p_s) = a p_s^b to level-1 data and then defines higher concatenation levels by composing this same function: P_L^{(l+1)} = P_L ∘ P_L^{(l)}. For b > 1, all such curves intersect at the fixed point p* = a^{-1/(b-1)}, which is exactly the threshold listed in Table I. Thus the threshold is mathematically forced by the two fit parameters a and b; the intersection of the three curves carries no independent information. The paper provides no level-2 or level-3 simulation and no test of the assumption that the level-1 logical error channel behaves like an independent per-step depolarizing channel at the next concatenation level. Because the abstract's central claim of 'comparable error thresholds of the order of 10^-4' depends on this extrapolation, the claim is not empirically established. I recommend either adding a direct simulation of at least one level of concatenation (even for a few physical error rates and one dimension, e.g., qubit with flag) to validate the recursion, or substantially reframing the result as an extrapolated estimate whose systematic uncertainty is discussed explicitly.
  2. [Sec. 5.4, Table I, q=5 no-flag row] The no-flag q=5 threshold of 4.36e-11 is an artifact of the fitted exponent b = 1.149 being close to 1. Since p* = a^{-1/(b-1)}, a small change in b produces an enormous change in the fixed point; the reported uncertainty of ±2.9e-11 from Monte Carlo propagation of the fit parameters reflects only the covariance of the two-parameter fit and not the structural sensitivity of the fixed-point formula to the chosen functional form. This extreme fragility is itself evidence that composing Eq. (16) is unreliable for threshold estimation in this regime. At minimum, the authors should check whether the power-law fit is valid over the range of p_s values that contribute to the fixed point, or they should omit this row from the headline comparison.
  3. [Abstract and Sec. 6] The abstract states that 'comparable error thresholds of the order of 10^-4 are obtained', and the conclusion repeats that 'we found that the thresholds for qutrits and ququints were comparable to those of qubits'. Given that the thresholds are extrapolations from a single-level fit rather than direct simulation results, this wording overstates the status of the numbers. The manuscript should either add validation or change the phrasing to something like 'estimated thresholds under the level-by-level power-law assumption' in both the abstract and the conclusion.
minor comments (4)
  1. [Sec. 5.2 and Fig. 7 caption] The text in Sec. 5.2 says the error bars represent 90% confidence intervals, while the Figure 7 caption says they correspond to 99% confidence intervals. Please make these consistent.
  2. [Sec. 5.3 and Fig. 9 caption] The text in Sec. 5.3 says 'Error bars represent 99% confidence intervals', but the Figure 9 caption says 'Error bars indicate 90% confidence intervals'. One of these is a typo; please correct it.
  3. [Table I vs. Fig. 11] Table I lists the qutrit flag threshold as 3.24e-4, while Figure 11 (right panel) labels the threshold star as 3.29e-4. The discrepancy should be resolved.
  4. [Sec. 5.3, sample-count description] The sentence describing the parameter A ('A is assigned a value of 20 for qubits, 5 for qutrits, and 2 for all data points corresponding to ququints with a flag qudit, except for the left-most point') is difficult to parse. Please clarify which data points use which number of samples and how A was chosen to balance statistics and runtime.

Circularity Check

1 steps flagged · score 6.0 of 10

The reported thresholds are fixed points of the fitted level-1 power law, not independently simulated concatenated thresholds.

  1. fitted input called prediction [Section 5.4, Eq. (16), Table I, Figures 10-12]
    "The curves for the subsequent concatenation levels on Figures 10, 11 and 12 are then obtained by plugging in the obtained logical probability as a per-step probability for the next concatenated level: P (l+1) L =P L(ps)◦P l L. By extrapolating these curves and determining the intersection point, we can get a good estimation of the threshold of these codes."

    Under Eq. (16), the level-l curve is the l-fold iterate P_L^{(l)}(p)=a^{1+b+...+b^{l-1}} p^{b^l}. For b>1, the l=1 and l=2 curves intersect at p_* = a^{-1/(b-1)}, the same fixed point of the single-level fit. Table I's 'Threshold' column is therefore not a measured threshold of the concatenated [[5^l,1,3^l]] code; it is this analytic function of the fitted parameters (a,b). The recursive 'concatenation' is a composition of the same fitted curve, with no l=2 or l=3 simulation, so the claimed 10^-4 values are forced by the fit and by the assumed power-law form rather than by an independent simulation of larger codes.

