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REVIEW 3 major objections 6 minor 78 references

SymFT makes exact sampling of Clifford-heavy fault-tolerant circuits fast by factoring out unitary frames and planning only the active non-stabilizer state once.

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2026-07-31 02:35 UTC pith:PEQOU4RN

load-bearing objection Solid systems paper: compile-once symbolic Clifford–Pauli factorization plus planned multi-coordinate dense kernels, with real measured speedups on the right FT workloads. the 3 major comments →

arxiv 2607.28600 v1 pith:PEQOU4RN submitted 2026-07-30 quant-ph

SymFT: Universal Fault-Tolerant Quantum Circuit Simulation via Symbolic Clifford--Pauli Frames and Stabilizer Coordinates

Wang Fang , Huazhe Lou , Riling Li This is my paper
classification quant-ph
keywords fault-tolerant quantum circuitsClifford simulationstabilizer formalismmagic-state cultivationsymbolic Pauli framesactive-state samplingquantum error correction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Fault-tolerant quantum protocols are mostly Clifford gates and Pauli measurements, with a thinner layer of non-Clifford rotations, noise, mid-circuit measurements, and classical Pauli feedback. Exact sampling of those circuits is expensive if every shot replays the full stabilizer evolution. This paper introduces SymFT, a simulator that compiles the shared Clifford structure once: it pulls rotations and measurements back through Clifford and Pauli frames so that noise and feedback become symbolic signs, then plans a single stabilizer-coordinate trajectory that only the dense active non-stabilizer amplitudes must follow shot by shot. Because residual frames are unitary, they drop out of branch probabilities and need not be applied per shot. On the tested surface-code and magic-state cultivation workloads, the resulting kernels beat leading pure-Clifford and near-Clifford samplers by clear margins on one CPU core, and beat the authors’ prior simulator by more than two orders of magnitude on cultivation.

Core claim

Exact branch-probability sampling for noisy adaptive Clifford-dominated circuits can be reduced to a one-time symbolic Clifford–Pauli factorization plus adaptive stabilizer-coordinate planning, so each shot only evaluates sparse symbolic signs and updates a dynamically sized dense active-state vector—without per-shot tableau updates or localization-induced Clifford sweeps—and this yields state-of-the-art throughput on the reported pure-Clifford and near-Clifford benchmarks.

What carries the argument

Symbolic Clifford–Pauli frame factorization K ≐ C E(s,m) O(s,m), which removes unitary residual frames from Pr(m|s), together with adaptive stabilizer-coordinate planning that shares one stabilizer–destabilizer tableau and emits direct multi-coordinate instructions on a dense active vector of width k.

Load-bearing premise

The speedups assume that, on the local fault-tolerant circuits people care about, symbolic signs stay sparse and the peak number of active non-stabilizer coordinates stays modest enough for dense 2^k kernels to win.

What would settle it

Re-run the same surface-code and magic-state cultivation circuits (and variants with denser noise or poorer scheduling) against Stim, Clifft, and SOFT under matched single-core and GPU settings: if SymFT’s reported shot rates and exact agreement on discard/logical flags do not hold, or if peak active width and sign density blow up so the complexity bound ceases to favor it, the central performance claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Magic-state cultivation and distillation campaigns can generate far more exact shots per core-hour without changing the circuit model.
  • Pure-Clifford surface-code memory sampling can run faster than Stim’s Pauli-frame path while staying exact.
  • GPU backends can keep full instruction programs on-device and terminate rejected shots early under detector postselection.
  • Further sequence rewrites that jointly cut peak active width and symbolic density would multiply the same sampling pipeline.
  • The same compiled sign-and-active-state interface can later host product-component, sparse, or tensor-network active representations without replaying Clifford gates.

Where Pith is reading between the lines

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

  • If symbolic signs remain local, detector-error-model style sparse maps and SymFT’s sign layer are essentially dual views of the same fault-to-measurement dependence, so hybrid rare-event estimators could attach with little glue.
  • Protocols whose non-Clifford support temporarily swells then collapses under measurement may favor active-width methods over global stabilizer-rank expansions even when total magic is large.
  • The main practical limit to watch is not qubit count but scheduling that keeps k_max and sign weight jointly small; compiler passes aimed at those two numbers are the natural next lever.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. SymFT is an exact classical sampler for noisy, adaptive Clifford-dominated circuits with Pauli rotations, stochastic Pauli noise, mid-circuit Pauli measurements, and measurement-record-controlled Pauli feedback. The method has two stages: (i) a one-pass symbolic Clifford–Pauli frame factorization KT ≃ CT ET(s,m) OT(s,m) that pulls rotations and projectors back through Clifford and symbolic Pauli frames so that residual unitary frames drop out of branch probabilities (Eqs. 3–4); and (ii) adaptive stabilizer-coordinate planning on a shared stabilizer–destabilizer tableau that compiles direct multi-coordinate instructions for a dynamically sized dense active-state vector of width k, avoiding per-shot tableau updates and Clifft-style localization of the dense vector. Sampling complexity is stated as O((nt + nS_m,active) 2^{kS_max} + (nt + nm)d + ne). C++/CUDA implementations are benchmarked against Stim, Clifft, Tsim, and SOFT on pure-Clifford surface-code memory and near-Clifford magic-state cultivation, coherent-noise, and distillation workloads, reporting single-core speedups of about 2.5× vs Stim and up to a few× vs Clifft, and >100× vs SOFT on cultivation GPUs.

