Pith. sign in

REVIEW 4 major objections 5 minor 75 references

Open-system quantum complexity is a sub-Finslerian geometry, not a Riemannian one.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 07:50 UTC pith:Q2OMYZDX

load-bearing objection A solid, honest extension of Nielsen complexity to Lindblad systems; the core geometry likely holds, but the flag-curvature plots and the handling of boundary controls need more rigor. the 4 major comments →

arxiv 2607.08411 v2 pith:Q2OMYZDX submitted 2026-07-09 quant-ph hep-th

The Geometry of Quantum Complexity in Open Systems

Ezra Acalapati , Kausik Ghosh , Giuseppe Policastro This is my paper
classification quant-ph hep-th
keywords quantum complexityopen quantum systemsLindbladian evolutionsub-Finsler geometryoptimal controlflag curvaturedissipative dynamicsquantum channels
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper extends the geometric recipe for quantum complexity from closed, unitary dynamics to open systems governed by a Markovian Lindbladian master equation. Its central claim is that the resulting complexity geometry is generically sub-Finslerian: the allowed tangent directions form a restricted cone, the cost function is a positively homogeneous Finsler norm on that cone, and geodesics are non-reversible because dissipation breaks time reversal. For a single qubit with one dissipative and one unitary control, the paper derives explicit Finsler functions for the driftless case (which is sub-Riemannian), the drifted case (genuinely conic sub-Finsler), and the singular or fully actuated case. It then shows, on depolarizing, amplitude-damping, and damped-oscillator examples, that changing the control penalty factors can flip the sign of the flag curvature—the same qualitative effect that in closed systems is read as a signature of chaotic divergence of nearby geodesics. If correct, this gives an operational, geometry-based measure of complexity for realistic dissipative platforms such as noisy intermediate-scale quantum devices and driven-dissipative systems.

Core claim

The paper's discovery is that the optimal-control problem defining complexity for a Lindbladian open system is exactly a sub-Finslerian geodesic problem on the manifold of mixed states. With coordinates given by Bloch-vector components and two controls, the paper derives three general forms of the Finsler function: equation (6.9) gives a quadratic, hence sub-Riemannian, norm when there is no drift; equation (6.12) gives a conic sub-Finsler function when a drift is present, with the admissibility constraint fixing a rescaling factor; equation (6.15) gives the function on the singular locus and in the fully actuated case, where the rescaling factor is instead optimized. The flag curvature comp

What carries the argument

The central object is the Finslerian function F(v), defined on the cone of admissible velocities: it is the factor that rescales any allowed velocity direction v until it lies on the unit-cost indicatrix, i.e. the boundary of the set of velocities reachable at cost one. This function is constructed from the optimal-control problem through Pontryagin's necessary conditions; its Hessian defines a direction-dependent fundamental tensor, and the associated spray and flag curvature give the geodesic equations and the divergence of nearby geodesics. The affine admissibility constraint that appears with drift is what turns a quadratic control cost into a genuinely conic sub-Finsler norm rather than

Load-bearing premise

The load-bearing premise is the footnote-5 assumption that true optimal trajectories are always captured by the normal case of Pontryagin's maximum principle; if abnormal extremals—which can be genuine minimizers in sub-Riemannian problems—arise for the Lindblad control system, the derived Finslerian function and curvature may describe the wrong geodesics.

What would settle it

Pick a qubit Lindbladian with drift and fixed endpoints, compute the full set of Pontryagin extremals including abnormal ones (costate normalization p0=0), and compare their costs against the normal-extremal Finsler geodesic of (6.12); an abnormal trajectory with strictly lower cost would disprove the claim that the Finslerian function gives the complexity. A more direct check: construct a two-control system whose admissibility cone has a boundary reachable only by abnormal extremals, and test whether (6.15) or (6.12) correctly predicts the minimizing velocity.

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

If this is right

  • Complexity distances for open systems are generically non-reversible: the cost of reaching a state is not the cost of returning, because dissipative controls cannot be inverted by completely positive trace-preserving operations.
  • In the absence of drift the complexity geometry is sub-Riemannian; adding an uncontrolled drift makes it genuinely conic sub-Finslerian, so geodesics are generally only piecewise smooth.
  • Penalty factors are not a minor tuning: they control the sign of the flag curvature, hence whether nearby optimal trajectories converge or diverge in state space.
  • The damped harmonic oscillator under thermal damping inherits the same qubit geometry after shifting to the thermal fixed point, so the framework applies to infinite-dimensional Gaussian systems.
  • The general formulas (6.9), (6.12), and (6.15) provide a unified starting point for computing complexity-geometry quantities in any Lindbladian system with one dissipative and one unitary control.

