Pith. sign in

REVIEW 4 major objections 3 minor 4 cited by

Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For odd-prime qudits, injected magic follows a two-term formula whose critical doping rate is set by the local dimension, and the chaos threshold sits at exactly twice that rate.

desk verdict T-gate branch is solid, but the generic Z_theta formulas in Eq. (20) are wrong at theta=0 and the paper needs an erratum before those claims can be trusted. read the letter →

arxiv 2506.02127 v5 pith:WEKR67BI submitted 2025-06-02 quant-ph cond-mat.stat-mech

Beatrice Magni , Xhek Turkeshi This is my paper
classification quant-phcond-mat.stat-mech PACS 03.67.-a03.67.Lx05.45.Mt
keywords dopedCliffordcircuitsquditmagicgeneralizedstabilizerentropiesWeingartencalculusreplicatensornetworksdynamicalphasetransitionout-of-time-orderedcorrelatorsnonstabilizerness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Across odd-prime-dimensional qudits, the paper aims to show that doping a random Clifford circuit with non-Clifford gates drives a universal transition in quantum magic, or non-stabilizerness. With global Clifford layers and a reference T-type gate, the averaged generalized stabilizer purity takes the two-term form $\zeta_\Omega \simeq \zeta_\Omega^{\mathrm{Haar}} + [\zeta_\Omega(|K\rangle)]^{N_T}$, so the magic density grows linearly with the doping ratio $q=N_T/N$ until $q_c=-\log d/\log \zeta_\Omega(|K\rangle)$, above which it saturates to $\log d$. The paper's distinctive claim is that locality is irrelevant to this threshold: brickwork and staircase qutrit circuits match the global-doping formula exactly. The same replica calculation shows that anticoncentration and R\'enyi entanglement approach Haar values once $N_T\sim \log N$, whereas the six-point out-of-time-order correlator needs an extensive number of dopings, with threshold $q_c^{\mathrm{OTOC}}=2q_c$. If correct, the work turns 'how many non-Clifford resources does a qudit circuit need?' into a parameter-free prediction from single-qudit data, and it separates the onset of maximal state magic from the onset of operator-level chaos.

What carries the argument

The carrying object is the replica transfer matrix $B_{\tau,\sigma}$ (Eq. 17), built from Clifford Weingarten functions, the Clifford commutant Gram matrix, and a doped Gram matrix that inserts the non-Clifford gate $K$ across $k$ replicas. For $k=3$, the smallest case distinguishing qudit Clifford dynamics from Haar dynamics, the Clifford commutant splits into permutation operators and intrinsic magic operators; the paper diagonalizes the transfer matrix using the large-$N$ quasi-orthogonality $G_{\pi\sigma}(D)\simeq D^k\delta_{\pi\sigma}+O(D^{k-1})$. This yields $B_{\pi\sigma}=\delta_{\pi\sigma}\{1\text{ for }\sigma\in S_k,\;\zeta_\Omega(|K\rangle)\text{ for intrinsic }\sigma\}$ (Eq. 24), so one sector preserves permutations with unit weight and the other multiplies by the single-gate magic at each doping. All later results, including magic saturation, $N_T\sim\log N$ anticoncentration, and the $2q_c$ OTOC threshold, follow from exponentiating this diagonal two-sector matrix.

What would settle it

An exact Clifford Weingarten computation without the diagonal approximation for a moderate system, e.g. $N=10$ qutrits with $T_3$ dopings, yields the parameter-free prediction in Eq. (20); comparing that prediction with the brute-force generalized stabilizer purity at several doping rates around $q_c$ would settle whether the $O(D^{k-1})$ off-diagonal terms are negligible. Alternatively, choose two $Z_\theta$ gates with equal single-qudit magic but different $\xi_d(\theta)$ and check whether the magic transition remains identical at finite $N$.

Watch

Extended reading notes

Core claim

In the paper's own terms, the central discovery is that for any generalized stabilizer purity $\zeta_\Omega$ in a doped Clifford circuit on odd-prime qudits, the circuit average obeys $\zeta_\Omega \simeq \zeta_\Omega^{\mathrm{Haar}} + [\zeta_\Omega(|K\rangle)]^{N_T}$ (Eq. 23) in the large-$N$ limit, where $\zeta_\Omega^{\mathrm{Haar}}$ is the Haar-random benchmark and $\zeta_\Omega(|K\rangle)$ is the single-qudit magic of the injected gate. From this formula the magic density $m_\Omega = M_\Omega/N$ is linear in the doping rate for $q<q_c$ and saturates to $\log d$ for $q>q_c$, with $q_c=-\log d/\log\zeta_\Omega(|K\rangle)$ (Eq. 28). The paper verifies the qutrit case against exact brickwork and staircase numerics, obtaining agreement that makes the magic growth independent of circuit geometry. It further derives exact expressions for inverse participation ratios and R\'enyi purities showing that $N_T\sim\log N$ non-Clifford gates approximate Haar values to precision $\varepsilon$, and for the averaged six-point OTOC showing a decay to Haar behavior only when the doping density is extensive, with threshold exactly $2q_c$. Together these results are presented as a unified, dimension-dependent picture of complexity growth in qudit doped Clifford circuits.

Load-bearing premise

The formulas assume that, for large systems, the replica-space overlaps between different Clifford-invariant operators are negligible, so the transfer matrix is diagonal; the only direct finite-size evidence offered is the qutrit brickwork and staircase match in Fig. 1(b). If those off-diagonal overlaps are not uniformly small at the sizes and observables probed, the universal formula, the critical doping rate, and the exact factor 2 in the chaos threshold would all need revision.

