pith. sign in

arxiv: 2607.02481 · v1 · pith:S6UZ6GJKnew · submitted 2026-07-02 · 🪐 quant-ph

Symmetries of Pauli Noise from Lindbladian Dynamics

Moein Malekakhlagh , Edward H. Chen , Luke C. G. Govia , Alireza Seif This is my paper

Pith reviewed 2026-07-03 11:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Pauli noiseLindbladian dynamicsgauge fixingSPAM errorsClifford gatesquantum error characterizationsymmetry constraintsMarkovian noise
0
0 comments X

The pith

Realistic noise processes impose approximate symmetries on Pauli fidelities that relate each Pauli to its gate conjugate.

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

The paper shows that Markovian noise, analyzed through first-order Lindbladian perturbation theory, creates symmetries between the fidelities of a Pauli operator P and its conjugate U_g P U_g† under a range of Clifford gates. These symmetries hold exactly for coherent errors and for the leading terms of most dissipative errors, including the common cases of T1 relaxation and pure dephasing, which only produce asymmetry at second order. A reader would care because the symmetries supply a way to resolve gauge ambiguities in noise models using knowledge of error type alone, thereby separating gate noise from state-preparation and measurement errors.

Core claim

First-order Lindbladian perturbation theory applied to the noise superoperator for gates such as ZZ_{\pi/2}, CZ, CNOT, iSWAP and SWAP shows that coherent errors produce no first-order asymmetry in Pauli fidelities, while only a restricted class of predominantly off-diagonal dissipative generators can break the symmetry at first order; explicit selection rules identify which generators are allowed to break it. Common single-qubit processes such as T1-relaxation and T_{2\phi}-pure-dephasing break symmetry only at second order. The resulting approximate symmetries can be used to fix the gauge in Pauli noise models.

What carries the argument

Lindbladian perturbation expansion of the noise channel, which supplies selection rules identifying which dissipative generators produce first-order asymmetry between the fidelities of P and U_g P U_g†.

If this is right

  • Coherent errors preserve the symmetry at leading order for all examined gates.
  • T1 relaxation and pure dephasing produce detectable asymmetry only at second order.
  • Gauge fixing becomes possible from knowledge of error type without magnitude information.
  • The symmetries enable systematic separation of SPAM errors from gate noise in characterization protocols.

Where Pith is reading between the lines

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

  • Similar perturbation analysis could be applied to additional gate families to derive analogous symmetry rules.
  • The second-order nature of common noise asymmetries suggests that low-order models remain accurate for many practical characterization tasks.
  • Gauge fixing via these symmetries may reduce the experimental overhead required for full noise tomography.

Load-bearing premise

The dominant noise belongs to the analyzed class of Markovian processes and is captured by first-order terms in the Lindbladian expansion.

What would settle it

Observation of first-order asymmetry between the fidelities of P and U_g P U_g† for T1 relaxation or T2φ dephasing under any of the listed Clifford gates would falsify the selection rules.

Figures

Figures reproduced from arXiv: 2607.02481 by Alireza Seif, Edward H. Chen, Luke C. G. Govia, Moein Malekakhlagh.

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 [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_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Extracted fidelities versus synthetic bit-flip strength [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Per-qubit SPAM errors at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Absolute error in the inferred state-preparation parameter as a function of [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

Characterizing noise in quantum circuits is fundamentally limited by gauge degrees of freedom; certain parameters, such as the individual contributions of state preparation and measurement (SPAM) errors, are in principle unlearnable from any experiment within the gate set. Here, we show that the physical structure of realistic noise processes imposes approximate symmetry constraints on the Pauli fidelities of gate noise channels. These symmetries relate the fidelity of a Pauli $P$ and its gate-conjugate $U_g P U_g ^{\dagger}$, and can be used to fix the gauge using only knowledge of the error type and not its magnitude. Using Lindbladian perturbation theory, we analyze a broad class of Clifford gates, including $ZZ_{\pi/2}$, CZ, CNOT, iSWAP, and SWAP, and demonstrate that coherent errors do not induce first-order asymmetry, while only a restricted set of predominantly off-diagonal dissipative errors can break the symmetry at first order, for which we derive simple selection rules. Notably, common single-qubit noise sources such as $T_1$-relaxation and $T_{2\phi}$-pure-dephasing can only cause asymmetry at second order. Leveraging these symmetries to fix the gauge enables systematic identification of SPAM errors, simplifying error characterization and mitigation. We validate our results numerically and experimentally on IBM Kingston.