Where Pith is reading between the lines

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

  • If abnormal Pontryagin extremals exist for the underlying Lindblad control problem, the normal-case derivation used here would need revision; abnormal geodesics can be true minimizers in sub-Riemannian geometry and are invisible to the Finsler functions derived in section 6.
  • Because the geometry is non-reversible and the evolution is a semigroup, complexity should inherit monotone-growth properties under composition of dissipative channels; this could connect to the linear-growth-then-saturation behavior expected of complexity and to holographic complexity for evaporating black holes.
  • The same sub-Finslerian scheme should apply to reduced dynamics that are approximately Markovian, such as a static patch of de Sitter space, with flag curvature measuring how horizon-induced dissipation affects the difficulty of preparing local states.
  • A natural test is whether the Bures-metric complexity of mixed states, which is Riemannian, emerges as a limit of the sub-Finsler norm when dissipative directions are penalized infinitely, paralleling the recovery of Fubini-Study in the closed case.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper extends Nielsen's geometric approach to quantum complexity to open quantum systems governed by Lindblad dynamics. The complexity of a mixed state is defined as the minimal cost of an optimal control problem on the Bloch ball, with quadratic cost in the controls and a constant terminal cost. Using Pontryagin's Maximum Principle, the authors derive Finslerian functions for a single-qubit system: in the driftless case the geometry is sub-Riemannian with F(v) proportional to sqrt(Γ_ij v^i v^j) (Eq. 6.9); with drift it is genuinely conic sub-Finslerian (Eq. 6.12); on the singular locus or in the fully actuated case it takes the form (6.15). The framework is illustrated on depolarizing and amplitude-damping channels with and without unitary/non-unitary drift, and on the damped harmonic oscillator. The paper reports numerical flag-curvature computations showing that penalty factors can change the sign of curvature, and plots optimal trajectories for the unitary-drift depolarizing example.

Significance. If the central reduction is valid, this is a useful and timely contribution: it gives a concrete geometric framework for complexity in dissipative systems, with explicit algebraic formulas rather than only existence arguments. The derivation of the Finsler functions from PMP is explicit and internally consistent, and the flatness check for the driftless depolarizing case is a good sanity check. The potential applications to holography and NISQ devices are plausible and clearly motivated. However, the exact identification of complexity with the Finsler distance relies on assumptions about normal extremals and interior controls that are not fully justified, and the curvature results are presented in a form that is hard to verify. These issues do not destroy the paper's value, but they need to be addressed before the central claim can be accepted.