Significance. If the reported throughputs and exactness claims hold, this is a substantial systems contribution to classical simulation of fault-tolerant protocols. The factorization identity is standard but cleanly specialized; the planning case analysis (dormant/active × diagonal/nondiagonal rotations and measurements) and direct multi-coordinate kernels are carefully worked out; and the implementation moves shared work into a compile-once plan with packed symbolic evaluation, SIMD/GPU kernels, and detector postselection. Concrete strengths include open test circuits, pinned single-core methodology, explicit complexity parameters (kS_max, d), and honest internal caveats (Tsim wins when ZX eliminates residual tensors; MSC d=7 exploratory; SMEM limit). The work is of clear practical value for QEC protocol design and decoder/postselection studies, and the algorithmic combination of symbolic frames with stabilizer-coordinate dense sampling is a credible advance over SOFT, Clifft, and Stim in the Clifford-dominated regime.

major comments (3)
  1. [Abstract; §7.1; Table 3] Abstract and §7 claim “state-of-the-art sampling performance” across tested pure- and near-Clifford circuits, but Table 3 shows Tsim faster by ~145× (coherent d=3,r=1) and ~3000× (d=5,r=1) when ZX reduction eliminates residual tensors and sampling collapses to a host map. The SOTA claim should be scoped explicitly to regimes with nontrivial residual active width / non-eliminated non-Clifford structure (as already discussed in §7.1 and §8), not left unqualified in the abstract.
  2. [§7.1; Table 1] Table 1’s MSC d=7 row (kS_max=19, 47.09 shots/s, 1.86× Clifft) is labeled exploratory: postselection is disabled and the authors state they have not validated the full output distribution or rare logical-error behavior. Including this row in the main performance narrative and in the abstract’s 1.86–3.51× range risks overstating a stress-test microbenchmark. Either move it to a clearly marked appendix/stress-test subsection or add a validated correctness check before using it for comparative claims.
  3. [§5; §7] §5’s transfer of the per-shot bound hinges on local FT circuits keeping symbolic-sign weight d ≪ nm+ne and peak active width modest. The paper asserts this for “local fault-tolerant circuits” but does not report measured d (or wt(λ) histograms) on the benchmark suite, nor how d scales with distance/rounds. Without those diagnostics, the claimed advantage over Clifft’s O((nt+nm+ne)n) Pauli-frame term remains only qualitatively supported. Please report d / expression sparsity for the main workloads.
minor comments (6)
  1. [Figure 1; §4] Figure 1 caption and factorization illustration are helpful; adding the corresponding OT sequence length / instruction mix for one MSC circuit would make the planning pass more concrete.
  2. [§6] §6 product-component cost model and adaptive batch-size policy are free parameters affecting throughput; document default thresholds and whether reported numbers use defaults only.
  3. [Table 3] Table 3 note that coherent d=3,r=3 discard rates differ between Clifft and SymFT is important; quantify the rates so readers can judge whether the 3.96× ratio is comparable.
  4. [§7.3; §8] Related-work §8 is thorough; a short explicit statement that SymFT does not currently support dynamic DEM construction / online decoding (noted in §7.3 vs SOFT) would help practitioners choose tools.
  5. [§3] Minor notation: “.=” for equality up to global phase is fine but should be defined once in a notation paragraph; ω(A,P) anticommutation indicator is standard—cite or define earlier when first used in Eq. (1).
  6. [Table 2; §7] GPU Table 2: Tsim path retains FP32/complex64 intermediates despite JAX x64; the comparison footnote is good—also state whether SymFT FP32 builds change the ranking on MSC d=3/5.

