Projective Approach to Dynamical Transport

J. Hajdu

Louis Nguyen

This paper develops a geometric stabilization framework for long-depth sequential quantum evolution through the introduction of the Recovery--Curvature Accumulation (RCA), a transport quantity measuring recovery-weighted projected commutator growth under ordered realization dynamics. Extending the classical threshold-theorem viewpoint beyond reduced logical-channel correctness, the theory analyzes how coherent noncommutative transport structure accumulates across sequential depth through projected Magnus geometry and asymptotic operator-flow behavior. A sharp curvature--recovery threshold is established, separating asymptotically stable transport from critical logarithmic scaling and persistent coherent amplification. Numerical simulations on toric-code realization architectures verify the predicted transport phases, higher-order Magnus suppression, and projected operator-flow convergence. The resulting framework connects threshold stabilization, Stinespring realization transport, Floquet dynamics, monitored systems, and noncommutative geometric transport through a unified asymptotic summability criterion for coherent sequential quantum evolution.

Byoungwoo Lee

## Overview This preprint studies antisymmetric Kossakowski transport generated by compact Lie-algebraic jump operators in finite-dimensional Gorini--Kossakowski--Sudarshan--Lindblad (GKLS) dynamics. The central question is whether directed transport between representation sectors can itself select an operator-space geometry in which the transport becomes reciprocal or skew-adjoint. For a compact simple Lie algebra, the antisymmetric coefficient tensor admits an exact orthogonal bracket--kernel decomposition. The bracket projection carries all nonunitality and satisfies a Casimir--Jordan identity, whereas the bracket-kernel component is unital and Hilbert--Schmidt skew-adjoint. ## Main results Projection between Casimir eigensectors gives the universal scaled-adjoint law $$F_{\nu\to\mu}=-\frac{\Delta_{\mu\nu}-C_{\rm ad}}{\Delta_{\mu\nu}+C_{\rm ad}}F_{\mu\to\nu}^{\dagger},$$ where $\Delta_{\mu\nu}=c_\nu-c_\mu>0$ is the Casimir gap and $C_{\rm ad}$ is the adjoint Casimir. The critical gap $\Delta_{\mu\nu}=C_{\rm ad}$ produces a one-way transport edge, including the universal scalar-to-adjoint boundary. For the multiplicity-free symmetric-power hierarchy $$V_p=\operatorname{Sym}^p\mathbb C^N,$$ every nonzero bracket direction activates every neighboring bond, and the forward and reverse neighboring blocks satisfy $$R_k=-\frac{k}{k+N}F_k^\dagger.$$ The resulting off-diagonal transport uniquely selects, up to an overall scale, the sector weights $$w_k\propto\frac{\Gamma(k)}{\Gamma(k+N)}.$$ Under the exact equivariant harmonic correspondence with $\mathbb CP^{N-1}$, this selected metric becomes a projective spectral multiplier. On nonconstant projective harmonics it is identified with the sign-corrected derivative, at the constant-kernel point, of a centered and rescaled projective cosine-transform intertwining family. Its inverse has the same scalar Gamma-ratio multiplier, when indexed by harmonic degree, as the corresponding order-$N$ conformal spherical intertwiner. For $N\ge3$, a $d_{abc}$-mediated same-sector Jordan block obstructs any positive $SU(N)$-invariant skew metric for the full traceless bracket transport. Consequently, within the symmetric-power family, $SU(2)$ is exceptional: the bracket kernel disappears, the same-sector obstruction is absent, and the transport-selected metric reduces to inverse Casimir and extends over the entire traceless internal bracket dynamics. ## Interpretation and scope The Gamma-function sequence, projective harmonic correspondence, cosine-transform spectrum, and conformal spherical multipliers are established mathematical structures. The new mechanism developed here is their dynamical selection by Lie-algebraic Lindblad transport, together with the representation-theoretic obstruction that prevents the $SU(2)$ inverse-Casimir geometry from extending generically to higher rank. The separate $SU(2)$ prototype, including its dissipative decomposition, contraction theory, exact spherical Green geometry, and complete-positivity extremality, is developed in: **Lee Byoungwoo, “Inverse-Casimir Geometry of Nonunital Multipole Transport in Linear-Spin Lindbladians,” Zenodo Version v1.0 (2026), DOI 10.5281/zenodo.22266973.** ## Version status This record is **Version v1.0** of the public preprint. The manuscript is presented without internal research-version or freeze labels.

Timothy Lovorn, Akihiko Sekine, Allan H. MacDonald

Ole Sönnerborn

Nonadiabatic holonomic quantum computation is a promising approach for implementing quantum gates that offers both efficiency and robustness against certain types of errors. A key element of this approach is a geometric constraint known as the parallel transport condition. According to the principle of covariance, this condition must be appropriately modified when changing reference frames. In this paper, we detail how to adjust the parallel transport condition when transitioning from the laboratory frame to a rotating reference frame. Furthermore, building on gauge invariance considerations, we develop a framework for nonadiabatic holonomic quantum computation with projective gates. The parallel transport condition of this framework effectively addresses the problem of global dynamical phases inherent in conventional nonadiabatic holonomic quantum computation. We extend the isoholonomic inequality, which provides a fundamental bound on the efficiency of protocols used to implement holonomic quantum gates, to encompass projective quantum gates. We also determine a minimum execution time for projective holonomic quantum gates and show that this time can be attained when the codimension of the computational space is sufficiently large.

Antti‐Pekka Jauho