major comments (4)
  1. [§5.2, §6] The reduction to a Finsler distance explicitly excludes abnormal extremals (p0=0). For the affine control systems considered here, abnormal extremals correspond to costates annihilating the control distribution. Such extremals exist in the examples: in §7.3 the surface x=0 satisfies p·Aγ=p·Aω=0 and H=0, and similarly in §7.4 the surfaces y=0 and z=0. The paper treats these as a 'singular locus' and proposes Eq. (6.15), but this formula is obtained by a separate minimization over the rescaling parameter, not by an abnormal PMP analysis. No proof is given that all abnormal minimizers lie on the singular locus or that Eq. (6.15) gives their true optimal cost. If an abnormal curve connects two endpoints with lower cost than any normal geodesic, the Finsler distance computed from the normal equations is only an upper bound, not the exact complexity. The manuscript should either prove absence
  2. [§7, Eqs. (7.20), (7.45), (7.66)] The admissible control set is repeatedly stated to include the constraint γ≥0, which restricts the admissible velocities to a half-plane (e.g., p. 18, p. 23, p. 31). However, the PMP stationarity condition (5.15) and the derived Finsler functions are obtained from unconstrained maximization over the controls. The paper does not solve the constrained PMP with the inequality γ≥0, nor does it verify that the optimal trajectories for the plotted endpoints satisfy γ>0 along the entire path. If an optimal path requires a boundary arc with γ=0, the correct complexity is larger than the unconstrained Finsler distance. This is not a minor technicality: for the driftless depolarizing case the quadratic F in Eq. (7.20) is even in v while the admissible cone is non-reversible, so the half-cone restriction and possible boundary arcs are essential. The authors should either incorporate the inequality
  3. [§7.3–§7.4] The flag-curvature plots are central to the claim that penalty factors can change the sign of the curvature, but no explicit formula for K is given, nor is the computational method (spray coefficients, curvature tensor, numerical integration) described. A reader cannot reproduce the curves in Figures 4 and 8. For a quantitative geometric claim, the paper should provide the explicit expression for the flag curvature in at least one of the two drifted examples, or a detailed description of the numerical procedure, including the definition of the admissible two-dimensional plane, the flagpole Y, and the chosen basis {e1,e2}.
  4. [§6] In the driftless case, the text states that the quadratic equation (6.6) 'determines a line in the space of \tilde p'. This is not correct: for two independent controls, the equation defines an ellipse (a quadric) in the two-dimensional \tilde p-space, which then gives an ellipse in control space and in the space of allowed velocities. The final formula (6.9) is consistent with an elliptic indicatrix, so this appears to be a typo, but it makes the derivation harder to follow and should be fixed.
minor comments (5)
  1. [§5.3] The definition of S0_x as the set of 'longest admissible velocities' (Eq. 5.12) is imprecise. It would be clearer to define S0_x as the set of maximal elements of the convex hull of S_x with respect to radial scaling, or to spell out the intended notion of 'longest'.
  2. [§7] The indicatrix plots (Figures 1–3, 7) do not indicate whether the full affine plane or the admissible half-plane (γ≥0) is displayed. Since the half-plane restriction is mentioned in the text, the figures should state explicitly what subset of the tangent space is drawn.
  3. [§7.5] The damped harmonic oscillator section is very brief and does not compute any Finsler function, curvature, or geodesics for this system. It only notes a formal equivalence to the depolarizing example. Either expand this section to give an explicit analysis or describe it explicitly as a reduction rather than a new example.
  4. [§7.3] The singular-locus Finsler function for x=0 (Eq. 7.46) appears with square roots and signs. The physical branch and domain of validity (e.g., y≠0, z≠0) are not stated. Please add the conditions under which this expression is real and positive.
  5. [Throughout] There are a few typographical inconsistencies, for example Eq. (7.17)–(7.18) uses ēL(v) in two different senses, and the phrase 'determines a line' in §6 should be corrected. A careful proofread would improve readability.

Circularity Check

0 steps flagged

No circularity found: the Finsler geometry is constructed from the declared control dynamics and cost; no fitted parameter is relabeled as a prediction.

full rationale

The central derivation (eqs. 6.9, 6.12, 6.15) follows from the control system (6.1) and quadratic cost (6.2) through the PMP stationarity condition (5.15); it is an exact Legendre-type transform, not an input equivalent to the output. The sub-Finslerian structure is forced by the dimension count (two controls, three state coordinates) and the admissibility constraint (6.3), not by a chosen ansatz. The examples evaluate these formulas with specific Lindbladians; no parameters are fitted to data. Flag-curvature sign changes are mathematical consequences of the stated penalties. The only self-citations ([4], [67]) are contextual (review and holographic application) and do not carry the derivation; the control-to-Finsler construction is taken from external ref. [17]. The restriction to normal PMP extremals (footnote 5) is an acknowledged technical assumption about the scope of the reduction, not a circular step.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The central construction rests on modeling choices: a control-affine Lindblad dynamics (6.1), a quadratic control cost (6.2)/(7.1) with hand-chosen weights and terminal cost, and the restriction to the normal PMP branch. No new physical entities are introduced. The mathematical machinery (Pontryagin, Finsler geometry) is standard, but the constant terminal cost (§5.4) and the normal-case restriction are ad hoc assumptions that shape the derived geometry.

