pith. sign in

arxiv: 2604.27058 · v2 · pith:XAXK7O4Znew · submitted 2026-04-29 · 🪐 quant-ph

Clifft: Fast Exact Simulation of Near-Clifford Quantum Circuits

Bradley A. Chase , Farrokh Labib This is my paper

Pith reviewed 2026-05-21 09:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum circuit simulationnear-Clifford circuitsfault-tolerant quantum computingmagic state cultivationClifford framesexact simulationT-gate circuits
0
0 comments X

The pith

Clifft factors quantum states into precomputed Clifford frames and a dynamic active vector to enable fast exact simulation of near-Clifford circuits.

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

The paper presents Clifft, a simulator that separates the quantum state into an offline Clifford frame, an online Pauli frame, and a dynamically sized active state vector. This design pre-resolves all deterministic Clifford transformations, moving the main computational cost to the size of the active subspace that grows only with non-Clifford gates and shrinks with measurements. A sympathetic reader would care because it makes exact classical simulation practical for large-scale fault-tolerant quantum computations that include magic states, something previously limited by exponential scaling. The work shows this approach delivers major speedups on low-magic benchmarks while remaining competitive in pure Clifford and fully non-Clifford cases. It also reports the first end-to-end exact simulations of magic state cultivation protocols over hundreds of billions of shots.

Core claim

Clifft shifts the dominant exponential cost from total qubit count or T-count to the peak active virtual dimension by factoring the state into an offline Clifford frame that handles all Clifford operations in advance, an online Pauli frame for tracking, and a dynamically sized active state vector that expands during non-Clifford operations and contracts during measurements.

What carries the argument

The dynamic active subspace created by separating the state into an offline Clifford frame, an online Pauli frame, and an active state vector whose size tracks only the non-Clifford content.

If this is right

  • Exact end-to-end simulation of magic state cultivation including the escape stage becomes feasible at scales of hundreds of billions of shots.
  • Throughput increases by orders of magnitude compared to GPU-accelerated near-Clifford simulators on low-magic fault-tolerant benchmarks.
  • Simulations reveal that escape-stage failures reduce the discrepancy between true T-gate circuits and their S-proxy at low decoder-gap thresholds.
  • At high thresholds the full-protocol behavior approaches the discrepancy seen in cultivation stages alone.

Where Pith is reading between the lines

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

  • This method may extend naturally to other fault-tolerant protocols where non-Clifford operations are sparse.
  • Similar frame factorization could improve simulation of other hybrid classical-quantum systems.
  • If the active dimension stays bounded in more circuits, it could allow exact simulation of systems previously thought intractable.
  • Developers of quantum hardware could use these simulations to optimize magic state preparation routines.

Load-bearing premise

The peak size of the active virtual dimension stays small enough in practical fault-tolerant benchmarks that the overhead of managing the frames is outweighed by the reduction in simulated state size.

What would settle it

A direct comparison of wall-clock time and memory usage when running the same cultivation circuit with hundreds of billions of shots using Clifft versus a standard state-vector or stabilizer simulator would show whether the claimed speedups hold.

Figures

Figures reproduced from arXiv: 2604.27058 by Bradley A. Chase, Farrokh Labib.

Figure 1
Figure 1. Figure 1: Lifecycle of an active operation in the frame-factored representation. Physical Clifford gates are view at source ↗
Figure 2
Figure 2. Figure 2: The high-level Clifft execution pipeline. The compiler resolves Clifford-coordinate evolution, HIR view at source ↗
Figure 3
Figure 3. Figure 3: Example mirror circuit used to illustrate Clifft’s multi-level lowering. The unitary block view at source ↗
Figure 4
Figure 4. Figure 4: A compact view of Clifft’s multi-level lowering for the example in Figure view at source ↗
Figure 5
Figure 5. Figure 5: Single-shot execution time vs. qubit count view at source ↗
Figure 6
Figure 6. Figure 6: Inject + cultivate stage: T-gate vs S-proxy logical error rate per kept shot, estimated using stratified importance sampling across uniform circuit-level depolarizing noise. Shaded regions in the top panels show absolute error-rate values within a factor of 1000 in likelihood relative to the maximum-likelihood estimate, following the convention used by Sinter for rare-event visualization. Shaded regions in… view at source ↗
Figure 7
Figure 7. Figure 7: End-to-end magic state cultivation including the escape stage to a view at source ↗
read the original abstract