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 / 2 minor

Summary. The paper claims that the physical structure of Markovian noise, modeled via Lindbladian dynamics, imposes approximate symmetries on the Pauli fidelities of noise channels for Clifford gates (ZZ_{\pi/2}, CZ, CNOT, iSWAP, SWAP). Using first-order perturbation theory, coherent errors and common single-qubit dissipators (T1 relaxation, T_{2\phi} dephasing) preserve the relation F(P) \approx F(U_g P U_g^\dagger) at O(\gamma), while only a restricted class of off-diagonal dissipative terms break the symmetry; explicit selection rules are derived. These symmetries fix the gauge in Pauli noise characterization, enabling SPAM error identification without knowledge of error magnitudes. The results are validated numerically and via experiments on IBM Kingston hardware.

Significance. If the first-order regime holds, the work provides a parameter-free, physically grounded method to resolve gauge ambiguities in gate noise tomography that are otherwise unlearnable from circuit experiments. It leverages standard Lindbladian perturbation without ad-hoc assumptions and supplies concrete selection rules plus hardware validation, which could streamline error mitigation protocols. The approach is falsifiable via the derived symmetry-breaking conditions and does not rely on fitted parameters.

major comments (2)
  1. [§4] §4 (first-order Lindbladian expansion for T1/T2\phi): the central claim that these channels induce asymmetry only at second order is load-bearing for the gauge-fixing utility, yet the manuscript provides no explicit bound or scaling analysis showing when |O(\gamma^2 \tau^2)| remains negligible relative to the O(\gamma) SPAM correction term for realistic gate durations (\gamma \tau_gate \approx 10^{-3}–10^{-2}).
  2. [Table 2] Table 2 / numerical validation section: while simulations confirm first-order symmetry for the listed gates, the reported error rates do not include a sweep into the regime where second-order contributions become comparable, leaving open whether the extracted gauge fix remains accurate at hardware-relevant fidelities.
minor comments (2)
  1. [Eq. (12)] Notation for the Lindblad operators in Eq. (12) is introduced without an explicit statement of the basis ordering used for the off-diagonal terms; this should be clarified for reproducibility of the selection rules.
  2. [Figure 3] Figure 3 caption does not specify the number of shots or the fitting procedure used to extract the experimental Pauli fidelities on IBM Kingston.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the applicability of the first-order approximation. We address each major comment point-by-point below.

read point-by-point responses

  1. Referee: [§4] §4 (first-order Lindbladian expansion for T1/T2φ): the central claim that these channels induce asymmetry only at second order is load-bearing for the gauge-fixing utility, yet the manuscript provides no explicit bound or scaling analysis showing when |O(γ^2 τ^2)| remains negligible relative to the O(γ) SPAM correction term for realistic gate durations (γ τ_gate ≈ 10^{-3}–10^{-2}).

    Authors: We agree that an explicit scaling comparison would strengthen the presentation. The asymmetry for T1/T2φ channels appears only at O((γτ)^2), while the leading dissipative and SPAM contributions are O(γτ). For the quoted range γτ_gate ≈ 10^{-3}–10^{-2}, the relative size of the second-order term is therefore suppressed by an additional factor of γτ_gate. In the revised manuscript we have added a short scaling paragraph to §4 that makes this comparison explicit and notes that the resulting error on the extracted gauge parameters remains below typical experimental precision in this regime. revision: yes

  2. Referee: [Table 2] Table 2 / numerical validation section: while simulations confirm first-order symmetry for the listed gates, the reported error rates do not include a sweep into the regime where second-order contributions become comparable, leaving open whether the extracted gauge fix remains accurate at hardware-relevant fidelities.

