pith. sign in

arxiv: 2511.21225 · v2 · submitted 2025-11-26 · 🪐 quant-ph

Phase Estimation with Compressed Controlled Time Evolution

Erenay Karacan This is my paper

Pith reviewed 2026-05-17 05:14 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum phase estimationcontrolled time evolutioncircuit compressionquantum simulationtranslationally invariant Hamiltoniansiterative phase estimationspin systems
0
0 comments X

The pith

A compression protocol allows controlled time evolution for translationally invariant local Hamiltonians to be encoded in quantum circuits with near-optimal depth scaling and additive control overhead.

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

The paper develops a method to compress the controlled time evolution operator used in quantum simulation algorithms. This compression targets Hamiltonians that are local and translationally invariant, which are common in quantum spin systems. By achieving a circuit depth that scales nearly optimally with time t as O(t polylog(t N/ε)), it addresses the bottleneck of controlled operations. The approach reduces the control overhead from multiplying the circuit size to adding only a small factor. This makes advanced techniques like iterative quantum phase estimation practical on current hardware for small lattices.

Core claim

The central claim is that a compression protocol encodes the controlled time evolution operator of translationally invariant, local Hamiltonians into a quantum circuit with depth O(t polylog(t N/ε)) while changing the control overhead from a multiplicative to an additive factor.

What carries the argument

The compression protocol for the controlled time evolution operator, which reduces control cost to additive while maintaining near-optimal time scaling.

If this is right

  • It enables Iterative Quantum Phase Estimation using as few as 414 CNOT gates for a 6x6 triangular lattice frustrated spin system.
  • It achieves ground state energy errors below 1% with about 1.5% variation on a 4x4 triangular lattice using noisy emulators of trapped-ion devices.
  • The reduced overhead allows efficient implementation of quantum algorithms that rely on controlled time evolution for larger systems.
  • The near-optimal scaling supports simulation times t with polylogarithmic factors in system size N and precision ε.

Where Pith is reading between the lines

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

  • Similar compression ideas could apply to other quantum algorithms beyond phase estimation that use controlled evolutions.
  • Testing on non-translationally invariant systems might reveal if the protocol can be generalized with additional overhead.
  • Hardware implementations on larger lattices could demonstrate scaling advantages in real devices.

Load-bearing premise

The compression protocol is designed specifically for translationally invariant and local Hamiltonians.

What would settle it

Directly comparing the gate count and depth of the compressed controlled evolution circuit against the uncompressed version on a small translationally invariant local Hamiltonian model would verify if the overhead is additive and the depth scaling holds.

Figures

Figures reproduced from arXiv: 2511.21225 by Erenay Karacan.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ansatz employed to optimize a TICC cir [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

Many optimally scaling quantum simulation algorithms employ controlled time evolution of the Hamiltonian, which is typically the major bottleneck for their efficient implementation. This work establishes a compression protocol for encoding the controlled time evolution operator of translationally invariant, local Hamiltonians into a quantum circuit. It achieves a near-optimal in time $t$ scaling for circuit depth $\mathcal{O}(t \text{ polylog}(t N/\epsilon))$, while reducing the control overhead from a multiplicative to an additive factor. We report that this compression protocol enables the implementation of Iterative Quantum Phase Estimation with as few as 414 CNOT gates for a frustrated quantum spin system on a 6$\times$6 triangular lattice and delivers ground state energy errors below 1% (with $\pm$ 1.5% variation, calculated with a hardware noise aware pipeline) on a 4$\times$4 triangular lattice using the noisy emulator of the Quantinuum H2 trapped ion device.

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

0 major / 3 minor

Summary. The manuscript presents a compression protocol for encoding the controlled time evolution operator of translationally invariant, local Hamiltonians into a quantum circuit. It claims a near-optimal circuit depth scaling of O(t polylog(t N/ε)) while reducing control overhead from multiplicative to additive. The protocol is applied to Iterative Quantum Phase Estimation, with concrete demonstrations reporting 414 CNOT gates for a 6×6 triangular lattice frustrated spin system and ground-state energy errors below 1% (with ±1.5% variation) on a 4×4 lattice using a hardware-noise-aware pipeline on the Quantinuum H2 emulator.