free parameters (5)
  • pγ = 1, 10, 100 in examples
    Penalty weight for the depolarizing control γ in the cost (7.1); chosen by hand in each example, and the geometry/curvature depends on it.
  • pω = 0.01, 1, 100 in examples
    Penalty weight for the unitary rotation control ω; chosen by hand, controls the relative cost of rotation vs. dissipation.
  • pγω = 0, 0.5, 1.9 in examples
    Cross-penalty between γ and ω in the cost (7.1); set by hand, skews the indicatrix and curvature.
  • Ct = 1 in all examples
    Constant terminal cost introduced in §5.4 to make the free-final-time problem well-posed; set to 1 in all numerical illustrations.
  • Δ = 1 in examples 3 and 4
    Strength of the uncontrolled drift term in §7.3/7.4; chosen by hand for the illustrations. Results are parametric in Δ but no fitting is done.
axioms (6)
  • domain assumption Lindblad (GKSL) master equation generates the open-system dynamics.
    Used throughout; restricts to Markovian, time-homogeneous CPTP evolutions (§2).
  • domain assumption The controlled velocity is affine and control-affine: ẋ = A(x)u + Δ(x) with two independent controls (eq. 6.1).
    This is the mathematical setting of the qubit examples; the number of controls is justified in §6.
  • ad hoc to paper The cost function is quadratic in the controls: L = Ct + (1/2) gαβ uα uβ (eq. 6.2), and in examples L = Ct + pγ γ² + pγω γω + pω ω² (eq. 7.1).
    The entire Finslerian-function derivation is based on this choice; other costs would give different geometric structures.
  • ad hoc to paper Only normal PMP extremals are considered (p0 = −1, footnote 5, §5.2).
    Abnormal extremals are excluded without justification; in sub-Riemannian problems they can be optimal.
  • ad hoc to paper A constant terminal cost Ct is added to regularize the free-final-time problem (§5.4).
    This prescription is introduced to make the action finite and is not derived from first principles.
  • standard math The Finsler function is defined on the admissible cone and is assumed to satisfy the standard smoothness/strong-convexity conditions where needed (§4, §5.3).
    Standard Finsler geometry definitions; for the conic sub-Finsler metric, strong convexity is assumed on the cone.

pith-pipeline@v1.3.0-alltime-deepseek · 27038 in / 14656 out tokens · 135454 ms · 2026-08-02T07:50:59.822337+00:00 · methodology

0 comments
read the original abstract

We extend Nielsen's geometric approach for quantum complexity from closed to open quantum systems, whose dynamics is governed by Lindbladian evolution. In this framework, complexity is defined through an optimal-control problem on the space of mixed states, with a cost assigned to both unitary and non-unitary generators. We show that the resulting geometric structure differs fundamentally from the Riemannian geometry that emerges in the case of unitary evolution. In the open-system setting, the natural geometry is typically sub-Finslerian. Dissipation makes the geodesics non-reversible, while the admissible tangent directions are restricted by the physically allowed controls. We analyze several physically motivated examples, including a single qubit subject to depolarizing and amplitude-damping channels, as well as the damped harmonic oscillator. We show that, similarly to the unitary case, varying the penalty factors in the cost functional modifies the geometric properties through changes in the flag curvature, the Finslerian analog of sectional curvature. Our results provide a geometric framework for quantifying the abstract notion of complexity in dissipative quantum systems, with potential connections to experimentally realizable setups.

Review history (2 revisions) →

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

75 extracted references · 53 linked inside Pith

  1. [1]

    Baiguera, V

    S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M.P. Heller et al., Quantum complexity in gravity, quantum field theory, and quantum information science, Phys. Rept.1159(2026) 1 [2503.10753]

    arXiv 2026

  2. [2]

    Nielsen, M.R

    M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty,Quantum Computation as Geometry, Science311(2006) 1133 [quant-ph/0603161]

    Pith/arXiv arXiv 2006

  3. [3]

    Dowling and M.A

    M.R. Dowling and M.A. Nielsen,The geometry of quantum computation,Quant. Inf. Comput.8(2008) 0861 [quant-ph/0701004]

    Pith/arXiv arXiv 2008

  4. [4]

    Chapman and G

    S. Chapman and G. Policastro,Quantum computational complexity from quantum information to black holes and back,Eur. Phys. J. C82(2022) 128 [2110.14672]

    Pith/arXiv arXiv 2022

  5. [5]

    Susskind,Computational Complexity and Black Hole Horizons,Fortsch

    L. Susskind,Computational Complexity and Black Hole Horizons,Fortsch. Phys.64(2016) 24 [1403.5695]

    Pith/arXiv arXiv 2016

  6. [6]