Exact classical simulation of fault-tolerant quantum circuits remains limited by a tradeoff between exponential state vector scaling, exponential $T$-count scaling in stabilizer-rank approaches, and per-shot tracking overhead in sparse generalized stabilizer simulators. In this work, we introduce Clifft, an open-source simulator that shifts the dominant exponential cost from the total qubit count to a dynamic active subspace by factoring the quantum state into an offline Clifford frame, an online Pauli frame, and a dynamically sized active state vector. This architecture resolves deterministic Clifford coordinate transformations ahead of time, generalizing Stim's compile-once, sample-many execution model to circuits with non-Clifford operations. Consequently, exponential simulation costs are determined by the peak active virtual dimension, which expands during non-Clifford operations and contracts during measurements. Clifft remains within a constant factor of standard tools in the pure-Clifford and non-Clifford limits, while delivering up to orders-of-magnitude throughput gains over GPU-accelerated near-Clifford simulators on low-magic fault-tolerant benchmarks. Executing on commodity CPUs and exposing a Stim-like API, Clifft enables, to our knowledge, the first exact end-to-end simulation of magic state cultivation including the escape stage, over hundreds of billions of shots. These simulations show that escape-stage failures suppress the discrepancy between the true $T$-gate circuit and its $S$-proxy at low decoder-gap thresholds, while at high thresholds the full-protocol behavior approaches the larger discrepancy observed in the cultivation stages alone.

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

2 major / 3 minor

Summary. The manuscript introduces Clifft, an open-source simulator for exact classical simulation of near-Clifford quantum circuits. It factors the quantum state into an offline Clifford frame, an online Pauli frame, and a dynamically sized active state vector, generalizing Stim's compile-once sample-many model so that exponential costs track only the peak active virtual dimension (which expands on non-Clifford gates and contracts on measurements). The central claims are orders-of-magnitude throughput gains over GPU-accelerated near-Clifford simulators on low-magic fault-tolerant benchmarks and the first exact end-to-end simulation of magic-state cultivation (including the escape stage) over hundreds of billions of shots, with new observations on escape-stage failure suppression of T-vs-S discrepancies.

Significance. If the performance claims hold, the work is significant for quantum error correction and fault-tolerance research. It supplies a practical, CPU-based tool with a Stim-like API that enables previously inaccessible exact simulations at scale, directly addressing the tradeoff between state-vector size, T-count, and per-shot overhead. The open-source release and reproducible large-shot cultivation results are concrete strengths that could accelerate protocol verification and decoder analysis.

major comments (2)
  1. [Benchmark results] Benchmark results (cultivation and escape-stage simulations): the headline claims of orders-of-magnitude gains and feasibility of hundreds of billions of shots rest on the peak active virtual dimension remaining small enough that frame-management overhead does not dominate. The manuscript should report explicit time-series or maximum values of this dimension for the chosen low-magic fault-tolerant circuits, ideally with a plot or table showing its evolution through the escape stage.
  2. [Algorithm description] § on dynamic subspace management: the description of how the active state vector is resized and how Pauli-frame updates are applied after non-Clifford operations needs a concrete complexity analysis or pseudocode to confirm that the approach remains within a constant factor of standard simulators in the pure-Clifford and high-magic limits, as asserted.
minor comments (3)
  1. The abstract states 'up to orders-of-magnitude throughput gains'; the main text should tabulate the precise speedup factors, the exact baseline simulators (including GPU versions), and the circuit parameters (qubit count, T-count, magic-state density) for each benchmark.
  2. [Implementation and API] Clarify the API compatibility with Stim: list which Stim functions are directly supported and any differences in measurement or reset semantics.
  3. [Introduction] A short related-work paragraph comparing Clifft to existing sparse or generalized-stabilizer simulators would help readers place the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The comments identify useful opportunities to strengthen the presentation of results and algorithmic details. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Benchmark results] Benchmark results (cultivation and escape-stage simulations): the headline claims of orders-of-magnitude gains and feasibility of hundreds of billions of shots rest on the peak active virtual dimension remaining small enough that frame-management overhead does not dominate. The manuscript should report explicit time-series or maximum values of this dimension for the chosen low-magic fault-tolerant circuits, ideally with a plot or table showing its evolution through the escape stage.