Significance. If the compression protocol and its scaling hold, the work provides a targeted optimization for a common bottleneck in quantum simulation and phase estimation algorithms for translationally invariant systems. The reduction to additive control overhead and the reported low gate counts on modest lattices could improve feasibility for near-term devices simulating many-body physics. The numerical results on triangular lattices supply concrete, hardware-relevant evidence, though the significance would increase with explicit comparisons to uncompressed baselines and broader Hamiltonian classes.

minor comments (3)
  1. Abstract: the numerical claims (414 CNOT gates, sub-1% error) are presented without any reference to the compression construction, error analysis, or circuit details; adding one sentence on the protocol's key steps would improve accessibility.
  2. The ±1.5% variation on the 4×4 energy error is reported but its origin (statistical sampling, noise model, or fitting) is not specified; a brief clarification in the results section would aid reproducibility.
  3. The scaling O(t polylog(t N/ε)) is stated as near-optimal; a short comparison to the information-theoretic lower bound or to standard Trotter-based controlled evolution would strengthen the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. We have addressed the suggestion to strengthen the manuscript with additional comparisons and clarifications on scope.

read point-by-point responses

  1. Referee: The significance would increase with explicit comparisons to uncompressed baselines and broader Hamiltonian classes.

    Authors: We agree that direct comparisons would improve clarity. In the revised version we have added a new subsection (Section IV.C) that reports circuit depth and CNOT counts for both the compressed protocol and the standard uncompressed controlled-evolution implementation on the same 4×4 and 6×6 triangular lattices, confirming the additive-control advantage. Regarding broader classes, the protocol fundamentally requires translational invariance and locality to achieve the stated compression; we have now stated this scope limitation explicitly in the introduction and added a short paragraph in the conclusions discussing why the same compression does not directly extend to non-translationally-invariant or long-range Hamiltonians. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents a constructive compression protocol for encoding the controlled time evolution operator of translationally invariant local Hamiltonians. The claimed near-optimal circuit depth scaling O(t polylog(t N/ε)) and reduction of control overhead to an additive factor are direct outcomes of this algorithmic construction rather than any fitted parameters or self-referential definitions. Numerical validations, such as gate counts on 6×6 lattices and energy errors on 4×4 lattices, serve as external checks on the protocol's performance and do not define or force the scaling claims. No load-bearing self-citations, uniqueness theorems imported from prior work, or ansatzes smuggled via citation are indicated; the derivation remains self-contained against the stated assumptions and benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the Hamiltonians are translationally invariant and local; no free parameters, invented entities, or additional axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Hamiltonians are translationally invariant and local
    Explicitly stated in the abstract as the setting for which the compression protocol is established.

pith-pipeline@v0.9.0 · 5449 in / 1353 out tokens · 85944 ms · 2026-05-17T05:14:59.313582+00:00 · methodology

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.

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. Tensor-based phase difference estimation on time series analysis

    quant-ph 2026-01 unverdicted novelty 6.0

    Tensor-network compression of nearest-neighbor circuits plus four-type measurements yields 0.4-4.7% error on 8-qubit Hubbard energy gaps and enables QPE-type runs on IBM devices up to 52 qubits with over 4000 two-qubit gates.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    TICC construction reduces control-overhead to an additive term: Sec. III

    work page

  2. [2]

    IV; Table II

    Practical efficiency, lower gate counts and/or higher accuracies: Sec. IV; Table II

    work page

  3. [3]

    QPE study: sub-percent accuracy energy esti- mates with quantified error bars: Sec. IV; Fig. 3

    work page

  4. [4]

    Scope and Limitations: Sec. V

    work page

  5. [5]

    small to large system trans- ferability

    Gate count scaling:O(tpolylog(tN/ϵ))with numerical support: App. A; Fig. A2. II. BACKGROUND In order to encode the time evolution operator of a generic Hamiltonian into a quantum circuit, there is a fundamental lower bound for the required num- ber of queries to be made to an oracle, encoding the Hamiltonian. This lower bound scales at least lin- early wi...

    work page 2000

  6. [6]

    S. R. White, Density matrix formulation for quan- tum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

    work page 1992

  7. [7]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science355, 602–606 (2017)

    work page 2017

  8. [8]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Computational com- plexity and fundamental limitations to fermionic quantum monte carlo simulations, Phys. Rev. Lett. 94, 170201 (2005)

    work page 2005

  9. [9]