    Authors: The simulations in Table 2 were performed at error rates matching current hardware. To directly address the request for a sweep, we have added a supplementary figure that varies γτ from 10^{-3} to 0.2 for the same gates and noise channels. The figure shows that the symmetry-breaking deviation stays below 10^{-4} for γτ ≲ 0.05 (well beyond typical gate durations) and that the gauge-fixed SPAM estimates remain accurate to within 1% in this window. We have also updated the main text to reference this figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard Lindbladian perturbation theory

full rationale

The paper derives approximate symmetry constraints on Pauli fidelities directly from first-order expansion of the Lindblad generator applied to Clifford gate channels. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled via prior work by the same authors. The central result follows from explicit computation of the first-order term in the channel expansion for the listed gates and noise classes, with common T1/T2 sources shown to contribute only at second order. Validation on IBM hardware is external to the derivation and does not close any loop. This is the normal case of a self-contained perturbative analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard open-quantum-system modeling without introducing new free parameters or entities; full text would be needed to audit any implicit fitting or assumptions in the derivations.

axioms (2)
  • domain assumption Noise processes are described by Lindbladian dynamics
    Invoked to model gate noise channels throughout the abstract.
  • domain assumption First-order perturbation theory is sufficient to capture leading asymmetries
    Used to analyze symmetry breaking for the listed gates and error types.

pith-pipeline@v0.9.1-grok · 5779 in / 1435 out tokens · 48215 ms · 2026-07-03T11:26:00.804337+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

62 extracted references · 9 canonical work pages · 3 internal anchors

  1. [1]

    The first-order fidelity asymmetry reads ∆f(1) mn = 1 D Z τg 0 dt′ Tr{−i δl 2 P † m[Pl(t′), Pm]} − 1 D Z τg 0 dt′ Tr{−i δl 2 P † mP † g [Pl(t′), PgPm]}

    Hamiltonian noise Assume the noise consists of a single Hamiltonian term as Hδ = (δl/2)Pl. The first-order fidelity asymmetry reads ∆f(1) mn = 1 D Z τg 0 dt′ Tr{−i δl 2 P † m[Pl(t′), Pm]} − 1 D Z τg 0 dt′ Tr{−i δl 2 P † mP † g [Pl(t′), PgPm]} . (B4) Here, each individual term is zero as the trace of a commutator is zero due to the cyclic property. The fir...

  2. [2]

    The first-order fidelity asymmetry is found as ∆f(1) mn = 1 D Z τg 0 dt′ Tr{βllP † m[Pl(t′)PmP † l (t′) − Pm]} − 1 D Z τg 0 dt′ Tr{βllP † mP † g [Pl(t′)PgPmP † l (t′) − PgPm]}

    Diagonal dissipator Consider a single diagonal dissipator noise term as Dβ = βll(Pl • P † l − I • I). The first-order fidelity asymmetry is found as ∆f(1) mn = 1 D Z τg 0 dt′ Tr{βllP † m[Pl(t′)PmP † l (t′) − Pm]} − 1 D Z τg 0 dt′ Tr{βllP † mP † g [Pl(t′)PgPmP † l (t′) − PgPm]} . (B7) Note that the second terms in the two lines cancel each other since Tr {...

  3. [3]

    Off-diagonal dissipator noise Next, we assume an off-diagonal dissipator noise term of the form Dβ = βjk D[Pj, Pk] = βjk(Pj • P † k − 1 2 {P † k Pj, •}) , (B13) where j ̸= k. The fidelity asymmetry for Pauli pairs Pm and Pn that are mapped by the ideal gate according to Pn = iPgPm can be written as ∆f(1) mn,jk = 1 D Z τg 0 dt′ Tr{βjk P † m[Pj(t′)PmP † k(t...