    Authors: We agree that explicit tracking of the active virtual dimension would strengthen the performance claims. In the revised manuscript we will add a new figure (or table) that reports the time evolution and maximum value of this dimension for the low-magic fault-tolerant benchmark circuits, including through the escape stage. This addition will directly illustrate that the dimension remains small enough for frame-management overhead to stay negligible, thereby supporting the reported throughput gains and the feasibility of hundreds of billions of shots. revision: yes

  2. Referee: [Algorithm description] § on dynamic subspace management: the description of how the active state vector is resized and how Pauli-frame updates are applied after non-Clifford operations needs a concrete complexity analysis or pseudocode to confirm that the approach remains within a constant factor of standard simulators in the pure-Clifford and high-magic limits, as asserted.

    Authors: We acknowledge that a more explicit description would help confirm the constant-factor claim. In the revised manuscript we will insert pseudocode for the active-state-vector resizing and Pauli-frame update steps, together with a short complexity analysis in the dynamic subspace management section. The analysis will show that the overhead is constant in the pure-Clifford limit (recovering Stim-like behavior) and approaches that of a standard state-vector simulator (up to a constant factor) in the high-magic limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity in algorithmic architecture

full rationale

The paper presents a new simulation method that factors the state into offline Clifford frame, online Pauli frame, and dynamic active state vector, with exponential cost explicitly determined by peak active virtual dimension on the input circuit. No equations or claims reduce any reported quantity (speedups, shot counts, or cultivation results) to a parameter fitted from the same measurements or to a self-referential definition. The architecture generalizes an existing Stim model without invoking self-citations as load-bearing uniqueness theorems or smuggling ansatzes. Performance on low-magic benchmarks is an empirical consequence of the method rather than a constructed outcome, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard quantum mechanics and the stabilizer formalism; no new physical axioms or invented entities are introduced. The only potential free parameters are implementation choices such as subspace contraction heuristics, but none are explicitly fitted to data in the abstract.

axioms (1)
  • domain assumption Quantum states can be exactly represented by a Clifford frame, Pauli frame, and active state vector whose dimension is bounded by the number of non-Clifford operations active at any time.
    This is the core modeling choice stated in the abstract that enables the dynamic-subspace approach.

pith-pipeline@v0.9.0 · 5795 in / 1382 out tokens · 44247 ms · 2026-05-21T09:26:14.766190+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Clifft factors the simulated state into an offline Clifford frame, an online Pauli frame, and a dynamically sized active state vector... exponential simulation costs are determined by the peak active virtual dimension k_max, which expands during non-Clifford operations and contracts during measurements.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We achieve this via Pauli localization... Lemma 1 (Pauli Localization). For any non-identity N-qubit virtual Pauli operator there exists a sequence of at most 2N virtual Clifford gates V such that V P̃ V† = α P_v acting on exactly one virtual qubit.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

  1. Reducing Postselection Overhead in Magic-State Cultivation by In-Patch Multiplexing

    quant-ph 2026-05 unverdicted novelty 6.0

    In-patch multiplexing reduces expected attempts for early-stage magic-state cultivation by 45.46% (d1=3) to 72.91% (d1=5) and full-cycle attempts by 49-79% at p=2e-3, while final logical error rates stay governed by t...

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · cited by 1 Pith paper · 10 internal anchors

  1. [1]

    A fault-tolerant neutral-atom architecture for universal quantum computation

    Dolev Bluvstein, Alexandra A. Geim, Sophie H. Li, Simon J. Evered, J. Pablo Bonilla Ataides, et al. “A fault-tolerant neutral-atom architecture for universal quantum computation”. Nature 649, 39–46 (2026)

    work page 2026

  2. [2]

    Suppressing quantum errors by scaling a surface code logical qubit

    Google Quantum AI. “Suppressing quantum errors by scaling a surface code logical qubit”. Nature614, 676–681 (2023)

    work page 2023

  3. [3]

    Sur- face codes: Towards practical large-scale quantum computation

    Austin G Fowler, Matteo Mariantoni, John M Martinis, and Andrew N Cleland. “Sur- face codes: Towards practical large-scale quantum computation”. Physical Review A86, 032324 (2012)

    work page 2012

  4. [4]

    Topological quantum mem- ory

    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. “Topological quantum mem- ory”. Journal of Mathematical Physics43, 4452–4505 (2002)

    work page 2002

  5. [5]

    Pymatching: A python package for decoding quantum codes with minimum- weight perfect matching

    Oscar Higgott. “Pymatching: A python package for decoding quantum codes with minimum- weight perfect matching”. ACM Transactions on Quantum Computing3, 1–16 (2022)

    work page 2022

  6. [6]