    E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Continuous- time monte carlo methods for quantum impurity models, Rev. Mod. Phys.83, 349 (2011)

    work page 2011

  10. [10]

    R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics21, 467 (1982)

    work page 1982

  11. [11]

    A. M. Dalzell, S. McArdle, M. Berta, P. Bienias, C.- F. Chen, A. Gilyén, C. T. Hann, M. J. Kastoryano, E.T.Khabiboulline, A.Kubica, G.Salton, S.Wang, and F. G. S. L. Brandão,Quantum Algorithms: A Survey of Applications and End-to-end Complexities (Cambridge University Press, 2025)

    work page 2025

  12. [12]

    Babbush, R

    R. Babbush, R. King, S. Boixo, W. Huggins, T. Khattar, G. H. Low, J. R. McClean, T. O’Brien, and N. C. Rubin, The grand challenge of quantum applications (2025), arXiv:2511.09124 [quant-ph]

    work page arXiv 2025

  13. [13]

    Arnault, P

    P. Arnault, P. Arrighi, S. Herbert, E. Kasnetsi, and T. Li, A typology of quantum algorithms (2024), arXiv:2407.05178 [quant-ph]

    work page arXiv 2024

  14. [14]

    Love, Alán Aspuru-Guzik, and Jeremy L

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a pho- tonic quantum processor, Nature Communications 5, 10.1038/ncomms5213 (2014)

    work page doi:10.1038/ncomms5213 2014

  15. [15]

    J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hy- brid quantum-classical algorithms, New Journal of Physics18, 023023 (2016)

    work page 2016

  16. [16]

    Quantum Computation by Adiabatic Evolution

    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum Computation by Adiabatic Evolution (2000), arXiv:quant-ph/0001106 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  17. [17]

    T.AlbashandD.A.Lidar,Adiabaticquantumcom- putation, Rev. Mod. Phys.90, 015002 (2018)

    work page 2018

  18. [19]

    G. H. Low and I. L. Chuang, Hamiltonian Simula- tion by Qubitization, Quantum3, 163 (2019)

    work page 2019

  19. [20]

    Farhi and S

    E. Farhi and S. Gutmann, An analog analogue of a digital quantum computation (1996), arXiv:quant- ph/9612026 [quant-ph]

    work page arXiv 1996

  20. [21]

    Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996)

    work page 1996

  21. [22]

    A. Y. Kitaev, Quantum measurements and the abelian stabilizer problem (1995), arXiv:quant- ph/9511026 [quant-ph]

    work page arXiv 1995

  22. [23]

    Cleve, A

    R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Quantum algorithms revisited, Proceedings of the 16 Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences454, 339–354 (1998)

    work page 1998

  23. [24]

    H. Ni, H. Li, and L. Ying, On low-depth algorithms for quantum phase estimation, Quantum7, 1165 (2023)

    work page 2023

  24. [25]

    Y. Dong, L. Lin, and Y. Tong, Ground-state prepa- ration and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue trans- formation of unitary matrices, PRX Quantum3, 040305 (2022)

    work page 2022

  25. [26]

    Karacan, Y

    E. Karacan, Y. Chen, and C. B. Mendl, Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering, Quantum9, 1624 (2025)

    work page 2025

  26. [27]

    Karacan, C

    E. Karacan, C. Mc Keever, M. Foss-Feig, D. Hayes, and M. Lubasch, Filter-enhanced adiabatic quan- tum computing on a digital quantum processor, Phys. Rev. Res.7, 033153 (2025)

    work page 2025

  27. [28]

    Gilyén, Y

    A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and be- yond: exponential improvements for quantum ma- trix arithmetics, inProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Comput- ing, STOC ’19 (ACM, 2019) p. 193–204

    work page 2019

  28. [29]

    Kimmel, C

    S. Kimmel, C. Y.-Y. Lin, G. H. Low, M. Ozols, and T. J. Yoder, Hamiltonian simulation with op- timal sample complexity, npj Quantum Information 3, 10.1038/s41534-017-0013-7 (2017)

    work page doi:10.1038/s41534-017-0013-7 2017

  29. [30]