Editorial extensions

If this is right

  • Below $q_c$ the magic density grows linearly with the doping ratio; above it the state's magic per qudit saturates to $\log d$, marking the crossover from stabilizer-simulable to Haar-typical behavior.
  • For the qudit $T$-gate, larger $d$ lowers $q_c$, so higher-dimensional qudits reach the Haar-typical magic regime with fewer non-Clifford gates per qudit.
  • State magic spreading is insensitive to circuit layout: a periodically doped brickwork or staircase circuit reproduces the all-to-all global Clifford prediction exactly.
  • Anticoncentration and entanglement purities reach Haar values with only logarithmically many non-Clifford gates, so a circuit can look pseudo-random in these probes while remaining far from operator-level chaos.
  • True chaos, diagnosed by the six-point OTOC, requires an extensive doping density, and the chaos threshold is exactly twice the magic threshold, leaving an intermediate phase with maximal magic but no OTOC scrambling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exact factor 2 between magic and OTOC thresholds suggests a general ladder: each additional replica-pair in a higher OTOC may push the chaos threshold further above $q_c$ in odd-prime qudits, though the paper does not state this generalization.
  • Equation (23) predicts that two different non-Clifford gates with the same single-qudit magic $\zeta_\Omega(|K\rangle)$ should produce identical large-$N$ magic dynamics; this is directly testable with $Z_\theta$ rotations tuned to equal $\zeta$.
  • Because the paper leaves open the effect of noise, a natural extension is to test whether local depolarizing or dephasing noise smears the magic transition into a crossover or shifts $q_c$; that would determine whether the threshold survives on near-term qudit hardware.
  • The locality-independence of state magic contrasts with operator-space magic, where the paper notes that gate location matters; measuring operator stabilizer entropy in the same qudit circuits would expose the two behaviors side by side.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The manuscript studies doped Clifford circuits acting on N qudits of odd prime dimension d, with T non-Clifford gates K interspersed among random Clifford layers. Using replica tensor networks and Clifford Weingarten calculus, it claims exact formulas for generalized stabilizer purities at k=3 for d=3,5,7, a universal large-N formula ζ_Ω ≃ ζ_Haar + [ζ_Ω(|K>)]^{NT}, a dynamical transition at q_c = -log d/log ζ_Ω(|K>), exact inverse participation ratios and Rényi purities, a connection to antiflatness in qutrits, and a chaos threshold q_c^{OTOC} = 2 q_c from a six-point OTOC. The paper also reports numerical agreement with qutrit brickwork and staircase circuits and deposits code and data.

Significance. If the claimed results are correct, the paper would significantly extend magic and chaos diagnostics from qubits to odd-prime qudit systems, providing explicit analytic predictions and a dimension-dependent threshold for the breakdown of classical simulability. The T-gate branch is supported by deposited numerics and by local-circuit simulations, and the reproduction of known qubit limits is a genuine strength. However, the exact generic-rotation formulas contain a demonstrable normalization error, and the universal formula needs qualification; the significance is therefore conditional on a corrected derivation.

major comments (4)
  1. [3.1, Eqs. (20)-(21)] Eq. (20) with Eq. (21) fails the identity limit θ=0. For K=Z^0=I, the circuit in Eq. (12) is a random Clifford circuit and |Ψ(NT)> is a uniformly random stabilizer state for every NT, so the averaged generalized stabilizer purity must be exactly 1. Inserting d=3 and θ=0 gives ξ3(0)=(1+2)^2=9, μ3(0)=29/3, β3(0)=−52/3, and the NT=1 large-N limit of Eq. (20) is approximately 29/3, which is larger than 1. This also contradicts the stated interpretation of μ_d(θ) as the single-qudit purity ζ_Ω(|θ>), which must equal 1 at θ=0. The same defect propagates into Eq. (34) and Eq. (44). The T3-gate benchmark in Fig. 1(b) uses the ξ=0 branch and is not directly affected, but the exact generic-rotation results for d=3,5,7 are ungrounded as written.
  2. [3.1, Eq. (23)] Eq. (23) has the wrong boundary value at NT=0. For NT=0, the evolution in Eq. (12) is one random Clifford gate applied to |0>, so the ensemble average of ζ_Ω is exactly 1, whereas Eq. (23) gives 1+ζ_Ω^Haar. If the formula is intended only for NT≥1, this needs to be stated and proven; more importantly, the failure suggests that the decomposition in Eq. (26) omits a contribution that is present at small NT and should be reconciled with the exact expression in Eq. (18).
  3. [3.1, Eq. (24)] The universal formulas, including Eqs. (35), (52), and the factor 2 in Eq. (53), rest on the quasi-orthogonality approximation G_{πσ}(D)≃D^k δ_{πσ}+O(D^{k-1}) and the diagonal form of B in Eq. (24). No control is provided for the accumulation of the O(D^{k-1}) off-diagonal entries after B^{NT} with NT=O(N); such corrections can in principle shift the apparent transition. Because the only finite-size check is the qutrit k=3 magic observable in Fig. 1(b), the OTOC and IPR predictions need a direct comparison with exact transfer-matrix diagonalization at finite N, or a bound on the off-diagonal part, before the threshold q_c^{OTOC}=2q_c can be considered established.
  4. [3.1 and 3.3, Eqs. (20) and (50)] The paper states that Eqs. (20) and (50) follow by 'straightforward but involved algebra' and 'straightforward algebraic manipulation' but does not display the intermediate sums. Given that Eq. (20) demonstrably violates the θ=0 limit, the omitted algebra cannot be accepted on assertion. The symbolic notebook in Ref. [99] should be used to provide the key intermediate expressions, or the derivation should be included in an appendix, with the normalization of ζ_Ω checked at each step.
minor comments (3)
  1. [Sec. 2] The name 'Gottesmann-Knill' should be 'Gottesman-Knill'.
  2. [Sec. 3.1] The definition of T3 as Z^{1/3} is inconsistent with setting ξ_d=0 for the T3 gate, since Eq. (21) gives ξ3(1/3)=(1+2 cos π)^2=1; please clarify the exact form of the qudit T-gate used in the ξ=0 branch.
  3. [Fig. 1(b) caption] The phrase 'exact superposition' is ambiguous; consider using 'agreement' or 'collapse' instead.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central magic-spreading formula is a parameter-free reduction to single-qudit magic, benchmarked against independent numerics; only minor methodological self-citations appear.