    Susskind,Entanglement is not enough,Fortsch

    L. Susskind,Entanglement is not enough,Fortsch. Phys.64(2016) 49 [1411.0690]

    Pith/arXiv arXiv 2016

  7. [7]

    Brown, D.A

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Holographic Complexity Equals Bulk Action?,Phys. Rev. Lett.116(2016) 191301 [1509.07876]

    Pith/arXiv arXiv 2016

  8. [8]

    Brown, L

    A.R. Brown, L. Susskind and Y. Zhao,Quantum Complexity and Negative Curvature,Phys. Rev. D95(2017) 045010 [1608.02612]

    Pith/arXiv arXiv 2017

  9. [9]

    Brown and L

    A.R. Brown and L. Susskind,Complexity geometry of a single qubit,Phys. Rev. D100 (2019) 046020 [1903.12621]

    Pith/arXiv arXiv 2019

  10. [10]

    Mori,Liouvillian-gap analysis of open quantum many-body systems in the weak dissipation limit,Phys

    T. Mori,Liouvillian-gap analysis of open quantum many-body systems in the weak dissipation limit,Phys. Rev. B109(2024) 064311 [2311.10304]

    Pith/arXiv arXiv 2024

  11. [11]

    Flory and M.P

    M. Flory and M.P. Heller,Geometry of Complexity in Conformal Field Theory,Phys. Rev. Res.2(2020) 043438 [2005.02415]

    Pith/arXiv arXiv 2020

  12. [12]

    Chagnet, S

    N. Chagnet, S. Chapman, J. de Boer and C. Zukowski,Complexity for Conformal Field Theories in General Dimensions,Phys. Rev. Lett.128(2022) 051601 [2103.06920]

    Pith/arXiv arXiv 2022

  13. [13]

    Lloyd and L

    S. Lloyd and L. Viola,Control of open quantum systems dynamics,quant-ph/0008101

    Pith/arXiv arXiv

  14. [14]

    Carlini, A

    A. Carlini, A. Hosoya, T. Koike and Y. Okudaira,Quantum Brachistochrone for Mixed States,J. Phys. A41(2008) 045303 [quant-ph/0703047]

    Pith/arXiv arXiv 2008

  15. [15]

    Morazotti, A.H

    N.A.d.C. Morazotti, A.H. da Silva, G. Audi, F.F. Fanchini and R.d.J. Napolitano,Optimized continuous dynamical decoupling via differential geometry and machine learning,Phys. Rev. A110(2024) 042601

    2024

  16. [16]

    Jefferson and R.C

    R. Jefferson and R.C. Myers,Circuit complexity in quantum field theory,JHEP10(2017) 107 [1707.08570]

    Pith/arXiv arXiv 2017

  17. [17]

    López and E

    C. López and E. Martínez,Sub-Finslerian Metric Associated to an Optimal Control System, SIAM J. Control Optim.39(2000) 798

    2000

  18. [18]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems, Oxford University Press (2002)

    2002

  19. [19]

    Rivas and S.F

    Á. Rivas and S.F. Huelga,Open Quantum Systems, Springer (2012), 10.1007/978-3-642-23354-8

    doi:10.1007/978-3-642-23354-8 2012

  20. [20]

    Stinespring,Positive functions on C*-algebras,Proceedings of the American Mathematical Society6(1955) 211

    W.F. Stinespring,Positive functions on C*-algebras,Proceedings of the American Mathematical Society6(1955) 211. – 39 –

    1955

  21. [21]

    Nielsen and I.L

    M.A. Nielsen and I.L. Chuang,Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 10th anniversary ed. (2010)

    2010

  22. [22]

    Watrous,The Theory of Quantum Information, Cambridge University Press, Cambridge (2018), 10.1017/9781316848142

    J. Watrous,The Theory of Quantum Information, Cambridge University Press, Cambridge (2018), 10.1017/9781316848142

    doi:10.1017/9781316848142 2018

  23. [23]

    Gorini, A

    V. Gorini, A. Kossakowski and E.C.G. Sudarshan,Completely Positive Dynamical Semigroups of N Level Systems,J. Math. Phys.17(1976) 821

    1976

  24. [24]

    Lindblad,On the Generators of Quantum Dynamical Semigroups,Commun

    G. Lindblad,On the Generators of Quantum Dynamical Semigroups,Commun. Math. Phys. 48(1976) 119

    1976

  25. [25]