Circularity Check

0 steps flagged

No significant circularity: algorithmic correctness plus external throughput benchmarks, not a self-referential derivation.

full rationale

SymFT’s load-bearing claims are (i) an exact factorization KT .= CT ET OT with unitary residual frames so Pr(m|s)=‖OT|0^n‖² (Eqs. 3–4), (ii) a case-split planner that emits direct multi-coordinate dense-state instructions (Section 4), and (iii) measured single-core/GPU shot rates against Stim, Clifft, Tsim, and SOFT (Section 7). The probability identity follows from unitarity of CE, not from defining probability in terms of the claimed speedups. Benchmark ratios are wall-clock measurements on fixed circuits, not parameters fitted to a subset and re-labeled as predictions. Self-citation to SOFT is predecessor engineering context and a comparison baseline; the new rates are not derived from SOFT’s theorems. No uniqueness theorem, ansatz-via-citation, or renaming of a known empirical law carries the central claim. Scope limits (sparse symbolic signs d, modest kS_max) are stated complexity/transfer conditions, not circular reductions. Finding: no circular steps.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 2 invented entities

Methods paper grounded in standard stabilizer/Pauli algebra and engineering benchmarks. Load-bearing background is Gottesman–Knill/tableau simulation, unitary invariance of measurement probabilities, and the supported noisy circuit model. No fitted physical constants. Main modeling bets are the Pauli-only noise/feedback fragment and sparsity/active-width behavior of FT workloads.

free parameters (2)
  • Adaptive batch size / cache footprint limit = circuit-dependent automatic choice
    Batch size is chosen from planned peak active width to limit B·2^k memory pressure; this is an implementation heuristic that can move absolute throughput though not exactness.
  • Product-component cost-model thresholds
    Exact product-component lowering is enabled only when a conservative cost model predicts benefit; thresholding affects performance path selection.
axioms (5)
  • standard math Clifford conjugation preserves Pauli operators; Pauli conjugation only contributes signs via the anticommutation indicator ω.
    Used throughout Section 3 to justify single-pass pullback of rotations and measurements.
  • standard math For KT ≃ CT ET(s,m) OT(s,m) with CT ET unitary, Pr(m|s)=tr(OT ρ0 OT†), so residual frames can be omitted from branch-probability sampling.
    Equation (4); central correctness justification for not applying frames per shot.
  • domain assumption Supported circuit model is Clifford + Hermitian Pauli rotations + Pauli measurements + stochastic Pauli noise + measurement-record-controlled Pauli feedback (parity controls).
    Section 2; non-Pauli feedback or arbitrary classical controls are acknowledged as less efficient extensions.
  • domain assumption In local FT circuits, affine symbolic signs are typically sparse so d is effectively small compared with nm+ne.
    Section 5; needed to prefer SymFT’s O((nt+nm)d) sign cost over Clifft-like O((·)n) Pauli-frame updates.
  • ad hoc to paper Benchmark suite (MSC injection/cultivation, coherent-noise surface code, distillation, pure-Clifford memory) is representative enough to support a state-of-the-art claim.
    Section 7 defines the compared workloads; SOTA is scoped to ‘tested’ circuits but marketed as general high-throughput FT sampling.
invented entities (2)
  • Symbolic Clifford–Pauli frame factorization (CT, ET(s,m), OT(s,m)) independent evidence
    purpose: Compile shared unitary frames once and reduce per-shot quantum work to pulled-back rotations/projectors with symbolic signs.
    Named organizational device built from standard conjugation identities; not a new physical object.
  • Adaptive stabilizer-coordinate planning with direct multi-coordinate dense instructions independent evidence
    purpose: Share one tableau trajectory across shots and avoid Clifft-style localization Cliffords on the active vector.
    Engineering compilation layer over stabilizer–destabilizer coordinates and active-state representation.

pith-pipeline@v1.2.0-daily-grok45 · 30582 in / 3565 out tokens · 64331 ms · 2026-07-31T02:35:39.479889+00:00 · methodology

0 comments
read the original abstract

Fault-tolerant protocols often consist largely of stabilizer subcircuits, yet the non-Clifford operations required for universality make exact sampling costly. We present SymFT, a high-throughput simulator for Clifford-dominated circuits with Pauli rotations, stochastic Pauli noise, mid-circuit Pauli measurements, and measurement-record-controlled Pauli feedback. It combines two ideas. First, symbolic Clifford--Pauli frame factorization reduces branch-probability sampling to Pauli rotations and measurement projectors, with noise and feedback represented by symbolic signs. Since the residual Clifford and Pauli frames are unitary, they do not affect branch probabilities and need not be applied in every shot. Second, adaptive stabilizer-coordinate planning uses a shared stabilizer--destabilizer tableau to define the basis and stores only the active non-stabilizer degrees of freedom in a dynamically sized dense active-state vector. It resolves basis changes once and emits direct multi-coordinate sampling instructions, thereby avoiding per-shot tableau updates and localization-induced Clifford transformations of the dense vector. Across the tested pure-Clifford and near-Clifford circuits, SymFT achieves state-of-the-art sampling performance. On a single CPU core, it is $2.51\text{--}2.56\times$ faster than Stim for surface-code circuits and $1.86\text{--}3.51\times$ faster than Clifft for magic-state cultivation and distillation circuits. For the tested cultivation circuits, its sampling throughput also exceeds that of our previous simulator, SOFT, by more than two orders of magnitude.