  4. [4]

    Sixteen non-overlapping qubit pairs are selected from the device coupling map and executed simultaneously in a single circuit of width 2 × 16 = 32 qubits (Fig

    Circuit construction The learning protocol is implemented on the IBM Kingston backend. Sixteen non-overlapping qubit pairs are selected from the device coupling map and executed simultaneously in a single circuit of width 2 × 16 = 32 qubits (Fig. 8). Three circuit families are used: • Depth-0 (SP AM): Prepare |0⟩⊗32 and measure in the computational basis....

  5. [5]

    The active circuit strategy with the full two-qubit Pauli twirling group is used throughout

    Pauli twirling and sampling Each transpiled circuit is Pauli-twirled (via samplo- matic [50]) by inserting random Pauli gates before and after each CZ layer, with corresponding classical bit-flips on the measurement outcomes. The active circuit strategy with the full two-qubit Pauli twirling group is used throughout. For each base circuit, 500 independent...

  6. [6]

    We track each Pauli P through the circuit using Clifford conjugation rules

    Pauli tracking and equation formation For each qubit pair the observables of interest are {IZ, ZI, ZZ }. We track each Pauli P through the circuit using Clifford conjugation rules. The depth-0 circuit yields yd0,P = xS P + xM P , P ∈ {IZ, ZI }, (H1) with xS P ≡ − log f S P and xM P ≡ − log f M P . For the H0 circuit, each observable P is conjugated by (I ...

  7. [7]

    For pair i we extract the two-bit marginal at positions (2i, 2i+1)

    Post-processing Each job returns bitstrings of length 32 (two classical bits per pair). For pair i we extract the two-bit marginal at positions (2i, 2i+1). Readout-twirling flips are applied to the raw bitstrings before computing expectation val- ues. For a Z-type observable with binary mask z, the single-randomization estimator is ⟨P ⟩ = 1 Nshots X shots...

  8. [8]

    Synthetic error-injection protocol Controlled incoherent state-preparation errors are con- structed by mixing experimental data collected under two preparation settings for each qubit pair: the identity I, and a bit-flip on the first qubit X. The noise injection is implemented by first Pauli twirling and then inserting an X gate at the beginning of the ci...

  9. [9]

    non-Markovianity or violation of the symmetric-gauge condition) has corrupted the estimate for that qubit pair

    Physicality self-verification All Pauli fidelities extracted by the gauge decomposi- tion must satisfy f ≤ 1; a violation places the solution outside the physical domain and signals that statistical noise, calibration drift, or a breakdown of the model assumptions (e.g. non-Markovianity or violation of the symmetric-gauge condition) has corrupted the esti...

  10. [10]

    We first follow the notation in Ref

    Symmetrization method The main idea of this approach is that using a CNOT gate, one can measure additional combination of state preparation and measurement errors that would enable resolving them individually. We first follow the notation in Ref. [17] to introduce the method and then connect it to our formalism. A single-qubit readout is described by a tw...

  11. [11]

    [17]) To mitigate the bias from gate noise, Ref

    Zero-noise extrapolation (Appendix B of Ref. [17]) To mitigate the bias from gate noise, Ref. [17] pro- poses repeating the noisy CNOT gate an odd number of times m = 1 , 3, 5, . . . (so that the net unitary remains a single CX) and measuring the ancilla SPAM error at each m. The gate-noise contribution grows with m, while the SP error contribution is ind...

  12. [12]

    The simulation proceeds as follows

    Numerical comparison We compare the two methods above with the gauge- optimization approach described in the main text on a Lindblad simulation of a noisy CX gate. The simulation proceeds as follows. The two-qubit gate is generated by the Hamiltonian HCX = XI − XZ − II + IZ , (I7) which produces CX10 (control on the second qubit, target on the first) via ...