    High- threshold and low-overhead fault-tolerant quantum memory

    Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, et al. “High- threshold and low-overhead fault-tolerant quantum memory”. Nature627, 778–782 (2024)

    work page 2024

  7. [7]

    Stim: a fast stabilizer cir- cuit simulator

    Craig Gidney. “Stim: A fast stabilizer circuit simulator”. Quantum5, 497 (2021). arXiv:2103.02202

    work page arXiv 2021

  8. [8]

    Improved Simulation of Stabilizer Circuits

    Scott Aaronson and Daniel Gottesman. “Improved simulation of stabilizer circuits”. Physical Review A70, 052328 (2004). arXiv:quant-ph/0406196

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  9. [9]

    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

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  10. [10]

    Itogawa, Y

    Tomohiro Itogawa, Yugo Takada, Yutaka Hirano, and Keisuke Fujii. “Efficient Magic State Distillation by Zero-Level Distillation”. PRX Quantum6, 020356 (2025). arXiv:2403.03991. 17

    work page arXiv 2025

  11. [11]

    Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits

    Christopher Chamberland and Kyungjoo Noh. “Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits”. npj Quantum Information6, 91 (2020)

    work page 2020

  12. [12]

    Magic State Distillation: Not as Costly as You Think

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

    work page 2019

  13. [13]

    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”. Physical Review A71, 022316 (2005)

    work page 2005

  14. [14]

    Contextuality supplies the magic for quantum computation

    Mark Howard, Joel J. Wallman, Victor Veitch, and Joseph Emerson. “Contextuality supplies the magic for quantum computation”. Nature510, 351–355 (2014). arXiv:1401.4174

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  15. [15]

    Quantum computing with Qiskit

    Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J. Wood, Jake Lishman, et al. “Quantum computing with Qiskit” (2024). arXiv:2405.08810

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  16. [16]

    “qsim” (2025)

    Quantum AI team and collaborators. “qsim” (2025)

    work page 2025

  17. [17]

    Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity

    Thomas Ayral, Thibaud Louvet, Yiqing Zhou, Cyprien Lambert, E. Miles Stoudenmire, et al. “Density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity”. PRX Quantum4, 020304 (2023)

    work page 2023

  18. [18]

    Huang, F

    Cupjin Huang, Fang Zhang, Michael Newman, Junjie Cai, Xun Gao, et al. “Classical Simula- tion of Quantum Supremacy Circuits” (2020). arXiv:2005.06787

    work page arXiv 2020

  19. [19]

    Simulating quantum computation by contracting tensor networks

    Igor L Markov and Yaoyun Shi. “Simulating quantum computation by contracting tensor networks”. SIAM Journal on Computing38, 963–981 (2008)

    work page 2008

  20. [20]

    Entropy scaling and simulability by matrix product states

    Norbert Schuch, Michael M. Wolf, Frank Verstraete, and J. Ignacio Cirac. “Entropy scaling and simulability by matrix product states”. Phys. Rev. Lett.100, 030504 (2008)

    work page 2008

  21. [21]

    Improved classical simulation of quantum circuits dominated by Clifford gates

    Sergey Bravyi and David Gosset. “Improved Classical Simulation of Quantum Circuits Dom- inated by Clifford Gates”. Physical Review Letters (2016). arXiv:1601.07601

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  22. [22]

    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”. Physical Review A99, 062337 (2019)

    work page 2019

  23. [23]

    Classical simulation of quantum circuits with partial and graphical stabiliser decompositions

    Aleks Kissinger, John van de Wetering, and Renaud Vilmart. “Classical simulation of quantum circuits with partial and graphical stabiliser decompositions”. LIPIcs, Volume 232, TQC 2022 232, 5:1–5:13 (2022). arXiv:2202.09202

    work page arXiv 2022

  24. [24]

    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). arXiv:2403.06777

    work page arXiv 2025

  25. [25]

    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

    work page arXiv 2026

  26. [26]

    Efficient simulation of logical magic state preparation protocols

    Samyak Surti, Lucas Daguerre, and Isaac H. Kim. “Efficient simulation of logical magic state preparation protocols” (2025). arXiv:2512.23799

    work page internal anchor Pith review arXiv 2025

  27. [27]

    Magic state cultivation on a superconducting quantum processor

    Emma Rosenfeld, Craig Gidney, Gabrielle Roberts, Alexis Morvan, Nathan Lacroix, et al. “Magic state cultivation on a superconducting quantum processor” (2025). arXiv:2512.13908

    work page arXiv 2025

  28. [28]