Figures

Figures reproduced from arXiv: 2607.28600 by Huazhe Lou, Riling Li, Wang Fang.

Figure 1
Figure 1. Figure 1: Illustration of symbolic Clifford–Pauli frame factorization. The original circuit (left) contains Clifford [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

78 extracted references · 18 linked inside Pith

  1. [1]

    Fault-tolerant quantum computation

    Peter W. Shor. “Fault-tolerant quantum computation”. In Proceedings of 37th Conference on Foundations of Computer Science. Pages 56–65. (1996). 1

    1996

  2. [2]

    Stabilizer codes and quantum error correction

    Daniel Gottesman. “Stabilizer codes and quantum error correction”. PhD thesis. California Institute of Technology. (1997). arXiv:quant-ph/9705052. 1 24

    Pith/arXiv arXiv 1997

  3. [3]

    The Heisenberg representation of quantum computers

    Daniel Gottesman. “The Heisenberg representation of quantum computers” (1998). arXiv:quant-ph/9807006. 1, 21

    Pith/arXiv arXiv 1998

  4. [4]

    Universal quantum computation with ideal Clifford gates and noisy ancillas

    Sergey Bravyi and Alexei Kitaev. “Universal quantum computation with ideal Clifford gates and noisy ancillas”. Phys. Rev. A71, 022316 (2005). 1

    2005

  5. [5]

    Efficient magic state factories with a catalyzed|CCZ⟩ to2|T⟩transformation

    Craig Gidney and Austin G. Fowler. “Efficient magic state factories with a catalyzed|CCZ⟩ to2|T⟩transformation”. Quantum3, 135 (2019)

    2019

  6. [6]

    Magic State Distillation: Not as Costly as You Think

    Daniel Litinski. “Magic State Distillation: Not as Costly as You Think”. Quantum3, 205 (2019)

    2019

  7. [7]

    Leveraging zero-level distillation to generate high-fidelity magic states

    Yutaka Hirano, Tomohiro Itogawa, and Keisuke Fujii. “Leveraging zero-level distillation to generate high-fidelity magic states”. In 2024 IEEE International Conference on Quantum Computing and Engineering (QCE). Volume 01, pages 843–853. (2024)

    2024

  8. [8]

    Constant-overhead magic state distil- lation

    Adam Wills, Min-Hsiu Hsieh, and Hayata Yamasaki. “Constant-overhead magic state distil- lation”. Nature Physics21, 1842–1846 (2024)

    2024

  9. [9]

    Low-overhead magic state distillation with color codes

    Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio, Benjamin J. Brown, and Stephen D. Bartlett. “Low-overhead magic state distillation with color codes”. PRX Quantum6, 030317 (2025)

    2025

  10. [10]

    Efficient magic state distillation by zero-level distillation

    Tomohiro Itogawa, Yugo Takada, Yutaka Hirano, and Keisuke Fujii. “Efficient magic state distillation by zero-level distillation”. PRX Quantum6, 020356 (2025). 1

    2025

  11. [11]

    Universal fault-tolerant quantum computation with only transversal gates and error correction

    Adam Paetznick and Ben W. Reichardt. “Universal fault-tolerant quantum computation with only transversal gates and error correction”. Phys. Rev. Lett.111, 090505 (2013). 1

    2013

  12. [12]

    Fault-tolerant conversion between the Steane and Reed-Muller quantum codes

    Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin. “Fault-tolerant conversion between the Steane and Reed-Muller quantum codes”. Phys. Rev. Lett.113, 080501 (2014)

    2014

  13. [13]

    Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes

    Héctor Bombín. “Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes”. New Journal of Physics17, 083002 (2015)

    2015

  14. [14]

    Universal transversal gates with color codes: A simplified approach

    Aleksander Kubica and Michael E. Beverland. “Universal transversal gates with color codes: A simplified approach”. Phys. Rev. A91, 032330 (2015). 1

    2015

  15. [15]

    Using concatenated quantum codes for universal fault-tolerant quantum gates

    Tomas Jochym-O’Connor and Raymond Laflamme. “Using concatenated quantum codes for universal fault-tolerant quantum gates”. Phys. Rev. Lett.112, 010505 (2014). 1

    2014

  16. [16]

    Thresholds for universal concatenated quantum codes

    Christopher Chamberland, Tomas Jochym-O’Connor, and Raymond Laflamme. “Thresholds for universal concatenated quantum codes”. Phys. Rev. Lett.117, 010501 (2016). 1