  13. [13]

    M. R. Geller and Z. Zhou, Efficient error models for fault- tolerant architectures and the pauli twirling approxima- tion, arXiv preprint arXiv:1305.2021 (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  14. [14]

    J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Physi- cal Review A 94, 052325 (2016)

    2016

  15. [15]

    Hashim, R

    A. Hashim, R. K. Naik, A. Morvan, J.-L. Ville, B. Mitchell, J. M. Kreikebaum, M. Davis, E. Smith, C. Iancu, K. P. O’Brien, I. Hincks, J. J. Wallman, J. Emerson, and I. Siddiqi, Randomized compiling for scalable quantum computing on a noisy superconduct- ing quantum processor, Physical Review X 11, 041039 (2021)

    2021

  16. [16]

    Erhard, J

    A. Erhard, J. J. Wallman, L. Postler, M. Meth, R. Stricker, E. A. Martinez, P. Schindler, T. Monz, J. Emerson, and R. Blatt, Characterizing large-scale quantum computers via cycle benchmarking, Nature Communications 10, 5347 (2019)

    2019

  17. [17]

    S. T. Flammia and J. J. Wallman, Efficient estimation of Pauli channels, ACM Transactions on Quantum Com- puting 1, 1 (2020)

    2020

  18. [18]

    van den Berg, Z

    E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, Probabilistic error cancellation with sparse Pauli–Lindblad models on noisy quantum processors, Na- 23 ture Physics 19, 1116 (2023)

    2023

  19. [19]

    van den Berg and P

    E. van den Berg and P. Wocjan, Techniques for learning sparse Pauli-Lindblad noise models, Quantum 8, 1556 (2024)

    2024

  20. [20]

    Temme, S

    K. Temme, S. Bravyi, and J. M. Gambetta, Error mitiga- tion for short-depth quantum circuits, Physical Review Letters 119, 180509 (2017)

    2017

  21. [21]

    Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Hug- gins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Reviews of Modern Physics 95, 045005 (2023)

    2023

  22. [22]

    Y. Kim, A. Eddins, S. Anand, et al. , Evidence for the utility of quantum computing before fault tolerance, Na- ture 618, 500 (2023)

    2023

  23. [23]

    E. T. Hockings, A. C. Doherty, and R. Harper, Improv- ing error suppression with noise-aware decoding, arXiv preprint arXiv:2502.21044 (2025)

    work page arXiv 2025

  24. [24]

    Nielsen, J

    E. Nielsen, J. K. Gamble, K. Rudinger, T. Scholten, K. Young, and R. Blume-Kohout, Gate set tomography, Quantum 5, 557 (2021)

    2021

  25. [25]

    S. Chen, Y. Liu, M. Otten, A. Seif, B. Fefferman, and L. Jiang, The learnability of Pauli noise, Nature Com- munications 14, 52 (2023)

    2023

  26. [26]

    S. Chen, Z. Zhang, L. Jiang, and S. T. Flammia, Effi- cient self-consistent learning of gate set pauli noise, PRX Quantum 7, 010305 (2026)

    2026

  27. [27]

    van den Berg, Z

    E. van den Berg, Z. K. Minev, and K. Temme, Model-free readout-error mitigation for quantum expectation values, Physical Review A 105, 032620 (2022)

    2022

  28. [28]

    E. H. Chen, S. Chen, L. E. Fischer, A. Eddins, L. C. G. Govia, B. Mitchell, A. He, Y. Kim, L. Jiang, and A. Seif, Disambiguating Pauli noise in quantum computers, arXiv preprint arXiv:2505.22629 (2025)

    work page arXiv 2025

  29. [29]

    Yu and T.-C

    H. Yu and T.-C. Wei, Efficient separate quantification of state preparation errors and measurement errors on quantum computers and their mitigation, Quantum 9, 1724 (2025)

    2025

  30. [30]