    Chruściński and S

    D. Chruściński and S. Pascazio,A Brief History of the GKLS Equation,Open Syst. Inf. Dyn. 24(2017) 1740001

    2017

  26. [26]

    Krantz, M

    P. Krantz, M. Kjaergaard, F. Yan, T.P. Orlando, S. Gustavsson and W.D. Oliver,A quantum engineer’s guide to superconducting qubits,Appl. Phys. Rev.6(2019) 021318 [1904.06560]

    Pith/arXiv arXiv 2019

  27. [27]

    Kubica, A

    A. Kubica, A. Haim, Y. Vaknin, H. Levine, F. Brandão and A. Retzker,Erasure Qubits: Overcoming theT 1 Limit in Superconducting Circuits,Phys. Rev. X13(2023) 041022 [2208.05461]

    Pith/arXiv arXiv 2023

  28. [28]

    Fisher, R

    K. Fisher, R. Prevedel, R. Kaltenbaek and K.J. Resch,Optimal linear optical implementation of a single-qubit damping channel,New J. Phys.14(2012) 033016 [1109.2070]

    Pith/arXiv arXiv 2012

  29. [29]

    Albert et al.,Performance and structure of single-mode bosonic codes,Phys

    V.V. Albert et al.,Performance and structure of single-mode bosonic codes,Phys. Rev. A97 (2018) 032346 [1708.05010]

    Pith/arXiv arXiv 2018

  30. [30]

    Stobbe, J

    S. Stobbe, J. Johansen, P.T. Kristensen, J.M. Hvam and P. Lodahl,Frequency dependence of the radiative decay rate of excitons in self-assembled quantum dots: Experiment and theory, Phys. Rev. B80(2009) 155307

    2009

  31. [31]

    Doherty, N.B

    M.W. Doherty, N.B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup and L.C.L. Hollenberg, The nitrogen-vacancy colour centre in diamond,Phys. Rept.528(2013) 1

    2013

  32. [32]

    Goldman, A

    M.L. Goldman, A. Sipahigil, M.W. Doherty, N.Y. Yao, S.D. Bennett, M. Markham et al., Phonon-Induced Population Dynamics and Intersystem Crossing in Nitrogen-Vacancy Centers,Phys. Rev. Lett.114(2015) 145502 [1406.4065]

    Pith/arXiv arXiv 2015

  33. [33]

    Życzkowski, R

    K. Życzkowski, R. Alicki and J. Emerson,Scalable noise estimation with random unitary operators,J. Opt. B7(2005) S347 [quant-ph/0503243]

    Pith/arXiv arXiv 2005

  34. [34]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson and E. Livine,Exact and approximate unitary 2-designs and their application to fidelity estimation,Phys. Rev. A80(2009) 012304 [quant-ph/0606161]

    Pith/arXiv arXiv 2009

  35. [35]

    Magesan, J.M

    E. Magesan, J.M. Gambetta and J. Emerson,Characterizing quantum gates via randomized benchmarking,Phys. Rev. A85(2012) 042311 [1109.6887]

    Pith/arXiv arXiv 2012

  36. [36]

    Wallman and J

    J.J. Wallman and J. Emerson,Noise tailoring for scalable quantum computation via randomized compiling,Phys. Rev. A94(2016) 052325 [1512.01098]

    Pith/arXiv arXiv 2016

  37. [37]

    Hashim et al.,Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor,Phys

    A. Hashim et al.,Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor,Phys. Rev. X11(2021) 041039 [2010.00215]

    Pith/arXiv arXiv 2021

  38. [38]

    Karpinski, C

    M. Karpinski, C. Radzewicz and K. Banaszek,Fiber-optic realization of anisotropic depolarizing quantum channels,J. Opt. Soc. Am. B25(2008) 668 [0707.3728]

    Pith/arXiv arXiv 2008

  39. [39]

    Shaham and H.S

    A. Shaham and H.S. Eisenberg,Realizing controllable noise in photonic quantum information channels,Phys. Rev. A83(2011) 022303 [1006.5795]. – 40 –

    Pith/arXiv arXiv 2011

  40. [40]

    Jeong, J.-C

    Y.-C. Jeong, J.-C. Lee and Y.-H. Kim,Experimental implementation of a fully controllable depolarizing quantum operation,Phys. Rev. A87(2013) 014301 [1204.0850]