    M. S. J. Tepaske, D. Hahn, and D. J. Luitz, Optimal compression of quantum many-body time evolution operatorsintobrickwallcircuits,SciPostPhysics14, 10.21468/scipostphys.14.4.073 (2023)

    work page doi:10.21468/scipostphys.14.4.073 2023

  30. [31]

    R.Mansuroglu, T.Eckstein, L.Nützel, S.A.Wilkin- son, and M. J. Hartmann, Variational hamiltonian simulation for translational invariant systems via classicalpre-processing,QuantumScienceandTech- nology8, 025006 (2023)

    work page 2023

  31. [32]

    S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Pollmann, Real- and imaginary-time evolution with compressed quantum circuits, PRX Quantum 2, 010342 (2021)

    work page 2021

  32. [33]

    Kökcü, D

    E. Kökcü, D. Camps, L. Bassman Oftelie, J. K. Fre- ericks, W. A. de Jong, R. Van Beeumen, and A. F. Kemper, Algebraic compression of quantum circuits forhamiltonianevolution,Phys.Rev.A105,032420 (2022)

    work page 2022

  33. [34]

    Kotil, R

    A. Kotil, R. Banerjee, Q. Huang, and C. B. Mendl, Riemannian quantum circuit optimization for Hamiltonian simulation, J. Phys. A: Math. Theor.57, 135303 (2024)

    work page 2024

  34. [35]

    I. N. M. Le, S. Sun, and C. B. Mendl, Riemannian quantum circuit optimization based on matrix prod- uct operators (2025), arXiv:2501.08872 [quant-ph]

    work page arXiv 2025

  35. [36]

    Barenco, C

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Physical Review A52, 3457–3467 (1995)

    work page 1995

  36. [37]

    Encoding electronic spectra in quantum circuits with linear t complexity

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear t complexity, Physical Review X8, 10.1103/physrevx.8.041015 (2018)

    work page doi:10.1103/physrevx.8.041015 2018

  37. [38]

    Balents, Spin liquids in frustrated magnets, Na- ture464, 199 (2010)

    L. Balents, Spin liquids in frustrated magnets, Na- ture464, 199 (2010)

    work page 2010

  38. [39]

    D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, Efficient quantum algorithms for simulat- ing sparse hamiltonians, Communications in Math- ematical Physics270, 359–371 (2006)

    work page 2006

  39. [40]

    S. Gu, R. D. Somma, and B. Şahinoğlu, Fast- forwarding quantum evolution, Quantum5, 577 (2021)

    work page 2021

  40. [41]

    Suzuki, General theory of fractal path integrals with applications to many-body theories and statis- tical physics, J

    M. Suzuki, General theory of fractal path integrals with applications to many-body theories and statis- tical physics, J. Math. Phys.32, 400 (1991)

    work page 1991

  41. [43]

    Wiebe, D

    N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders, Higher order decompositions of ordered operator ex- ponentials, Journal of Physics A: Mathematical and Theoretical43, 065203 (2010)

    work page 2010

  42. [44]

    A. A. Avtandilyan and W. V. Pogosov, Optimal- order trotter–suzuki decomposition for quantum simulation on noisy quantum computers, Quan- tum Information Processing24, 10.1007/s11128- 024-04627-z (2024)

    work page doi:10.1007/s11128- 2024

  43. [45]

    E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Communications in Mathematical Physics28, 251 (1972)

    work page 1972

  44. [47]

    M. A. Nielsen and I. L. Chuang,Quantum Compu- tation and Quantum Information, 10th ed. (Cam- bridge University Press, Cambridge, 2010)

    work page 2010

  45. [48]

    B. F. Schiffer, D. S. Wild, N. Maskara, M. D. Lukin, and J. I. Cirac, Hardware-efficient quan- tum phase estimation via local control (2025), arXiv:2506.18765 [quant-ph]

    work page arXiv 2025

  46. [49]

    R. Iten, R. Colbeck, E. Malvetti,et al., Uni- versalqcompiler: Synthesizing arbitrary quantum computations,https://github.com/Q-Compiler/ UniversalQCompiler(2020), accessed: 2025-11-14

    work page 2020

  47. [50]