full rationale

The derivation chain is self-contained in the relevant sense. The circuit-averaged generalized stabilizer purity is built from the replica transfer matrix B_{πσ} in Eq. (17), and the large-N result Eq. (23) follows from the quasi-orthogonality G_{πσ}(D) ≃ D^k δ_{πσ} + O(D^{k-1}) attributed to the independent commutant theory [54], not to a fit. The only input in Eq. (23) is the single-qudit purity ζ_Ω(|K>), which is a fixed property of the injected gate and is not obtained from the many-body data being predicted; the transition threshold q_c = -log d / log ζ_Ω(|K>) in Eq. (28) is a crossing condition between the Haar term and the doping term, not a fitted parameter. The paper is benchmarked externally: Eq. (20) with ξ_d=0 is compared to brickwork and staircase numerics in Fig. 1(b), with code and data deposited (Ref. [99]). Refs. [28] and [30] are self-citations for the replica tensor-network formalism and the GSE definitions, but they are methodological and are complemented by independent Schur-Weyl/commutant results [54,55]; they do not carry the load of the new prediction. A separate, non-circular concern is that the generic-rotation exact formula Eq. (20) is stated without derivation and appears internally inconsistent at θ=0: for K=I the circuit is Clifford so any generalized stabilizer purity should remain 1, whereas inserting ξ_3(0)=9 into Eqs. (20)-(21) gives ζ ≃ 29/3 for N→∞ and N_T≥1. That is a correctness/finite-size issue, not a circularity, and it mainly affects the generic-rotation branch; the T_d-gate branch (ξ=0) is supported by the deposited numerics. Overall, no equation reduces by construction to its own conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to the numerical data: d and the gate K are inputs, and ξ_d(θ) and β_d(θ) are presented as exact commutant evaluations. The load-bearing inputs are the Clifford commutant structure from prior work and the large-N quasi-orthogonality approximation. No new particles, forces, or entities are postulated.

assumptions (4)
  • standard math Schur-Weyl duality for the Clifford group: moments of random Clifford unitaries are controlled by the Clifford commutant spanned by stochastic Lagrangian subspaces Σ_k(d).
    Invoked throughout Sec. 2 and used in Eq. (7) to write the replica transfer matrix; sourced from Refs. [54,55].
  • domain assumption For odd prime d and k=3, the intrinsic commutant elements Ω_d distinguish Clifford from Haar behavior, so GSE and OTOC6 are valid diagnostics.
    Used in Secs. 3.1 and 3.3; relies on the known fact that the Clifford group is a 2-design but not a 3-design for d≥3.
  • domain assumption Large-N quasi-orthogonality of the Clifford commutant: G_{πσ}(D) ≃ D^k δ_{πσ} + O(D^{k-1}), yielding the diagonal transfer matrix in Eq. (24).
    This approximation underpins the universal formulas Eq. (23), Eq. (35), and Eq. (52); its validity at all relevant N is not proven in the text.
  • domain assumption Initialization in any stabilizer state is equivalent to |0> after Clifford averaging.
    State invariance is stated in Sec. 3 before Eq. (12); it is standard but fixes the boundary condition for all replica contractions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits." pith.science (2026). https://pith.science/paper/WEKR67BI

@misc{pith2026250602127,
  author       = {Pith},
  title        = {Pith review of: Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WEKR67BI}},
  note         = {Machine review of arXiv:2506.02127}
}
abstract

We investigate the emergence of quantum complexity and chaos in doped Clifford circuits acting on qudits of odd prime dimension $d$. Using doped Clifford Weingarten calculus and a replica tensor network formalism, we derive exact results and perform large-scale simulations in regimes challenging for tensor network and Pauli-based methods. We begin by analyzing generalized stabilizer entropies, computable magic monotones in many-qudit systems, and identify a dynamical phase transition in the doping rate, marking the breakdown of classical simulability and the onset of Haar-random behavior. The critical behavior is governed by the qudit dimension and the magic content of the non-Clifford gate. Using the qudit $T$-gate as a benchmark, we show that higher-dimensional qudits converge faster to Haar-typical stabilizer entropies. For qutrits ($d=3$), analytical predictions match numerics on brickwork circuits, showing that locality plays a limited role in magic spreading. We also examine anticoncentration and entanglement growth, showing that $O(\log N)$ non-Clifford gates suffice for approximating Haar expectation values to precision $\varepsilon$, and relate antiflatness measures to stabilizer entropies in qutrit systems. Finally, we analyze out-of-time-order correlators and show that a finite density of non-Clifford gates is needed to induce chaos, with a sharp transition fixed by the local dimension, twice that of the magic transition. Altogether, these results establish a unified framework for diagnosing complexity in doped Clifford circuits and deepen our understanding of resource theories in multiqudit systems.

