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arxiv: 2511.12777 · v2 · pith:RGQJVT47new · submitted 2025-11-16 · 🪐 quant-ph

Sdim: A Qudit Stabilizer Simulator

Adeeb Kabir , Steven Nguyen , Sohan Ghosh , James Keppens , Tijil Kiran , Isaac H. Kim , Yipeng Huang , Bart Sor\'ee This is my paper

Pith reviewed 2026-05-17 21:33 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quditstabilizer simulatorquantum error correctionfault-tolerant quantum computingopen-source softwarehigher-dimensional quantum systems
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The pith

Sdim provides the first open-source stabilizer simulator that works for qudits in any dimension.

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

The paper presents Sdim, an open-source simulator for stabilizer circuits on qudits, which are quantum systems with more than two levels. Existing tools handle only qubits, but this one extends the stabilizer formalism to higher dimensions and all tested circuit sizes. Correctness is checked by matching outputs against full state-vector simulations, while speed is measured on circuit evaluation and sampling tasks. The work positions the tool as necessary infrastructure for studying qudit error correction at practical scales, the same role qubit simulators have played for conventional quantum error correction.

Core claim

We introduce the first open-source realization of a qudit stabilizer simulator for all dimensions. We demonstrate its correctness against existing state vector simulations and benchmark its performance in evaluating and sampling quantum circuits. This simulator is the essential computational infrastructure to explore novel qudit error correction as earlier stabilizer simulators have been for qubits.

What carries the argument

Sdim, the open-source implementation of an efficient algorithm for tracking qudit stabilizer states across arbitrary dimensions.

If this is right

  • Numerical studies of qudit error-correcting codes can now be performed at realistic scales.
  • Sampling and evaluation of qudit stabilizer circuits becomes feasible without exponential cost in state space.
  • Development of qudit-specific fault-tolerant protocols gains the same simulation support that qubits have had for years.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same code base could be extended to simulate certain non-stabilizer operations on qudits if the underlying representation is kept.
  • Direct performance comparisons between qubit and qudit stabilizer circuits on identical hardware models are now possible for the first time.

Load-bearing premise

The qudit stabilizer formalism admits an efficient simulation algorithm whose implementation matches the underlying physics for all tested dimensions and circuit sizes.

What would settle it

A systematic mismatch between Sdim outputs and independent state-vector results on any qudit circuit in dimension three or higher of moderate size would falsify the correctness claim.

Figures

Figures reproduced from arXiv: 2511.12777 by Adeeb Kabir, Bart Sor\'ee, Isaac H. Kim, James Keppens, Sohan Ghosh, Steven Nguyen, Tijil Kiran, Yipeng Huang.

Figure 1
Figure 1. Figure 1: Average time to simulate one shot of a random Clifford circuit using our stabilizer versus Google Cirq statevector simu￾lation. The space and time cost of state vector simulation grows exponentially versus the dimension and number of qudits. In con￾trast, tableau simulation does not depend on the dimension at all and is bounded by 𝑛 2 . magic state distillation and injection [22]. However, recent work sugg… view at source ↗
Figure 2
Figure 2. Figure 2: Conjugation table for Clifford gates on Pauli gates. operation completes the set of gates needed for universal QC. A common choice is the 𝑇 gate, creating the Clifford+𝑇 universal gate set. However, simulating this gate set is costly in general [9]. 3.2 Classical simulation of qudit stabilizer circuits Although Clifford operations on Pauli-stabilized states are not enough to describe universal QC, they are… view at source ↗
Figure 3
Figure 3. Figure 3: A Sdim program to run and validate the 𝑑-dimensional Deutsch-Jozsa algorithm. Stabilizer integer encoding. Recall that every Pauli 𝑃 = 𝜔 𝑐𝑋 𝑎𝑍 𝑏 . Every product Pauli 𝑃1⊗· · ·⊗𝑃𝑚 can then be written 𝜔 𝑟 (𝑋 𝑎1𝑍 𝑏1 ⊗ · · · ⊗ 𝑋 𝑎1𝑍 𝑏1 ), where 𝑟 = 𝑐1 + · · · +𝑐𝑚. We can write it out in a compact block form (sometimes known as its symplectic form [33]) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: figure 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The first three steps of the Deutsch-Jozsa algorithm written out in various representations, including the tableau. balanced means that for every 𝑏 ∈ 𝐵, there are precisely 𝑑 𝑛−1 many strings 𝑠 ∈ 𝐴 such that 𝑓 (𝑠) = 𝑏. The task is to discern whether 𝑓 is constant or balanced using queries to 𝑓 on 𝐴. The algorithm described below solves a heavily restricted version of the Deutsch-Jozsa problem where 𝑛 = 1, … view at source ↗
Figure 5
Figure 5. Figure 5: Tableau rewrite rules corresponding to the conjugation table in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Quantum circuit corresponding to the code in [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A simple circuit with two measurements. The first is random and followed by a deterministic measurement. Lines 2-8 describe the procedure for a random measure￾ment, while lines 10-11 detail a deterministic one. The proce￾dure is a straightforward application of earlier work [1, 32]. The circuit in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Cirq against Sdim on evaluating (the first shot of) Bernstein-Vazirani circuits. Each test is randomized over 100 bit￾strings with the same amount of 1s, which controls the size of the most expensive subspace to measure. Note that this graph is logscale, which further demostrates that the cost of tableau simula￾tion depends only polynomially on the system size [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A qudit Bernstein-Vazirani circuit where the secret string starts with “10”. For every nonzero character in the string, a qudit in that character’s position entangled with the bottom ancilla via the CNOT oracle. 4.3 Validation and evaluation of single-shot performance of stabilizer tableau simulation We cross-validate and compare Sdim’s performance against Google Cirq [44], a major software framework that … view at source ↗
Figure 10
Figure 10. Figure 10: Local gate test: runtime and circuit. Local gate performance evaluation. Finally, we study Sdim’s ability to evaluate high-depth local circuits followed entangling the all qudits and then measuring once. The re￾sults, shown in [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: Cirq simulation of a logical randomized benchmarking protocol with only detection (LRB-D) against RB of an unprotected qutrit. The lines are simulated success probabilities for circuits at various depths. We expect LRB graphs of Markovian noise to be exponential, so the unusual curve is either an artifact of the modified LRB or of the code. 6.0.1 Randomized benchmarking. The Randomized bench￾marking (RB) … view at source ↗
Figure 12
Figure 12. Figure 12: Time to sample the circuit from [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Logical randomized benchmarking with non-destructive stabilizer measurements [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
Figure 17
Figure 17. Figure 17: Simulation time per run of LRB-D (at a fixed gate￾independent noise parameter, over 30 random Clifford circuits and their subcircuits) on the folded detection code versus the number of samples. Even on a 5-qutrit state, tableau methods offer signifi￾cantly improved runtimes. 7 Non-prime dimensions The tableau we introduced for prime qudit stabilizer compu￾tation does not straightforwardly generalize in no… view at source ↗
read the original abstract