    Kanno, K

    S. Kanno, K. Sugisaki, H. Nakamura, H. Yamauchi, R. Sakuma, T. Kobayashi, Q. Gao, and N. Ya- mamoto, Tensor-based quantum phase difference estimation for large-scale demonstration, Proceed- ings of the National Academy of Sciences122, e2425026122 (2025), arXiv:2408.04946 [quant-ph]

    work page arXiv 2025

  48. [51]

    J.Zhang, J.Vala, S.Sastry,andK.B.Whaley,Mini- mumconstructionoftwo-qubitquantumoperations, Phys. Rev. Lett.93, 020502 (2004)

    work page 2004

  49. [52]

    K. X. Wei, I. Lauer, E. Pritchett, W. Shanks, D. C. McKay, and A. Javadi-Abhari, Native two-qubit gates in fixed-coupling, fixed-frequency transmons beyond cross-resonance interaction, PRX Quantum 5, 020338 (2024). 17

    work page 2024

  50. [53]

    J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, Quantum algorithm for simulating real time evolution of lattice hamiltonians, SIAM Journal on Computing52, FOCS18 (2021)

    work page 2021

  51. [55]

    M. B. Hastings and T. Koma, Spectral gap and ex- ponential decay of correlations, Communications in Mathematical Physics265, 781–804 (2006)

    work page 2006

  52. [56]

    M. B. Hastings, Locality in quantum systems (2010), arXiv:1008.5137 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  53. [57]

    D. W. Berry, A. M. Childs, and R. Kothari, Hamil- tonian simulation with nearly optimal dependence on all parameters, in2015 IEEE 56th Annual Sym- posium on Foundations of Computer Science(IEEE,

    work page

  54. [58]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre,Opti- mization Algorithms on Matrix Manifolds(Prince- ton University Press, Princeton, NJ, 2008)

    work page 2008

  55. [59]

    Okazaki, Y

    Y. Atia and D. Aharonov, Fast-forwarding of hamiltonians and exponentially precise measure- ments, Nature Communications8, 10.1038/s41467- 017-01637-7 (2017)

    work page doi:10.1038/s41467- 2017

  56. [60]

    Explicit quantum circuits for block encodings of certain sparse matrices.arXiv preprint arXiv:2203.10236, 2022

    D. Camps, L. Lin, R. V. Beeumen, and C. Yang, Ex- plicitquantumcircuitsforblockencodingsofcertain sparse matrices (2023), arXiv:2203.10236 [quant- ph]

    work page arXiv 2023

  57. [61]

    Gilyén, Y

    A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and be- yond: Exponential improvements for quantum ma- trix arithmetics, inProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC ’19)(ACM, 2019) pp. 193–204

    work page 2019

  58. [62]

    B. D. Clader, A. M. Dalzell, N. Stamatopoulos, G. Salton, M. Berta, and W. J. Zeng, Quantum re- sources required to block-encode a matrix of classi- cal data, IEEE Transactions on Quantum Engineer- ing3, 1–23 (2022)

    work page 2022

  59. [63]

    Ding and L

    Z. Ding and L. Lin, Simultaneous estimation of mul- tiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers, Quan- tum7, 1136 (2023)

    work page 2023

  60. [65]

    M. R. Geller and Z. Zhou, Efficient error models for fault-tolerant architectures and the pauli twirling approximation, Phys. Rev. A88, 012314 (2013)

    work page 2013

  61. [66]

    J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Physical Review A94, 10.1103/phys- reva.94.052325 (2016)

    work page doi:10.1103/phys- 2016

  62. [67]

    G. Liu, Z. Xie, Z. Xu, and X. Ma, Group twirling and noise tailoring for multiqubit controlled phase gates, Phys. Rev. Res.6, 043221 (2024)

    work page 2024

  63. [68]

    Ryan-Anderson, J

    C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernogu- zov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, Realization of real-time fault-tolerant quantum error correction, Phys. Rev. X11, 041058 (2021)

    work page 2021

  64. [69]

    Quantinuum Ltd., System Model H2 Emulators, https://docs.quantinuum.com/systems/user_ guide/emulator_user_guide/emulators/h2_ emulators.html(2025), accessed: 2025-09-27

    work page 2025