Figures

Figures reproduced from arXiv: 2506.02127 by the authors.

Figure 1
Figure 1. (a) GSE densities for different local dimensions (d=3,5,11) and system sizes computed with the reference Td-gate. The initial linear growth culminates in a transition, at the value qc in Eq. (28), that becomes sharper as the system size increases. After this point, the universal value log(d) is reached for N → ∞. (b) We observe the exact superposition of the numerical value of the GSE at d = 3 and k = 3 obtained fro… view at source ↗
Figure 2
Figure 2. We illustrate the scaling of Eq. (54) with the number of injected non-Clifford gates NT for local dimensions (a) d = 3 and (b) d = 5. Across different system sizes, we observe an exponential decay of ∆OTOC6, which approaches the Haar-random value OTOCHaar 6 at a characteristic doping rate q = NT /N ∼ O(1). Evaluating the critical density marking the transition from the non-chaotic to the chaotic regime (dashed lines… view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

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

  1. Higher-order Symmetric Quantum Mpemba Effect in Fragmented Systems

    quant-ph 2026-06 unverdicted novelty 7.0 of 10

    In charge- and dipole-conserving fragmented systems, charge and dipole asymmetries exhibit Mpemba-like crossings on parametrically distinct timescales, driven by frozen sectors retaining asymmetry and active sectors relaxing.

  2. Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks

    quant-ph 2025-12 conditional novelty 6.0 of 10

    Real stabilizer states follow a new "orthogonal Clifford Porter–Thomas" overlap distribution, reached by shallow real-Clifford circuits in log depth; one imaginary state recovers unitary-Clifford statistics, polylog m...

  3. Emergence of Generic Entanglement Structure in Doped Matchgate Circuits

    quant-ph 2025-07 conditional novelty 6.0 of 10

    Non-Gaussian doping of matchgate circuits restores generic entanglement growth in unitary evolution and, at extensive per-time injection rates, stabilizes a volume-law phase under measurements.

  4. Magic phase transitions in monitored gaussian fermions

    quant-ph 2025-07 conditional novelty 6.0 of 10

    Measurement-induced transitions in monitored free-fermion systems appear in the subleading logarithmic corrections to stabilizer Renyi entropies, not in the leading extensive magic.

Reference graph

Works this paper leans on

98 extracted references · 56 canonical work pages · cited by 4 Pith papers

  1. [99]

    Data and code for:

    Beatrice Magni and Xhek Turkeshi. “Data and code for: "Quantum Complexity and Chaos in Many-Qudit Doped Clifford Circuits"” (2025). (Zenodo) doi: 10.5281/zenodo.17201804. Accepted in Quantum 2025-12-21, click title to verify. Published under CC-BY 4.0.20

    work page doi:10.5281/zenodo.17201804 2025

  2. [1]

    Quantum Computing in the NISQ era and beyond

    John Preskill. “Quantum Computing in the NISQ era and beyond”. Quantum2, 79 (2018)

    2018

  3. [2]

    Random quantum circuits

    Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. “Random quantum circuits”. Annu. Rev. Condens. Matter Phys.14, 335–379 (2023)

    2023

  4. [3]

    Colloquium: Synthetic quantum matter in nonstandard geometries

    Tobias Grass, Dario Bercioux, Utso Bhattacharya, Maciej Lewenstein, Hai Son Nguyen, and Christof Weitenberg. “Colloquium: Synthetic quantum matter in nonstandard geometries”. Rev. Mod. Phys.97, 011001 (2025)

    2025

  5. [4]

    The density-matrix renormalization group in the age of matrix product states

    Ulrich Schollwöck. “The density-matrix renormalization group in the age of matrix product states”. Ann. Phys.326, 96–192 (2011)

    2011

  6. [5]

    Colloquium: Many- body localization, thermalization, and entanglement

    Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. “Colloquium: Many- body localization, thermalization, and entanglement”. Rev. Mod. Phys.91, 021001 (2019)

    2019

  7. [6]

    Many-body localization in the age of classical computing*

    Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio, Lev Vidmar, and Jakub Za- krzewski. “Many-body localization in the age of classical computing*”. Rep. Prog. Phys. 88, 026502 (2025)

    2025

  8. [7]

    Stabilizer codes and quantum error correction

    Daniel Gottesman. “Stabilizer codes and quantum error correction” (1997). arXiv:quant- ph/9705052

    arXiv 1997

Show all 98 references
  1. [8]

    Theory of fault-tolerant quantum computation

    Daniel Gottesman. “Theory of fault-tolerant quantum computation”. Phys. Rev. A57, 127– 137 (1998)

    1998

  2. [9]

    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)

    2005

  3. [10]

    Coherent errors make magic

    Xhek Turkeshi. “Coherent errors make magic”. Nat. Phys.20, 1696–1697 (2024)

    2024

  4. [11]

    Quantum entanglement growth under random unitary dynamics

    Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. “Quantum entanglement growth under random unitary dynamics”. Phys. Rev. X7, 031016 (2017)

    2017

  5. [12]

    Entanglement growth and minimal membranes in (d+ 1) random unitary circuits

    Piotr Sierant, Marco Schirò, Maciej Lewenstein, and Xhek Turkeshi. “Entanglement growth and minimal membranes in (d+ 1) random unitary circuits”. Phys. Rev. Lett.131, 230403 (2023)

    2023

  6. [13]