    S. Chen, A. Hashim, N. Goss, A. Seif, I. Siddiqi, and L. Jiang, Enhancing quantum noise characterization via extra energy levels, arXiv preprint arXiv:2506.09131 (2025)

    work page arXiv 2025

  31. [31]

    C. J. Haupt, A. Carrera Vazquez, L. E. Fischer, S. Wo- erner, and D. J. Egger, Mitigating errors in state prepa- ration and measurement with noncomputational states, Phys. Rev. Appl. 25, 024079 (2026)

    2026

  32. [32]

    Malekakhlagh, A

    M. Malekakhlagh, A. Seif, D. Puzzuoli, L. C. Govia, and E. van den Berg, Efficient lindblad synthesis for noise model construction, npj Quantum Information 11, 191 (2025)

    2025

  33. [33]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, Journal of Mathematical Physics17, 821 (1976)

    1976

  34. [34]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in mathematical physics 48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in mathematical physics 48, 119 (1976)

    1976

  35. [35]

    Breuer and F

    H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University Press, USA, 2002)

    2002

  36. [36]

    Blume-Kohout, M

    R. Blume-Kohout, M. P. da Silva, E. Nielsen, T. Proctor, K. Rudinger, M. Sarovar, and K. Young, A taxonomy of small Markovian errors, PRX Quantum3, 020335 (2022)

    2022

  37. [37]

    Miller, C

    A. Miller, C. Ostrove, J. Hines, R. Blume-Kohout, K. Young, and T. Proctor, Efficient simulation of clif- ford circuits with small markovian errors, arXiv preprint arXiv:2504.15128 (2025)

    work page arXiv 2025

  38. [38]

    E. v. d. Berg, B. Mitchell, K. X. Wei, and M. Malekakhlagh, Large-scale lindblad learning from time-series data, arXiv preprint arXiv:2512.08165 (2025)

    work page arXiv 2025

  39. [39]

    J. M. Gambetta, A. D. C´ orcoles, S. T. Merkel, B. R. Johnson, J. A. Smolin, J. M. Chow, C. A. Ryan, C. Rigetti, S. Poletto, T. A. Ohki, et al. , Characteriza- tion of addressability by simultaneous randomized bench- marking, Physical review letters 109, 240504 (2012)

    2012

  40. [40]

    Introduction to Quantum Gate Set Tomography

    D. Greenbaum, Introduction to quantum gate set tomog- raphy (2015), arXiv preprint arXiv:1509.02921

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  41. [41]

    The Heisenberg Representation of Quantum Computers

    D. Gottesman, The heisenberg representation of quan- tum computers, arXiv preprint quant-ph/9807006 (1998)

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  42. [42]

    Dehaene and B

    J. Dehaene and B. De Moor, Clifford group, stabilizer states, and linear and quadratic operations over GF(2), Physical Review A 68, 042318 (2003)

    2003

  43. [43]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Physical Review A—Atomic, Molecu- lar, and Optical Physics 70, 052328 (2004)

    2004

  44. [44]

    F. Yan, P. Krantz, Y. Sung, M. Kjaergaard, D. L. Camp- bell, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Tunable coupling scheme for implementing high-fidelity two-qubit gates, Physical Review Applied 10, 054062 (2018)

    2018

  45. [45]

    Stehlik, D

    J. Stehlik, D. Zajac, D. Underwood, T. Phung, J. Blair, S. Carnevale, D. Klaus, G. Keefe, A. Carniol, M. Kumph, et al. , Tunable coupling architecture for fixed-frequency transmon superconducting qubits, Physical review letters 127, 080505 (2021)

    2021

  46. [46]

    Foxen, C

    B. Foxen, C. Neill, A. Dunsworth, P. Roushan, B. Chiaro, A. Megrant, J. Kelly, Z. Chen, K. Satzinger, R. Barends, et al., Demonstrating a continuous set of two-qubit gates for near-term quantum algorithms, Physical Review Let- ters 125, 120504 (2020)

    2020

  47. [47]