    2016

  17. [17]

    Universal fault-tolerant gates on concatenated stabilizer codes

    Theodore J. Yoder, Ryuji Takagi, and Isaac L. Chuang. “Universal fault-tolerant gates on concatenated stabilizer codes”. Phys. Rev. X6, 031039 (2016). 1

    2016

  18. [18]

    Error rates and resource overheads of encoded three-qubit gates

    Ryuji Takagi, Theodore J. Yoder, and Isaac L. Chuang. “Error rates and resource overheads of encoded three-qubit gates”. Phys. Rev. A96, 042302 (2017). 1

    2017

  19. [19]

    Magic state cultivation: growing T states as cheap as CNOT gates

    Craig Gidney, Noah Shutty, and Cody Jones. “Magic state cultivation: growing T states as cheap as CNOT gates” (2024). arXiv:2409.17595. 1, 19

    Pith/arXiv arXiv 2024

  20. [20]

    Efficient magic state cultivation with lattice surgery

    Yutaka Hirano, Riki Toshio, Tomohiro Itogawa, and Keisuke Fujii. “Efficient magic state cultivation with lattice surgery” (2025). arXiv:2510.24615

    arXiv 2025

  21. [21]

    Efficient magic state cultivation onRP 2

    Zi-Han Chen, Ming-Cheng Chen, Chao-Yang Lu, and Jian-Wei Pan. “Efficient magic state cultivation onRP 2”. PRX Quantum7, 010315 (2026)

    2026

  22. [22]

    High rate magic state cultivation on the surface code

    Yotam Vaknin, Shoham Jacoby, Arne Grimsmo, and Alex Retzker. “High rate magic state cultivation on the surface code”. PRX Quantum7, 010353 (2026)

    2026

  23. [23]

    Fold-transversal surface code cultivation

    Kaavya Sahay, Pei-Kai Tsai, Kathleen (Katie) Chang, Qile Su, Thomas B. Smith, Shrad- dha Singh, and Shruti Puri. “Fold-transversal surface code cultivation”. PRX Quantum7, 033006 (2026)

    2026

  24. [24]

    Constant depth magic state cultivation with Clifford measurements by gauging

    Bence Hetényi, Benjamin J. Brown, and Dominic J. Williamson. “Constant depth magic state cultivation with Clifford measurements by gauging” (2026). arXiv:2603.05429

    Pith/arXiv arXiv 2026

  25. [25]

    Efficient magic state culti- vation for √ Tgates

    I-Chi Chen, Matheus da Silva Fonseca, and Andrew Sornborger. “Efficient magic state culti- vation for √ Tgates” (2026). arXiv:2606.10430. 1

    Pith/arXiv arXiv 2026

  26. [26]

    Stim: a fast stabilizer circuit simulator

    Craig Gidney. “Stim: a fast stabilizer circuit simulator”. Quantum5, 497 (2021). 2, 16, 18, 21

    2021

  27. [27]

    Improvedclassicalsimulationofquantumcircuitsdominated by clifford gates

    SergeyBravyiandDavidGosset. “Improvedclassicalsimulationofquantumcircuitsdominated by clifford gates”. Phys. Rev. Lett.116, 250501 (2016). 2, 23, 24

    2016

  28. [28]

    Simulation of quantum circuits by low-rank stabilizer decompositions

    Sergey Bravyi, Dan Browne, Padraic Calpin, Earl Campbell, David Gosset, and Mark Howard. “Simulation of quantum circuits by low-rank stabilizer decompositions”. Quantum 3, 181 (2019)

    2019

  29. [29]

    Improved upper bounds on the stabilizer rank of magic states

    Hammam Qassim, Hakop Pashayan, and David Gosset. “Improved upper bounds on the stabilizer rank of magic states”. Quantum5, 606 (2021). 2, 23 25

    2021

  30. [30]

    Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions

    Aleks Kissinger and John van de Wetering. “Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions”. Quantum Science and Technology7, 044001 (2022). 2, 23

    2022

  31. [31]

    Classical Simulation of Quan- tum Circuits with Partial and Graphical Stabiliser Decompositions

    Aleks Kissinger, John van de Wetering, and Renaud Vilmart. “Classical Simulation of Quan- tum Circuits with Partial and Graphical Stabiliser Decompositions”. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Volume 232, pages 5:1–5:13. (2022). 22, 24

    2022

  32. [32]

    Procedurally optimised ZX-diagram cutting for ef- ficient T-decomposition in classical simulation

    Matthew Sutcliffe and Aleks Kissinger. “Procedurally optimised ZX-diagram cutting for ef- ficient T-decomposition in classical simulation”. Electronic Proceedings in Theoretical Com- puter Science406, 63–78 (2024)