    Improved simulation of stabilizer circuits

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

    2004

  7. [14]

    Improvedclassicalsimulationofquantumcircuitsdominated by clifford gates

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

    2016

  8. [15]

    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

  9. [16]

    Quantum resource theories

    Eric Chitambar and Gilad Gour. “Quantum resource theories”. Rev. Mod. Phys.91, 025001 (2019)

    2019

  10. [17]

    Many-body quantum magic

    Zi-Wen Liu and Andreas Winter. “Many-body quantum magic”. PRX Quantum3, 020333 (2022)

    2022

  11. [18]

    Stabilizer rényi entropy

    Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. “Stabilizer rényi entropy”. Phys. Rev. Lett.128, 050402 (2022)

    2022

  12. [19]

    Learningt-doped stabilizer states

    Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. “Learningt-doped stabilizer states”. Quantum8, 1361 (2024)

    2024

  13. [21]

    Quantum state designs with clifford- enhanced matrix product states

    Guglielmo Lami, Tobias Haug, and Jacopo De Nardis. “Quantum state designs with clifford- enhanced matrix product states”. PRX Quantum6, 010345 (2025)

    2025

  14. [22]

    Unveiling the stabilizer group of a matrix product state

    Guglielmo Lami and Mario Collura. “Unveiling the stabilizer group of a matrix product state”. Phys. Rev. Lett.133, 010602 (2024)

    2024

  15. [23]

    Hybrid stabilizer matrix product operator

    Antonio Francesco Mello, Alessandro Santini, and Mario Collura. “Hybrid stabilizer matrix product operator”. Phys. Rev. Lett.133, 150604 (2024)

    2024

  16. [24]

    Probing quantum complexity via universal saturation of stabilizer entropies

    Tobias Haug, Leandro Aolita, and M.S. Kim. “Probing quantum complexity via universal saturation of stabilizer entropies”. Quantum9, 1801 (2025). Accepted in Quantum 2025-12-21, click title to verify. Published under CC-BY 4.0.16

    2025

  17. [25]

    Disentangling unitary dynamics with classically simulable quantum circuits

    Gerald E. Fux, Benjamin Béri, Rosario Fazio, and Emanuele Tirrito. “Disentangling unitary dynamics with classically simulable quantum circuits” (2025). arXiv:2410.09001

    2025

  18. [26]

    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

  19. [27]

    Classical simulability of clifford+t circuits with clifford- augmented matrix product states

    Zejun Liu and Bryan K. Clark. “Classical simulability of clifford+t circuits with clifford- augmented matrix product states” (2024). arXiv:2412.17209

    2024 arXiv

  20. [28]

    Anticoncen- tration in clifford circuits and beyond: From random tensor networks to pseudomagic states

    Beatrice Magni, Alexios Christopoulos, Andrea De Luca, and Xhek Turkeshi. “Anticoncen- tration in clifford circuits and beyond: From random tensor networks to pseudomagic states”. Phys. Rev. X15, 031071 (2025)

    2025

  21. [29]

    The non-clifford cost of random unitaries

    Lorenzo Leone, Salvatore F. E. Oliviero, Alioscia Hamma, Jens Eisert, and Lennart Bittel. “The non-clifford cost of random unitaries” (2025). arXiv:2505.10110

    2025 arXiv

  22. [30]

    Magic spreading in random quantum circuits

    Xhek Turkeshi, Emanuele Tirrito, and Piotr Sierant. “Magic spreading in random quantum circuits”. Nat. Commun.16, 2575 (2025)

    2025

  23. [31]

    Qudit shadow estimation based on the clifford group and the power of a single magic gate

    Chengsi Mao, Changhao Yi, and Huangjun Zhu. “Qudit shadow estimation based on the clifford group and the power of a single magic gate”. Phys. Rev. Lett.134, 160801 (2025)

    2025

  24. [32]

    Mpemba effects in quantum complexity

    Sreemayee Aditya, Alessandro Summer, Piotr Sierant, and Xhek Turkeshi. “Mpemba effects in quantum complexity” (2025). arXiv:2509.22176

    2025

  25. [33]

    Emergence of generic entanglement structure in doped match- gate circuits

    Alessio Paviglianiti, Luca Lumia, Emanuele Tirrito, Alessandro Silva, Mario Collura, Xhek Turkeshi, and Guglielmo Lami. “Emergence of generic entanglement structure in doped match- gate circuits” (2025). arXiv:2507.12526

    2025 arXiv

  26. [34]

    Quantum state designs via magic teleportation

    Hugo Lóio, Guglielmo Lami, Lorenzo Leone, Max McGinley, Xhek Turkeshi, and Jacopo De Nardis. “Quantum state designs via magic teleportation” (2025). arXiv:2510.13950

    2025

  27. [35]

    Growth and spreading of quantum resources under random circuit dynamics

    Sreemayee Aditya, Xhek Turkeshi, and Piotr Sierant. “Growth and spreading of quantum resources under random circuit dynamics” (2025). arXiv:2512.14827

    2025

  28. [36]

    Anticoncentration and state design of doped real clifford circuits and tensor networks

    Beatrice Magni, Markus Heinrich, Lorenzo Leone, and Xhek Turkeshi. “Anticoncentration and state design of doped real clifford circuits and tensor networks” (2025). arXiv:2512.15880

    2025 arXiv

  29. [37]

    Impact of clifford oper- ations on non-stabilizing power and quantum chaos

    Naga Dileep Varikuti, Soumik Bandyopadhyay, and Philipp Hauke. “Impact of clifford oper- ations on non-stabilizing power and quantum chaos” (2025). arXiv:2505.14793