    Pith/arXiv arXiv 2013

  41. [41]

    Denis and J

    J. Denis and J. Martin,Extreme depolarization for any spin,Phys. Rev. Res.4(2022) 013178 [2106.11680]

    Pith/arXiv arXiv 2022

  42. [42]

    Lidar, A

    D.A. Lidar, A. Shabani and R. Alicki,Conditions for strictly purity-decreasing quantum Markovian dynamics,Chem. Phys.322(2006) 82 [quant-ph/0411119]

    Pith/arXiv arXiv 2006

  43. [43]

    Bao, S.-S

    D. Bao, S.-S. Chern and Z. Shen,An Introduction to Riemann-Finsler Geometry, Springer, New York (2000)

    2000

  44. [44]

    Boscain, M

    U. Boscain, M. Sigalotti and D. Sugny,Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control,PRX Quantum2(2021) 030203 [2010.09368]

    Pith/arXiv arXiv 2021

  45. [45]

    Pontryagin, V.G

    L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko,The Mathematical Theory of Optimal Processes, John Wiley and Sons, New York (1962)

    1962

  46. [46]

    Liberzon,Calculus of Variations and Optimal Control Theory, Princeton University Press (2012)

    D. Liberzon,Calculus of Variations and Optimal Control Theory, Princeton University Press (2012)

    2012

  47. [47]

    Rivas, A.D.K

    Á. Rivas, A.D.K. Plato, S.F. Huelga and M.B. Plenio,Markovian master equations: a critical study,New J. Phys.12(2010) 113032 [1006.4666]

    Pith/arXiv arXiv 2010

  48. [48]

    Caceres, S

    E. Caceres, S. Chapman, J.D. Couch, J.P. Hernandez, R.C. Myers and S.-M. Ruan, Complexity of Mixed States in QFT and Holography,JHEP03(2020) 012 [1909.10557]

    Pith/arXiv arXiv 2020

  49. [49]

    Bhattacharyya, T

    A. Bhattacharyya, T. Hanif, S.S. Haque and M.K. Rahman,Complexity for an open quantum system,Phys. Rev. D105(2022) 046011 [2112.03955]

    Pith/arXiv arXiv 2022

  50. [50]

    Bhattacharyya, T

    A. Bhattacharyya, T. Hanif, S.S. Haque and A. Paul,Decoherence, entanglement negativity, and circuit complexity for an open quantum system,Phys. Rev. D107(2023) 106007 [2210.09268]

    Pith/arXiv arXiv 2023

  51. [51]

    Bhattacharya, A

    A. Bhattacharya, A. Bhattacharyya and S. Maulik,Pseudocomplexity of purification for free scalar field theories,Phys. Rev. D106(2022) 086010 [2209.00049]

    Pith/arXiv arXiv 2022

  52. [52]

    Bhattacharyya, S

    A. Bhattacharyya, S. Brahma, S.S. Haque, J.S. Lund and A. Paul,The early universe as an open quantum system: complexity and decoherence,JHEP05(2024) 058 [2401.12134]

    Pith/arXiv arXiv 2024

  53. [53]

    Ruan,Purification Complexity without Purifications,JHEP01(2021) 092 [2006.01088]

    S.-M. Ruan,Purification Complexity without Purifications,JHEP01(2021) 092 [2006.01088]

    Pith/arXiv arXiv 2021

  54. [54]

    Lumia, E

    L. Lumia, E. Tirrito, M. Collura, F.H.L. Essler and R. Fazio,Complexity of Quantum Trajectories,2602.00232

    Pith/arXiv arXiv

  55. [55]

    Craps, O

    B. Craps, O. Evnin and G. Pascuzzi,A relation between krylov and nielsen complexity,Phys. Rev. Lett.132(2024) 160402

    2024

  56. [56]

    Craps, G

    B. Craps, G. Pascuzzi, J.F. Pedraza, L.-C. Qu and S.-M. Ruan,Explicit Connections Between Krylov and Nielsen Complexity,2511.15799

    arXiv

  57. [57]

    Craps, O

    B. Craps, O. Evnin and G. Pascuzzi,Multiseed krylov complexity,Phys. Rev. Lett.134 (2025) 050402

    2025

  58. [58]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu,Operator growth and Krylov construction in dissipative open quantum systems,JHEP12(2022) 081 [2207.05347]