    2024

  33. [33]

    Fast classical simulation of quantum circuits via parametric rewriting in the zx-calculus

    Matthew Sutcliffe and Aleks Kissinger. “Fast classical simulation of quantum circuits via parametric rewriting in the zx-calculus”. Electronic Proceedings in Theoretical Computer Science426, 247–269 (2025). 23, 24

    2025

  34. [34]

    Tsim: Fast universal simulator for quantum error correction

    Rafael Haenel, Xiuzhe Luo, and Chen Zhao. “Tsim: Fast universal simulator for quantum error correction” (2026). arXiv:2604.01059. 2, 16, 18, 21, 22, 23, 24

    arXiv 2026

  35. [35]

    SOFT: a high-performance simulator for universal fault-tolerant quantum cir- cuits

    Riling Li, Keli Zheng, Yiming Zhang, Huazhe Lou, Shenggang Ying, Ke Liu, and Xiaom- ing Sun. “SOFT: a high-performance simulator for universal fault-tolerant quantum cir- cuits” (2025). arXiv:2512.23037. 2, 3, 6, 18, 21, 22, 23

    arXiv 2025

  36. [36]

    A generalization of the stabilizer formalism for simulating arbitrary quantum circuits

    Theodore J Yoder. “A generalization of the stabilizer formalism for simulating arbitrary quantum circuits” (2012). 2, 23

    2012

  37. [37]

    Clifft: Fast exact simulation of near-clifford quantum circuits

    Bradley A. Chase and Farrokh Labib. “Clifft: Fast exact simulation of near-clifford quantum circuits” (2026). arXiv:2604.27058. 2, 3, 6, 15, 18, 21, 24

    Pith/arXiv arXiv 2026

  38. [38]

    Clifford Circuit Opti- mization with Templates and Symbolic Pauli Gates

    Sergey Bravyi, Ruslan Shaydulin, Shaohan Hu, and Dmitri Maslov. “Clifford Circuit Opti- mization with Templates and Symbolic Pauli Gates”. Quantum5, 580 (2021). 2, 23

    2021

  39. [39]

    Symbolic execution for quantum error correction programs

    Wang Fang and Mingsheng Ying. “Symbolic execution for quantum error correction programs”. Proc. ACM Program. Lang.8(2024). 23

    2024

  40. [40]

    SymPhase: Phase symbolization for fast simulation of stabilizer circuits

    Wang Fang and Mingsheng Ying. “SymPhase: Phase symbolization for fast simulation of stabilizer circuits”. In Proceedings of the 61st ACM/IEEE Design Automation Conference. DAC ’24 (2024). 16, 21, 23

    2024

  41. [41]

    SyQMA: A memory-efficient, symbolic and exact universal simulator for quantum error correction

    George Umbrarescu and David Amaro. “SyQMA: A memory-efficient, symbolic and exact universal simulator for quantum error correction” (2026). arXiv:2604.15043. 2, 23, 24

    Pith/arXiv arXiv 2026

  42. [42]

    GeForce RTX 4090 specifications

    NVIDIA Corporation. “GeForce RTX 4090 specifications”.https://www.nvidia.com/en-us/ geforce/graphics-cards/40-series/rtx-4090/(2026). Accessed 2026-07-19. 18

    2026

  43. [43]

    CUDA C++ Best Practices Guide: instruction throughput

    NVIDIA Corporation. “CUDA C++ Best Practices Guide: instruction throughput”. (2026). url:https://docs.nvidia.com/cuda/cuda-c-best-practices-guide/. 18

    2026

  44. [44]

    Improved simulation of stabilizer circuits

    Scott Aaronson and Daniel Gottesman. “Improved simulation of stabilizer circuits”. Phys. Rev. A70, 052328 (2004). 21

    2004

  45. [45]

    Fast simulation of stabilizer circuits using a graph-state representation

    Simon Anders and Hans J. Briegel. “Fast simulation of stabilizer circuits using a graph-state representation”. Phys. Rev. A73, 022334 (2006). 21

    2006

  46. [46]

    Fast Stabiliser Simulation with Quadratic Form Expansions

    Niel de Beaudrap and Steven Herbert. “Fast Stabiliser Simulation with Quadratic Form Expansions”. Quantum6, 803 (2022). 21

    2022

  47. [47]

    Simulation of noisy Clifford circuits without fault propagation

    Nicolas Delfosse and Adam Paetznick. “Simulation of noisy Clifford circuits without fault propagation” (2023). arXiv:2309.15345. 21

    Pith/arXiv arXiv 2023

  48. [48]

    Analytical and compressed simulation of noisy stabilizer circuits

    Paul Aigner, Jasmin Matti, Maria Flors Mor-Ruiz, Julius Wallnöfer, and Wolfgang Dür. “Analytical and compressed simulation of noisy stabilizer circuits” (2026). arXiv:2604.22588. 21, 24