    2025

  30. [38]

    Complexity transitions in chaotic quantum systems

    Gopal Chandra Santra, Alex Windey, Soumik Bandyopadhyay, Andrea Legramandi, and Philipp Hauke. “Complexity transitions in chaotic quantum systems” (2025). arXiv:2505.09707

    2025

  31. [39]

    Efficient unitary designs with a system-size independent number of non-clifford gates

    J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth. “Efficient unitary designs with a system-size independent number of non-clifford gates”. Commun. Math. Phys.397, 995–1041 (2022)

    2022

  32. [40]

    Robustness of Magic and Symmetries of the Stabiliser Polytope

    Markus Heinrich and David Gross. “Robustness of Magic and Symmetries of the Stabiliser Polytope”. Quantum3, 132 (2019)

    2019

  33. [41]

    Transitions in Entanglement Complexity in Random Cir- cuits

    Sarah True and Alioscia Hamma. “Transitions in Entanglement Complexity in Random Cir- cuits”. Quantum6, 818 (2022)

    2022

  34. [42]

    Single t gate in a clifford circuit drives transition to universal entanglement spectrum statistics

    Shiyu Zhou, Zhicheng Yang, Alioscia Hamma, and Claudio Chamon. “Single t gate in a clifford circuit drives transition to universal entanglement spectrum statistics”. SciPost Phys. 9, 087 (2020)

    2020

  35. [43]

    Stabilizerentropiesandnonstabilizernessmonotones

    TobiasHaugandLorenzoPiroli. “Stabilizerentropiesandnonstabilizernessmonotones”. Quan- tum7, 1092 (2023)

    2023

  36. [44]

    Efficient witnessing and testing of magic in mixed quantum states

    Tobias Haug and Poetri Sonya Tarabunga. “Efficient witnessing and testing of magic in mixed quantum states” (2025). arXiv:2504.18098

    2025 arXiv

  37. [45]

    Nonstabilizerness via perfect pauli sampling of matrix product states

    Guglielmo Lami and Mario Collura. “Nonstabilizerness via perfect pauli sampling of matrix product states”. Phys. Rev. Lett.131, 180401 (2023)

    2023

  38. [46]

    Stabilizer rényi entropy and confor- mal field theory

    Masahiro Hoshino, Masaki Oshikawa, and Yuto Ashida. “Stabilizer rényi entropy and confor- mal field theory” (2025). arXiv:2503.13599

    2025

  39. [47]

    Stabilizer entanglement as a magic highway

    Zong-Yue Hou, ChunJun Cao, and Zhi-Cheng Yang. “Stabilizer entanglement as a magic highway” (2025). arXiv:2503.20873

    2025 arXiv

  40. [48]

    Spectral properties versus magic generation int-doped random clifford circuits

    Dominik Szombathy, Angelo Valli, Cătălin Pa şcu Moca, János Asbóth, Lóránt Farkas, Tibor Rakovszky, and Gergely Zaránd. “Spectral properties versus magic generation int-doped random clifford circuits”. Phys. Rev. Res.7, 043080 (2025). Accepted in Quantum 2025-12-21, click titl...

    2025

  41. [49]

    Asymptotically independent fluctuations of stabilizer rényi entropy and entanglement in ran- dom unitary circuits

    Dominik Szombathy, Angelo Valli, Cătălin Pa şcu Moca, Lóránt Farkas, and Gergely Zaránd. “Asymptotically independent fluctuations of stabilizer rényi entropy and entanglement in ran- dom unitary circuits”. Phys. Rev. Res.7, 043072 (2025)

    2025

  42. [50]

    Dynamical magic transitions in monitored clifford+tcircuits

    Mircea Bejan, Campbell McLauchlan, and Benjamin Béri. “Dynamical magic transitions in monitored clifford+tcircuits”. PRX Quantum5, 030332 (2024)

    2024

  43. [51]

    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)

    2024

  44. [52]

    Disentangling scrambling and decoherence via quantum teleportation

    Beni Yoshida and Norman Y. Yao. “Disentangling scrambling and decoherence via quantum teleportation”. Phys. Rev. X9, 011006 (2019)

    2019

  45. [53]

    Quantum Chaos is Quantum

    Lorenzo Leone, Salvatore F. E. Oliviero, You Zhou, and Alioscia Hamma. “Quantum Chaos is Quantum”. Quantum5, 453 (2021)

    2021

  46. [54]

    A complete theory of the clifford commutant

    Lennart Bittel, Jens Eisert, Lorenzo Leone, Antonio A. Mele, and Salvatore F. E. Oliviero. “A complete theory of the clifford commutant” (2025). arXiv:2504.12263

    2025 arXiv

  47. [55]

    Schur–weyl duality for the clifford group with applications: Property testing, a robust hudson theorem, and de finetti representations

    David Gross, Sepehr Nezami, and Michael Walter. “Schur–weyl duality for the clifford group with applications: Property testing, a robust hudson theorem, and de finetti representations”. Commun. Math. Phys.385, 1325–1393 (2021)

    2021

  48. [56]

    Magic resources of the heisenberg picture

    Neil Dowling, Pavel Kos, and Xhek Turkeshi. “Magic resources of the heisenberg picture”. Phys. Rev. Lett.135, 050401 (2025)

    2025

  49. [57]

    Bridging entanglement and magic resources within operator space

    Neil Dowling, Kavan Modi, and Gregory A. L. White. “Bridging entanglement and magic resources within operator space”. Phys. Rev. Lett.135, 160201 (2025)