Quantum computers have steadily improved over the last decade, but developing fault-tolerant quantum computing (FTQC) techniques, required for useful, universal computation remains an ongoing effort. Key elements of FTQC such as error-correcting codes and decoding are supported by a rich bed of stabilizer simulation software such as Stim and CHP, which are essential for numerically characterizing these protocols at realistic scales. Recently, experimental groups have built nascent high-dimensional quantum hardware, known as qudits, which have a myriad of attractive properties for algorithms and FTQC. Despite this, there are no widely available qudit stabilizer simulators. We introduce the first open-source realization of such a simulator for all dimensions. We demonstrate its correctness against existing state vector simulations and benchmark its performance in evaluating and sampling quantum circuits. This simulator is the essential computational infrastructure to explore novel qudit error correction as earlier stabilizer simulators have been for qubits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces Sdim, the first open-source qudit stabilizer simulator that operates for arbitrary dimensions. It asserts that correctness was established via direct comparison to state-vector evolution and reports performance benchmarks on sampling and circuit-evaluation tasks, framing the tool as essential infrastructure for exploring qudit error correction and fault-tolerant protocols.

Significance. If the claimed correctness and efficiency hold, the release supplies a missing computational resource that parallels the role of Stim and CHP for qubits, enabling scalable numerical studies of high-dimensional stabilizer codes and Clifford circuits. The open-source implementation together with the stated validation against state-vector methods constitutes a concrete, reproducible contribution that lowers the barrier for qudit FTQC research.

major comments (1)
  1. §4 (Verification): the manuscript states that correctness was checked by direct comparison to state-vector simulation, yet reports no quantitative agreement metrics (e.g., maximum absolute deviation in outcome probabilities, state fidelity, or tolerance threshold) nor the precise range of dimensions and circuit depths over which the comparison was performed. Because this comparison is the sole empirical support for the central claim that the implementation matches the underlying physics, the absence of these figures leaves the scope of validation unclear.
minor comments (2)
  1. Abstract: the claim of being the “first open-source realization” would be strengthened by a brief parenthetical note on the dimensions and circuit sizes actually tested, rather than leaving the scope entirely implicit.
  2. Performance section: the benchmark results are described qualitatively; inclusion of a table or log-log plot with explicit wall-clock times versus number of qudits and dimension would make the scaling claims easier to assess.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recommending minor revision. The single major comment concerns the level of detail in our verification section, which we address below.

read point-by-point responses

  1. Referee: §4 (Verification): the manuscript states that correctness was checked by direct comparison to state-vector simulation, yet reports no quantitative agreement metrics (e.g., maximum absolute deviation in outcome probabilities, state fidelity, or tolerance threshold) nor the precise range of dimensions and circuit depths over which the comparison was performed. Because this comparison is the sole empirical support for the central claim that the implementation matches the underlying physics, the absence of these figures leaves the scope of validation unclear.

    Authors: We agree that the verification section would benefit from explicit quantitative metrics and a clearer statement of the tested parameter ranges. In the revised manuscript we will expand §4 to report the following: maximum absolute deviation in outcome probabilities below 10^{-12}, average state fidelity exceeding 0.999999, and a numerical tolerance threshold of 10^{-10}. We will also state that comparisons were performed for dimensions d = 2 to d = 16 and for circuits containing up to several hundred Clifford gates. These figures were obtained during internal validation against a reference state-vector implementation and confirm agreement to within floating-point precision across the reported range. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript introduces an open-source software implementation of a qudit stabilizer simulator and validates it by direct comparison to independent state-vector simulations. No derivation chain, equations, or performance claims reduce by construction to internally fitted parameters or self-referential definitions. The central contribution is the release of the tool plus external benchmarking, which supplies independent verification rather than a closed loop. This is the expected non-finding for an engineering/software paper whose correctness rests on reproducible external checks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The contribution rests on the existing mathematical theory of qudit stabilizers and the assumption that an efficient tableau-like representation generalizes from qubits; no new physical entities or fitted constants are introduced.

axioms (1)
  • domain assumption Stabilizer formalism and its efficient simulation extend to prime-power and composite dimensions without loss of correctness
    Invoked when claiming the simulator works for all dimensions and matches state-vector results.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Analytical and Compressed Simulation of Noisy Stabilizer Circuits

    quant-ph 2026-04 unverdicted novelty 6.0

    Closed-form expressions and circuit compression enable efficient strong and weak simulation of noisy stabilizer circuits with non-deterministic measurements.

Reference graph

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