full rationale

The only load-bearing circularity is in the threshold estimation. The raw circuit-level simulations (Figures 7 and 9) and decoder comparisons are self-contained; the code construction, syndrome circuits, and flag-qudit modification are not circular and do not rely on self-citations. The headline claim, however, comes from Section 5.4, where the higher-concatenation curves in Figures 10-12 are generated by recursively composing the same fitted power law P_L(p_s)=a p_s^b. Consequently the 'intersection point' that defines each threshold is the analytic fixed point p_* = a^{-1/(b-1)}, a direct function of the fitted parameters in Table I. The paper explicitly declines to re-run simulations at concatenation levels 2 or 3, so the l=2 and l=3 curves carry no new information; the threshold is a property of the fit and the assumed level-by-level ansatz taken from ref. [44]. Under the rubric this is a fitted input called a prediction rather than a self-citation chain. It is a partial circularity: the central quantitative claim (order-10^-4 thresholds, comparable across dimensions) reduces by construction to the fitted a and b, while the supporting circuit simulations remain independent evidence for the error-suppression effect of the flag qudit.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The quantitative threshold claim depends on the two fitted power-law parameters a and b. The main modeling assumptions are the depolarizing circuit-level noise model, the flag-qudit timing assumption, the error-free final measurement, and the level-by-level concatenation approximation. No new physical entities are introduced; the flag qudit is an established fault-tolerance technique from [42].

free parameters (2)
  • Power-law prefactor a (Eq. 16) = q=2: 36.7 (no flag), 766 (flag); q=3: 58.7 (no flag), 1116 (flag); q=5: 35.3 (no flag), 792 (flag)
    Fitted to the simulated 3-cycle logical error rates in Sec 5.3. The threshold is the fixed point p* = (1/a)^(1/(b-1)), so every tabulated threshold depends directly on this fitted value.
  • Power-law exponent b (Eq. 16) = q=2: 1.264 (no flag), 1.873 (flag); q=3: 1.288 (no flag), 1.870 (flag); q=5: 1.149 (no flag), 1.798 (flag)
    Fitted rather than fixed to the ideal value 2. The q=5 no-flag value 1.149 is close to 1, making the extrapolated threshold 4.36e-11 extremely sensitive to tiny changes in b.
assumptions (6)
  • standard math Prime-dimension qudit stabilizer codes have n-k generators and correctable error set P_q = {omega^a X^r Z^s}.
    Used in Sec 2.2 to define the 5-qudit code and its syndrome structure; follows from Gottesman [28,29].
  • standard math The five-qudit code with stabilizers generated by I Z X X^dagger Z^dagger is a distance-3 perfect code that corrects all single-qudit errors.
    Basis for the circuits in Sec 3; from quantum Hamming and Singleton bounds [23,29,31].
  • domain assumption Circuit-level noise is well modeled by single- and two-qudit depolarizing channels at each time step, plus ancilla bit-flip before readout.
    Sec 5.1 and 5.3. Real qudit devices have biased errors, leakage, and crosstalk; all threshold numbers are relative to this model.
  • domain assumption The flag qudit can be measured and reinitialized fast enough that no additional idle periods are introduced on data qudits.
    Sec 5.3, following the flag-QEC convention [42]. If false, the flag circuit adds noise that the simulations do not capture.
  • domain assumption The logical error rate of a concatenation level can be treated as an independent physical error rate for the next level, and the level-1 power-law fit P_L=a p_s^b can be composed to predict higher levels.
    Sec 5.4, Eq (16), using the level-by-level method of [44]. This is the load-bearing assumption behind the headline thresholds; no l=2 or l=3 simulation validates it in this paper.
  • domain assumption Ancilla readout in the final syndrome extraction cycle is error-free.
    Sec 5.1; a standard idealization in threshold simulations, but it makes the last cycle less noisy than earlier cycles.

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Cite this review

Pith. "Pith review of Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions." pith.science (2026). https://pith.science/paper/ABAGBW3Q

@misc{pith2026250205992,
  author       = {Pith},
  title        = {Pith review of: Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ABAGBW3Q}},
  note         = {Machine review of arXiv:2502.05992}
}
abstract

Qudits can be described by a state vector in a $q$-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, we investigate the viability of qudits in error correction codes by creating and simulating the quantum circuitry for the smallest qudit error correction code with a multidimensional circuit-level noise model and specifically adapted decoders. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of $10^{-4}$ are obtained for qudits of dimensions $2$, $3$ and $5$.

Figures

Figures reproduced from arXiv: 2502.05992 by the authors.

Figure 1
Figure 1. FIG. 1. Encoding circuit of the 5 qudit code, valid for all prime dimensions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quantum circuit representing a single cycle of error syndromes of the 5 qudit code, valid for all prime dimensions [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Quantum circuit representing the encoded implementation of the transversal 3-qudit gate ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Matching graph of a single cycle 5 qubit code. All [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Matching graph of a single cycle 5 ququint ( [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Logical error rates obtained in simulations with standard depolarization noise of the 3-cycle 5-qudit code using qudits [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Example of a syndrome extraction quantum circuit [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Simulated logical error probabilities for the 5-qudit code under circuit-level noise, using qudits of dimensions 2, 3, and [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Simulated logical error probabilities for the 5-qubit code decoded by the BM decoder with fitting curves for three [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Simulated logical error probabilities for the 5-qutrit code decoded by the BM decoder with fitting curves for three [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Simulated logical error probabilities for the 5-ququint code decoded by the BM decoder with fitting curves for three [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Labeled nodes during the creation of the matching [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Labeled nodes, with dangling edges representing [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

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Forward citations

Cited by 1 Pith paper

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Reference graph

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