    Pith/arXiv arXiv 2022

  59. [59]

    Caputa and J.M

    P. Caputa and J.M. Magan,Quantum Computation as Gravity,Phys. Rev. Lett.122(2019) 231302 [1807.04422]. – 41 –

    Pith/arXiv arXiv 2019

  60. [60]

    Baiguera, N

    S. Baiguera, N. Chagnet, S. Chapman and O. Shoval,CFT complexity and penalty factors, JHEP02(2026) 247 [2507.22118]

    Pith/arXiv arXiv 2026

  61. [61]

    Demulder,Non-invertible circuit complexity from fusion operations,2601.09535

    S. Demulder,Non-invertible circuit complexity from fusion operations,2601.09535

    arXiv

  62. [62]

    Q. Tang, R. Barad and X. Wen,Exact operator dynamics in Lindbladian Wess-Zumino-Witten conformal field theories,2606.19465

    Pith/arXiv arXiv

  63. [63]

    Bai,Exact Lindbladian Dynamics from Conformal Embeddings and Topological Defects in Conformal Field Theory,2607.08827

    C. Bai,Exact Lindbladian Dynamics from Conformal Embeddings and Topological Defects in Conformal Field Theory,2607.08827

    Pith/arXiv arXiv

  64. [64]

    C. Bai, W. Mao, M. Nozaki, M.T. Tan and X. Wen,Relaxation Process During Complex Time Evolution In Two-Dimensional Integrable and Chaotic CFTs,2601.09290

    arXiv

  65. [65]

    Jørstad, R.C

    E. Jørstad, R.C. Myers and S.-M. Ruan,Complexity=anything: singularity probes,JHEP07 (2023) 223 [2304.05453]

    Pith/arXiv arXiv 2023

  66. [66]

    Cáceres, Á.J

    E. Cáceres, Á.J. Murcia, A.K. Patra and J.F. Pedraza,Kasner eons with matter: holographic excursions to the black hole singularity,JHEP12(2024) 077 [2408.14535]

    Pith/arXiv arXiv 2024

  67. [67]

    Policastro and S

    G. Policastro and S. Wittum,Probing the singularity of scalar-haired black holes with holographic complexity,JHEP05(2026) 116 [2512.07403]

    arXiv 2026

  68. [68]

    Almheiri, N

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield,The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,JHEP12(2019) 063 [1905.08762]

    Pith/arXiv arXiv 2019

  69. [69]

    Almheiri, R

    A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao,The Page curve of Hawking radiation from semiclassical geometry,JHEP03(2020) 149 [1908.10996]

    Pith/arXiv arXiv 2020

  70. [70]

    L. Sá, P. Ribeiro and T. Prosen,Lindbladian dissipation of strongly-correlated quantum matter,Phys. Rev. Res.4(2022) L022068 [2112.12109]

    Pith/arXiv arXiv 2022

  71. [71]

    Kulkarni, T

    A. Kulkarni, T. Numasawa and S. Ryu,Lindbladian dynamics of the Sachdev-Ye-Kitaev model,Phys. Rev. B106(2022) 075138 [2112.13489]

    Pith/arXiv arXiv 2022

  72. [72]

    Alicki, G

    R. Alicki, G. Barenboim and A. Jenkins,Quantum thermodynamics of de Sitter space,Phys. Rev. D108(2023) 123530 [2307.04800]

    Pith/arXiv arXiv 2023

  73. [73]

    Marquet et al.,Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s,Phys

    A. Marquet et al.,Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s,Phys. Rev. X14(2024) 021019 [2307.06761]

    Pith/arXiv arXiv 2024

  74. [74]

    Marquet et al.,Harnessing two-photon dissipation for enhanced quantum measurement and control,Phys

    A. Marquet et al.,Harnessing two-photon dissipation for enhanced quantum measurement and control,Phys. Rev. Applied22(2024) 034053 [2403.07744]

    Pith/arXiv arXiv 2024

  75. [75]

    Gustiani, D

    C. Gustiani, D. Leichtle, J. Miller, R. Grassie, D. Mills and E. Kashefi,On-Chip Verified Quantum Computation with an Ion-Trap Quantum Processing Unit,Phys. Rev. Lett.135 (2025) 160801 [2410.24133]. – 42 –

    Pith/arXiv arXiv 2025