    2025

  50. [58]

    Quan- tifying nonstabilizerness through entanglement spectrum flatness

    Emanuele Tirrito, Poetri Sonya Tarabunga, Gugliemo Lami, Titas Chanda, Lorenzo Leone, Salvatore F. E. Oliviero, Marcello Dalmonte, Mario Collura, and Alioscia Hamma. “Quan- tifying nonstabilizerness through entanglement spectrum flatness”. Phys. Rev. A109, L040401 (2024)

    2024

  51. [59]

    Measuring nonstabilizerness via multifractal flatness

    Xhek Turkeshi, Marco Schirò, and Piotr Sierant. “Measuring nonstabilizerness via multifractal flatness”. Phys. Rev. A108, 042408 (2023)

    2023

  52. [60]

    Quantum computation and quantum information

    Michael A. Nielsen and Isaac L. Chuang. “Quantum computation and quantum information”. Cambridge University Press. Cambridge, New York (2010)

    2010

  53. [61]

    Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations

    Daniel Gottesman and Isaac L. Chuang. “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”. Nature402, 390–393 (1999)

    1999

  54. [62]

    Qudit versions of the qubitπ/8gate

    Mark Howard and Jiri Vala. “Qudit versions of the qubitπ/8gate”. Phys. Rev. A86, 022316 (2012)

    2012

  55. [63]

    The clifford hierarchy for one qubit or qudit

    Nadish de Silva and Oscar Lautsch. “The clifford hierarchy for one qubit or qudit” (2025). arXiv:2501.07939

    2025 arXiv

  56. [64]

    Introduction to Haar Measure Tools in Quantum Information: A Be- ginner’s Tutorial

    Antonio Anna Mele. “Introduction to Haar Measure Tools in Quantum Information: A Be- ginner’s Tutorial”. Quantum8, 1340 (2024)

    2024

  57. [65]

    Universality in the anticoncentration of chaotic quantum circuits

    Arman Sauliere, Beatrice Magni, Guglielmo Lami, Xhek Turkeshi, and Jacopo De Nardis. “Universality in the anticoncentration of chaotic quantum circuits”. Phys. Rev. B112, – (2025)

    2025

  58. [66]

    Quantum calculus

    Victor Kac and Pokman Cheung. “Quantum calculus”. Springer New York. (2002)

    2002

  59. [67]

    Third moments of qudit clifford orbits and 3-designs based on magic orbits

    Huangjun Zhu, Chengsi Mao, and Changhao Yi. “Third moments of qudit clifford orbits and 3-designs based on magic orbits” (2024). arXiv:2410.13575

    2024 arXiv

  60. [68]

    Multipartite entanglement in stabilizer tensor networks

    Sepehr Nezami and Michael Walter. “Multipartite entanglement in stabilizer tensor networks”. Phys. Rev. Lett.125, 241602 (2020)

    2020

  61. [69]

    Normal form for single-qutrit clifford+Toperators and synthesis of single-qutrit gates

    Shiroman Prakash, Akalank Jain, Bhakti Kapur, and Shubangi Seth. “Normal form for single-qutrit clifford+Toperators and synthesis of single-qutrit gates”. Phys. Rev. A98, 032304 (2018)

    2018

  62. [70]

    Fluctuations of nuclear reaction widths

    C. E. Porter and R. G. Thomas. “Fluctuations of nuclear reaction widths”. Phys. Rev.104, 483–491 (1956)

    1956

  63. [71]

    Participation spectroscopy and entangle- ment hamiltonian of quantum spin models

    David J Luitz, Nicolas Laflorencie, and Fabien Alet. “Participation spectroscopy and entangle- ment hamiltonian of quantum spin models”. J. Stat. Mech. Theory Exp.2014, P08007 (2014)

    2014

  64. [72]

    Multifractal scalings across the many- body localization transition

    Nicolas Macé, Fabien Alet, and Nicolas Laflorencie. “Multifractal scalings across the many- body localization transition”. Phys. Rev. Lett.123, 180601 (2019)

    2019

  65. [73]

    Universal behavior beyond multifrac- tality in quantum many-body systems

    David J. Luitz, Fabien Alet, and Nicolas Laflorencie. “Universal behavior beyond multifrac- tality in quantum many-body systems”. Phys. Rev. Lett.112, 057203 (2014). Accepted in Quantum 2025-12-21, click title to verify. Published under CC-BY 4.0.18

    2014

  66. [74]

    Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions

    Piotr Sierant and Xhek Turkeshi. “Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions”. Phys. Rev. Lett.128, 130605 (2022)

    2022

  67. [75]

    Anticoncentration and nonstabilizerness spreading under ergodic quantum dynamics

    Emanuele Tirrito, Xhek Turkeshi, and Piotr Sierant. “Anticoncentration and nonstabilizerness spreading under ergodic quantum dynamics”. Phys. Rev. Lett.135, 220401 (2025)

    2025

  68. [76]

    Random quantum circuits anticoncentrate in log depth

    Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G. S. L. Brandão. “Random quantum circuits anticoncentrate in log depth”. PRX Quantum3, 010333 (2022)

    2022

  69. [77]

    Random unitaries in extremely low depth

    Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. “Random unitaries in extremely low depth”. Science389, 92–96 (2025)

    2025

  70. [78]

    Anti-concentration for the unitary haar mea- sure and applications to random quantum circuits

    Bill Fefferman, Soumik Ghosh, and Wei Zhan. “Anti-concentration for the unitary haar mea- sure and applications to random quantum circuits” (2024). arXiv:2407.19561