    Pith/arXiv arXiv 2026

  49. [49]

    STABSim: A parallelized Clifford simulator with features beyond direct simulation

    Sean Garner, Chenxu Liu, Meng Wang, Samuel Stein, and Ang Li. “STABSim: A parallelized Clifford simulator with features beyond direct simulation” (2025). arXiv:2507.03092. 21

    Pith/arXiv arXiv 2025

  50. [50]

    GPU-accelerated quantum simulation of stabilizer circuits

    Muhammad Osama, Dimitrios Thanos, and Alfons Laarman. “GPU-accelerated quantum simulation of stabilizer circuits” (2026). arXiv:2603.14641. 21

    arXiv 2026

  51. [51]

    Efficient inner-product algorithm for stabilizer states

    Hector J. Garcia, Igor L. Markov, and Andrew W. Cross. “Efficient inner-product algorithm for stabilizer states” (2013). arXiv:1210.6646. 23

    Pith/arXiv arXiv 2013

  52. [52]

    Simulation of quantum circuits via stabilizer frames

    Héctor J. García and Igor L. Markov. “Simulation of quantum circuits via stabilizer frames”. IEEE Transactions on Computers64, 2323–2336 (2015). 23

    2015

  53. [53]

    Computing logical error thresholds with the pauli frame sparse representation

    Thomas Tuloup and Thomas Ayral. “Computing logical error thresholds with the pauli frame sparse representation” (2026). arXiv:2603.14670. 23 26

    Pith/arXiv arXiv 2026

  54. [54]

    Stabilizer tensor networks: Universal quantum simulator on a basis of stabilizer states

    Sergi Masot-Llima and Artur Garcia-Saez. “Stabilizer tensor networks: Universal quantum simulator on a basis of stabilizer states”. Phys. Rev. Lett.133, 230601 (2024). 23, 24

    2024

  55. [55]

    Stabilizer tensor networks with magic state injection

    Azar C. Nakhl, Ben Harper, Maxwell West, Neil Dowling, Martin Sevior, Thomas Quella, and Muhammad Usman. “Stabilizer tensor networks with magic state injection”. Phys. Rev. Lett. 134, 190602 (2025)

    2025

  56. [56]

    MPStab: an hybrid stabilizers tensor-network quantum circuit simulator

    Giulio Crognaletti, Mattia Robbiano, Michele Grossi, and Matteo Robbiati. “MPStab: an hybrid stabilizers tensor-network quantum circuit simulator” (2026). arXiv:2607.24258. 23, 24

    Pith/arXiv arXiv 2026

  57. [57]

    On the stabilizer formalism and its generalization

    Éloi Descamps and Borivoje Dakić. “On the stabilizer formalism and its generalization”. Journal of Physics A: Mathematical and Theoretical57, 455301 (2024). 23

    2024

  58. [58]

    Qimax: Efficient quantum simulation via GPU-accelerated extended stabilizer formalism

    Vu Tuan Hai, Bui Cao Doanh, Le Vu Trung Duong, Pham Hoai Luan, and Yasuhiko Nakashima. “Qimax: Efficient quantum simulation via GPU-accelerated extended stabilizer formalism” (2026). arXiv:2505.03307. 23

    Pith/arXiv arXiv 2026

  59. [59]

    Quadratic lower bounds on the approximate sta- bilizer rank: A probabilistic approach

    Saeed Mehraban and Mehrdad Tahmasbi. “Quadratic lower bounds on the approximate sta- bilizer rank: A probabilistic approach”. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing. Pages 608–619. (2024). 23

    2024

  60. [60]

    Approximate stabilizer rank and improved weak simulation of clifford-dominated circuits for qudits

    Yifei Huang and Peter Love. “Approximate stabilizer rank and improved weak simulation of clifford-dominated circuits for qudits”. Phys. Rev. A99, 052307 (2019)

    2019

  61. [61]

    Quantifying quantum speedups: Improved classical simulation from tighter magic mono- tones

    James R. Seddon, Bartosz Regula, Hakop Pashayan, Yingkai Ouyang, and Earl T. Campbell. “Quantifying quantum speedups: Improved classical simulation from tighter magic mono- tones”. PRX Quantum2, 010345 (2021)

    2021

  62. [62]

    Fast estimation of outcome probabilities for quantum circuits

    Hakop Pashayan, Oliver Reardon-Smith, Kamil Korzekwa, and Stephen D. Bartlett. “Fast estimation of outcome probabilities for quantum circuits”. PRX Quantum3, 020361 (2022)

    2022

  63. [63]