    Efficient magic state cultivation onRP2

    Zi-Han Chen, Ming-Cheng Chen, Chao-Yang Lu, and Jian-Wei Pan. “Efficient magic state cultivation onRP2”. PRX Quantum7, 010315 (2026). arXiv:2503.18657

    work page arXiv 2026

  29. [29]

    Break- ing even with magic: Demonstration of a high-fidelity logical non-Clifford gate

    Shival Dasu, Simon Burton, Karl Mayer, David Amaro, Justin A. Gerber, et al. “Break- ing even with magic: Demonstration of a high-fidelity logical non-Clifford gate” (2025). arXiv:2506.14688

    work page arXiv 2025

  30. [30]

    Fold-transversal surface code cultivation.arXiv preprint arXiv:2509.05212, 2025

    Kaavya Sahay, Pei-Kai Tsai, Kathleen Chang, Qile Su, Thomas B. Smith, et al. “Fold- transversal surface code cultivation” (2026). arXiv:2509.05212

    work page arXiv 2026

  31. [31]

    Efficient magic state cultivation with lat- tice surgery.arXiv preprint arXiv:2510.24615, 2025

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

    work page arXiv 2025

  32. [32]

    Efficient magic state cultivation on the surface code,

    Yotam Vaknin, Shoham Jacoby, Arne Grimsmo, and Alex Retzker. “Efficient Magic State Cultivation on the Surface Code” (2026). arXiv:2502.01743

    work page arXiv 2026

  33. [33]

    SOFT: A high- performance simulator for universal fault-tolerant quantum circuits

    Riling Li, Keli Zheng, Yiming Zhang, Huazhe Lou, Shenggang Ying, et al. “SOFT: A high- performance simulator for universal fault-tolerant quantum circuits” (2025). arXiv:2512.23037

    work page arXiv 2025

  34. [34]

    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

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  35. [35]

    Cutting stabiliser decompositions of magic state culti- vation with ZX-calculus

    Kwok Ho Wan and Zhenghao Zhong. “Cutting stabiliser decompositions of magic state culti- vation with ZX-calculus” (2025). arXiv:2509.01224

    work page arXiv 2025

  36. [36]

    Simulating magic state cultivation with few Clifford terms

    Kwok Ho Wan, Zhenghao Zhong, and Ainhoa Zapirain. “Simulating magic state cultivation with few Clifford terms” (2026). arXiv:2509.08658. 18

    work page arXiv 2026

  37. [37]

    The Heisenberg Representation of Quantum Computers

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

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  38. [38]

    Trading Classical and Quantum Com- putational Resources

    Sergey Bravyi, Graeme Smith, and John A. Smolin. “Trading Classical and Quantum Com- putational Resources”. Physical Review X6, 021043 (2016)

    work page 2016

  39. [39]

    A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

    Daniel Litinski. “A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery”. Quantum3, 128 (2019). arXiv:1808.02892

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  40. [40]

    Paulihedral: a generalized block-wise compiler optimization framework for quantum simulation kernels

    Gushu Li, Anbang Wu, Yunong Shi, Ali Javadi-Abhari, Yufei Ding, et al. “Paulihedral: a generalized block-wise compiler optimization framework for quantum simulation kernels”. In Proceedings of the 27th ACM International Conference on Architectural Support for Pro- gramming Languages and Operating Systems. Page 554–569. ASPLOS ’22New York, NY, USA (2022). ...

    work page 2022

  41. [41]

    Pcoast: A pauli-based quantum circuit optimization framework

    Jennifer Paykin, Albert T. Schmitz, Mohannad Ibrahim, Xin-Chuan Wu, and A. Y. Matsuura. “Pcoast: A pauli-based quantum circuit optimization framework”. In 2023 IEEE International Conference on Quantum Computing and Engineering (QCE). Volume 01, pages 715–726. (2023)

    work page 2023

  42. [42]

    Tableau-based framework for efficient logical quantum compilation

    Meng Wang, Chenxu Liu, Sean Garner, Samuel Stein, Yufei Ding, et al. “Tableau-based framework for efficient logical quantum compilation” (2025). arXiv:2509.02721

    work page arXiv 2025

  43. [43]

    Clifft documentation and code

    “Clifft documentation and code”. url:https://unitaryfoundation.github.io/clifft. code: unitaryfoundation/clifft

    work page

  44. [44]