    2024

  71. [79]

    Phase transitions in random circuit sampling

    A. Morvan, B. Villalonga, et al. “Phase transitions in random circuit sampling”. Nature634, 328–333 (2024)

    2024

  72. [80]

    Quantum supremacy using a programmable superconducting processor

    Frank Arute, Kunal Arya, et al. “Quantum supremacy using a programmable superconducting processor”. Nature574, 505–510 (2019)

    2019

  73. [81]

    Characterizing quantum supremacy in near-term devices

    Sergio Boixo, Sergei V. Isakov, Vadim N. Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J. Bremner, John M. Martinis, and Hartmut Neven. “Characterizing quantum supremacy in near-term devices”. Nat. Phys.14, 595–600 (2018)

    2018

  74. [82]

    Efficient quantum algorithms for stabilizer en- tropies

    Tobias Haug, Soovin Lee, and M. S. Kim. “Efficient quantum algorithms for stabilizer en- tropies”. Phys. Rev. Lett.132, 240602 (2024)

    2024

  75. [83]

    Quantum entanglement

    Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. “Quantum entanglement”. Rev. Mod. Phys.81, 865–942 (2009)

    2009

  76. [84]

    Entanglement in many- body systems

    Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. “Entanglement in many- body systems”. Rev. Mod. Phys.80, 517–576 (2008)

    2008

  77. [85]

    Entanglement entropy and conformal field theory

    Pasquale Calabrese and John Cardy. “Entanglement entropy and conformal field theory”. J. Phys. A: Math. Theor.42, 504005 (2009)

    2009

  78. [86]

    Stabilizer entropy dynamics after a quantum quench

    Davide Rattacaso, Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. “Stabilizer entropy dynamics after a quantum quench”. Phys. Rev. A108, 042407 (2023)

    2023

  79. [87]

    Non- stabilizerness and violations of chsh inequalities

    Stefano Cusumano, Lorenzo Campos Venuti, Simone Cepollaro, Gianluca Esposito, Daniele Iannotti, Barbara Jasser, Jovan Odavić, Michele Viscardi, and Alioscia Hamma. “Non- stabilizerness and violations of chsh inequalities” (2025). arXiv:2504.03351

    2025 arXiv

  80. [88]

    Interplay of entanglement structures and stabilizer entropy in spin models

    Michele Viscardi, Marcello Dalmonte, Alioscia Hamma, and Emanuele Tirrito. “Interplay of entanglement structures and stabilizer entropy in spin models” (2025). arXiv:2503.08620

    2025

  81. [89]

    Stabilizer entropy and entanglement complexity in the sachdev-ye-kitaev model

    Barbara Jasser, Jovan Odavić, and Alioscia Hamma. “Stabilizer entropy and entanglement complexity in the sachdev-ye-kitaev model”. Phys. Rev. B112, 174204 (2025)

    2025

  82. [90]

    En- tanglement and Stabilizer entropies of random bipartite pure quantum states

    Daniele Iannotti, Gianluca Esposito, Lorenzo Campos Venuti, and Alioscia Hamma. “En- tanglement and Stabilizer entropies of random bipartite pure quantum states”. Quantum9, 1797 (2025)

    2025

  83. [91]

    Chaos and complexity by design

    Daniel A. Roberts and Beni Yoshida. “Chaos and complexity by design”. J. High Energ. Phys. 2017, 121 (2017)

    2017

  84. [92]

    The clifford group fails gracefully to be a unitary 4-design

    Huangjun Zhu, Richard Kueng, Markus Grassl, and David Gross. “The clifford group fails gracefully to be a unitary 4-design” (2016). arXiv:1609.08172

    2016 arXiv

  85. [93]

    Pauli spectrum and nonstabilizerness of typical quantum many-body states

    Xhek Turkeshi, Anatoly Dymarsky, and Piotr Sierant. “Pauli spectrum and nonstabilizerness of typical quantum many-body states”. Phys. Rev. B111, 054301 (2025)

    2025

  86. [94]

    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” (2025). arXiv:2505.21606

    2025 arXiv

  87. [95]

    Classically estimating observables of noiseless quantum circuits

    Armando Angrisani, Alexander Schmidhuber, Manuel S. Rudolph, M. Cerezo, Zoë Holmes, and Hsin-Yuan Huang. “Classically estimating observables of noiseless quantum circuits”. Phys. Rev. Lett.135, 170602 (2025)

    2025

  88. [96]

    Efficient quantum-enhanced classical simulation for patches of quantum landscapes

    Sacha Lerch, Ricard Puig, Manuel S. Rudolph, Armando Angrisani, Tyson Jones, M. Cerezo, Supanut Thanasilp, and Zoë Holmes. “Efficient quantum-enhanced classical simulation for patches of quantum landscapes” (2024). arXiv:2411.19896

    2024 arXiv

  89. [97]

    Augmenting density matrix renormalization group with matchgates and clifford circuits

    Jiale Huang, Xiangjian Qian, Zhendong Li, and Mingpu Qin. “Augmenting density matrix renormalization group with matchgates and clifford circuits” (2025). arXiv:2505.08635

    2025

  90. [98]

    Augmenting density matrix renormalization group with clifford circuits

    Xiangjian Qian, Jiale Huang, and Mingpu Qin. “Augmenting density matrix renormalization group with clifford circuits”. Phys. Rev. Lett.133, 190402 (2024). Accepted in Quantum 2025-12-21, click title to verify. Published under CC-BY 4.0.19

    2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.