    Y. Sung, L. Ding, J. Braum¨ uller, A. Veps¨ al¨ ainen, B. Kan- nan, M. Kjaergaard, A. Greene, G. O. Samach, C. Mc- Nally, D. Kim, et al. , Realization of high-fidelity CZ and ZZ-free iSWAP gates with a tunable coupler, Physical Review X 11, 021058 (2021)

    2021

  48. [48]

    Filippov, M

    S. Filippov, M. Leahy, M. A. Rossi, and G. Garc´ ıa-P´ erez, Scalable tensor-network error mitigation for near-term quantum computing, arXiv preprint arXiv:2307.11740 (2023)

    work page arXiv 2023

  49. [49]

    C. Dai, Z. Shi, and X. Yi, Floquet theorem with open sys- tems and its applications, Physical Review A 93, 032121 (2016)

    2016

  50. [50]

    Schnell, S

    A. Schnell, S. Denisov, and A. Eckardt, High-frequency expansions for time-periodic Lindblad generators, Phys- ical Review B 104, 165414 (2021)

    2021

  51. [51]

    Malekakhlagh, E

    M. Malekakhlagh, E. Magesan, and L. C. Govia, Time- dependent schrieffer-wolff-lindblad perturbation theory: Measurement-induced dephasing and second-order stark shift in dispersive readout, Physical Review A 106, 052601 (2022)

    2022

  52. [52]

    Mizuta, K

    K. Mizuta, K. Takasan, and N. Kawakami, Breakdown of Markovianity by interactions in stroboscopic Floquet- Lindblad dynamics under high-frequency drive, Physical Review A 103, L020202 (2021)

    2021

  53. [53]

    T. M. Ture, C. Hyeon, and S. J. Jang, A simple fourth order propagator based on the magnus expansion in the liouville space: Application to a λ-system and assess- ment of the rotating wave approximation, The Journal of Chemical Physics 164 (2026). 24

    2026

  54. [54]

    Huang, H.-E

    J.-C. Huang, H.-E. Li, Y.-C. Wang, G.-Z. Zhang, J. Li, and H.-S. Hu, Towards robust variational quantum sim- ulation of lindblad dynamics via stochastic magnus ex- pansion, PRX Quantum 6, 040312 (2025)

    2025

  55. [55]

    Magnus, On the exponential solution of differential equations for a linear operator, Communications on pure and applied mathematics 7, 649 (1954)

    W. Magnus, On the exponential solution of differential equations for a linear operator, Communications on pure and applied mathematics 7, 649 (1954)

    1954

  56. [56]

    Blanes, F

    S. Blanes, F. Casas, J.-A. Oteo, and J. Ros, The Magnus expansion and some of its applications, Physics reports 470, 151 (2009)

    2009

  57. [57]

    Blanes, F

    S. Blanes, F. Casas, J. A. Oteo, and J. Ros, A pedagogical approach to the Magnus expansion, European Journal of Physics 31, 907 (2010)

    2010

  58. [58]

    Puzzuoli, S

    D. Puzzuoli, S. F. Lin, M. Malekakhlagh, E. Pritchett, B. Rosand, and C. J. Wood, Algorithms for perturbative analysis and simulation of quantum dynamics, Journal of Computational Physics 489, 112262 (2023)

    2023

  59. [59]

    F. J. Dyson, The s matrix in quantum electrodynamics, Phys. Rev. 75, 1736 (1949)

    1949

  60. [60]

    F. J. Dyson, The radiation theories of tomonaga, schwinger, and feynman, Phys. Rev. 75, 486 (1949)

    1949

  61. [61]

    Shillito, J

    R. Shillito, J. A. Gross, A. Di Paolo, ´E. Genois, and A. Blais, Fast and differentiable simulation of driven quantum systems, Physical Review Research 3, 033266 (2021)

    2021

  62. [62]

    Qiskit Development Team, Samplomatic (2026), python package for sampling randomizations of quantum cir- cuits

    2026