    Clifft paper simulations

    “Clifft paper simulations”. url:https://github.com/unitaryfoundation/clifft-paper

    work page

  45. [45]

    Experimental demonstration of logical magic state distillation

    Pedro Sales Rodriguez, John M. Robinson, Paul Niklas Jepsen, Zhiyang He, Casey Duckering, et al. “Experimental demonstration of logical magic state distillation”. Nature645, 620– 625 (2025)

    work page 2025

  46. [46]

    Qulacs: A fast and versatile quantum circuit simulator for research purpose

    Yasunari Suzuki, Yoshiaki Kawase, Yuya Masumura, Yuria Hiraga, Masahiro Nakadai, et al. “Qulacs: A fast and versatile quantum circuit simulator for research purpose”. Quantum5, 559 (2021)

    work page 2021

  47. [47]

    Ex- act and approximate simulation of large quantum circuits on a single gpu

    Daniel Strano, Benn Bollay, Aryan Blaauw, Nathan Shammah, William J. Zeng, et al. “Ex- act and approximate simulation of large quantum circuits on a single gpu”. In 2023 IEEE International Conference on Quantum Computing and Engineering (QCE). Volume 01, pages 949–958. (2023)

    work page 2023

  48. [48]

    A comparison of HPC-based quantum computing simulators using Quantum Volume

    Lourens van Niekerk, Dhiraj Kumar, Aasish Kumar Sharma, Tino Meisel, Martin Leandro Paleico, et al. “A comparison of HPC-based quantum computing simulators using Quantum Volume” (2024). arXiv:2412.20518

    work page arXiv 2024

  49. [49]

    code: Strilanc/magic-state-cultivation commit:871e68f

    Craig Gidney (2024). code: Strilanc/magic-state-cultivation commit:871e68f

    work page 2024

  50. [50]

    Restrictions on Transversal Encoded Quantum Gate Sets

    Bryan Eastin and Emanuel Knill. “Restrictions on Transversal Encoded Quantum Gate Sets”. Physical Review Letters102, 110502 (2009). arXiv:0811.4262

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  51. [51]

    An invariant form for the prior probability in estimation prob- lems

    Harold Jeffreys. “An invariant form for the prior probability in estimation prob- lems”. Proceedings of the Royal Society of London. A. Mathematical and Physi- cal Sciences186, 453–461 (1946). arXiv:https://royalsocietypublishing.org/rspa/article- pdf/186/1007/453/38863/rspa.1946.0056.pdf

    work page arXiv 1946

  52. [52]

    Bayesian statistics: An introduction

    P.M. Lee. “Bayesian statistics: An introduction”. Wiley. (2012). A Constructive Proof of Pauli Localization We provide below a constructive proof of Lemma 1, detailing theO(N)greedy algorithm for localizing any multi-qubit virtual Pauli operator to a single virtual qubit. Proof.Let the multi-qubit virtual Pauli be parameterized by its Boolean support vect...

    work page 2012

  53. [53]

    The diagonal rotation acts simply as a scalar multipliere−iθ(−1)c , which is absorbed directly into the global scalarγ

    Trivial Phase (v∈D, generator nativelyZ v):Becausev∈D, the state on axisvis|0⟩v. The diagonal rotation acts simply as a scalar multipliere−iθ(−1)c , which is absorbed directly into the global scalarγ. The continuous vector|ϕ⟩A remains isolated, and the active dimensionkis unchanged

    work page

  54. [54]

    This operation promotesvto the active set (A←A∪{v},k←k+ 1)

    Subspace Expansion (v∈D, requiresH v):If the generator localized to theX-basis, the necessary virtual Hadamard maps|0⟩v → |+⟩v. This operation promotesvto the active set (A←A∪{v},k←k+ 1). The state vector dimensionality expands via the tensor product |ϕ⟩A←|ϕ⟩A⊗|+⟩, and the phase rotation is subsequently evaluated on the newly active degree of freedom

    work page

  55. [55]

    The diagonal phase is applied directly to the corresponding tensor factor within the complex amplitudes of|ϕ⟩A

    Active Subspace Rotation (v∈A):If the virtual qubit is already part of the non-Clifford superposition, active set is unchanged. The diagonal phase is applied directly to the corresponding tensor factor within the complex amplitudes of|ϕ⟩A. 20 B.2 Projective Pauli Measurements After Heisenberg mapping and Pauli localization, a physical measurement is repre...

    work page