    Improved strong simulation of universal quantum circuits

    Lucas Kocia. “Improved strong simulation of universal quantum circuits” (2022). arXiv:2012.11739. 23

    Pith/arXiv arXiv 2022

  64. [64]

    Trading classical and quantum compu- tational resources

    Sergey Bravyi, Graeme Smith, and John A. Smolin. “Trading classical and quantum compu- tational resources”. Phys. Rev. X6, 021043 (2016). 23

    2016

  65. [65]

    Optimizing T gates in Clifford+T circuit asπ/4rotations around paulis

    Fang Zhang and Jianxin Chen. “Optimizing T gates in Clifford+T circuit asπ/4rotations around paulis” (2019). arXiv:1903.12456. 23, 24

    Pith/arXiv arXiv 2019

  66. [66]

    Simulation of qubit quantum circuits via pauli propagation

    Patrick Rall, Daniel Liang, Jeremy Cook, and William Kretschmer. “Simulation of qubit quantum circuits via pauli propagation”. Phys. Rev. A99, 062337 (2019). 23

    2019

  67. [67]

    Pauli propagation: A computational framework for simulating quantum systems

    Manuel S. Rudolph, Tyson Jones, Yanting Teng, Armando Angrisani, and Zoë Holmes. “Pauli propagation: A computational framework for simulating quantum systems” (2026). arXiv:2505.21606

    Pith/arXiv arXiv 2026

  68. [68]

    Sim- ulating quantum circuits with arbitrary local noise using pauli propagation

    Armando Angrisani, Antonio A. Mele, Manuel S. Rudolph, M. Cerezo, and Zoë Holmes. “Sim- ulating quantum circuits with arbitrary local noise using pauli propagation”. PRX Quantum 7, 020313 (2026). 23

    2026

  69. [69]

    Efficient simulation of clifford circuits with small markovian errors

    Ashe Miller, Corey Ostrove, Jordan Hines, Robin Blume-Kohout, Kevin Young, and Tim- othy Proctor. “Efficient simulation of clifford circuits with small markovian errors” (2025). arXiv:2504.15128. 23

    Pith/arXiv arXiv 2025

  70. [70]

    Simulating quantum error correction beyond pauli stochastic errors

    Jordan Hines, Corey Ostrove, Kenneth Rudinger, Stefan Seritan, Kevin Young, Robin Blume- Kohout, and Timothy Proctor. “Simulating quantum error correction beyond pauli stochastic errors” (2026). arXiv:2603.18457. 23

    arXiv 2026

  71. [71]

    De- signing fault-tolerant circuits using detector error models

    Peter-Jan H.S. Derks, Alex Townsend-Teague, Ansgar G. Burchards, and Jens Eisert. “De- signing fault-tolerant circuits using detector error models”. Quantum9, 1905 (2025). 23

    1905

  72. [72]

    Simulation of rare events in quantum error correction

    Sergey Bravyi and Alexander Vargo. “Simulation of rare events in quantum error correction”. Phys. Rev. A88, 062308 (2013). 23, 24

    2013

  73. [73]

    Rare event simulation of quantum error-correcting circuits

    Carolyn Mayer, Anand Ganti, Uzoma Onunkwo, Tzvetan Metodi, Benjamin Anker, and Jacek Skryzalin. “Rare event simulation of quantum error-correcting circuits” (2025). arXiv:2509.13678. 23

    arXiv 2025

  74. [74]

    Dynamical subset sampling of quantum error-correcting protocols

    Sascha Heußen, Don Winter, Manuel Rispler, and Markus Müller. “Dynamical subset sampling of quantum error-correcting protocols”. Phys. Rev. Res.6, 013177 (2024). 23

    2024

  75. [75]

    Fail fast: techniques to probe rare events in quantum error correction

    Michael E. Beverland, Malcolm Carroll, Andrew W. Cross, and Theodore J. Yoder. “Fail fast: techniques to probe rare events in quantum error correction” (2025). arXiv:2511.15177. 23

    arXiv 2025

  76. [76]

    Scalable testing of quantum error correction

    John Zhuoyang Ye and Jens Palsberg. “Scalable testing of quantum error correction” (2026). arXiv:2602.04921. 23, 24 27

    arXiv 2026

  77. [77]

    Simulating magic state cultivation with few Clifford terms

    Kwok Ho Wan, Zhenghao Zhong, and Ainhoa Zapirain. “Simulating magic state cultivation with few Clifford terms”. Quantum10, 2134 (2026). 23

    2026

  78. [78]

    Efficient simulation of logical magic state preparation protocols

    Samyak Surti, Lucas Daguerre, and Isaac H. Kim. “Efficient simulation of logical magic state preparation protocols”. PRX Quantum7, 020329 (2026). 24 28

    2026