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arXiv:2601.00735

Geometric Complexity of Quantum Channels via Unitary Dilations

Alberto Acevedo, Antonio Falcó

quant-phmath-phSubmitted Jan 2, 2026• Announced Jan 5, 2026• 0 citations
v1 · Jan 2, 2026v2 · Oct 24, 2026
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Abstract

Nielsen's geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary (closed--system) dynamics. For open dynamics, however, the reduced evolution is described by quantum channels and admits many inequivalent Stinespring realizations, so any meaningful complexity notion must specify which microscopic resources are counted as accessible and which transformations are regarded as gauge. We introduce and analyze a geometric complexity functional for families of quantum channels based on unitary dilations. We distinguish an implementation-dependent complexity, defined relative to explicit dilation data, from an intrinsic channel complexity obtained by minimizing over a physically motivated class of admissible dilations (e.g. bounded environment dimension, energy or norm constraints, and penalty structures). The functional has a subtractive form: it compares the geometric cost of the total unitary realization with a canonical surrogate term that removes purely environmental contributions. We justify this subtraction from concise postulates, including closed-system consistency, environment-only neutrality, and invariance under dilation gauge transformations that leave the channel unchanged. This leads to a companion quantity, noise complexity, quantifying the loss of geometric complexity relative to a prescribed ideal closed evolution. We establish a coherence-based lower bound for unitary geometric complexity, derive structural properties such as linear time scaling under time-homogeneous dilations, and obtain dissipator--controlled bounds in the Markovian (GKSL/Lindblad) regime under a standard dilation construction. Finally, we illustrate the framework on canonical benchmark noise models, including dephasing, amplitude damping, and depolarizing (Pauli) channels.

Journal reference
Alberto Acevedo and Antonio Falcó 2026 J. Phys. A: Math. Theor. 59 205303

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Nielsen's geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary (closed--system) dynamics. For open dynamics, however, the reduced evolution is described by quantum channels and admits many inequivalent Stinespring realizations, so any meaningful complexity notion must specify which microscopic resources are counted as accessible and which transformations are regarded as gauge. We introduce and analyze a geometric complexity functional for families of quantum channels based on unitary dilations. We distinguish an implementation-dependent complexity, defined relative to explicit dilation data, from an intrinsic channel complexity obtained by minimizing over a physically motivated class of admissible dilations (e.g. bounded environment dimension, energy or norm constraints, and penalty structures). The functional has a subtractive form: it compares the geometric cost of the total unitary realization with a canonical surrogate term that removes purely environmental contributions. We justify this subtraction from concise postulates, including closed-system consistency, environment-only neutrality, and invariance under dilation gauge transformations that leave the channel unchanged. This leads to a companion quantity, noise complexity, quantifying the loss of geometric complexity relative to a prescribed ideal closed evolution. We establish a coherence-based lower bound for unitary geometric complexity, derive structural properties such as linear time scaling under time-homogeneous dilations, and obtain dissipator--controlled bounds in the Markovian (GKSL/Lindblad) regime under a standard dilation construction. Finally, we illustrate the framework on canonical benchmark noise models, including dephasing, amplitude damping, and depolarizing (Pauli) channels.
Abstract Nielsen’s geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary closed-system dynamics. For open evolutions, the reduced dynamics is described by time-dependent families of quantum channels { Λ t } t ⩾ 0 admitting many inequivalent Stinespring realizations, so any meaningful complexity notion must specify which microscopic resources are counted and which transformations are treated as gauge. We develop a dilation-based geometric complexity framework that distinguishes implementation-dependent complexity, attached to a specific realization, from intrinsic channel complexity obtained by minimizing over a physically admissible class of dilations. The proposed functional is subtractive: it removes purely environmental contributions from the geometric cost of the joint unitary realization. The subtraction is derived from concise structural postulates—closed-system consistency, environment-only neutrality, and invariance under dilation gauge transformations—and admits a natural variational and geometric interpretation. We also define noise complexity as a companion quantity measuring the geometric deviation of a noisy implementation from a prescribed ideal closed-system evolution. In the Hilbert–Schmidt setting, we establish coherence-based lower bounds for unitary and noise complexity, prove linear time scaling under time-homogeneous dilation models, and obtain dissipator-controlled bounds in the Markovian GKSL regime. We illustrate the framework on canonical single-qubit benchmark channels: dephasing, amplitude damping, and Pauli/depolarizing noise.
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.
The unitary dynamics of quantum systems can be modeled as a trajectory on a Riemannian manifold. This theoretical framework naturally yields a purely geometric interpretation of computational complexity for quantum algorithms, a notion originally developed by Michael Nielsen (Circa, 2007). However, for nonunitary dynamics, it is unclear how one can recover a completely geometric characterization of Nielsen-like geometric complexity. The main obstacle to overcome is that nonunitary dynamics cannot be characterized by Lie groups (which are Riemannian manifolds), as is the case for unitary dynamics. Building on Nielsen's work, we present a definition of geometric complexity for a fairly generic family of quantum channels. These channels are useful for modeling noise in quantum circuits, among other things, and analyze the geometric complexity of these quantum channels.
We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions ---many examples of which have been recently proposed in the literature--- are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd's bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov's coadjoint orbit action, providing an explicit realization of the ``complexity equals action'' idea; iii) it arises from a Hamilton-Jacobi analysis of the ``quantum action'' describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.
In the study of open quantum systems, one commonly describes the evolution of a system of interest through reduced dynamics, obtained by treating the environment indirectly rather than as a part of the full model. This thesis presents an expository account of an alternative, dilation-based viewpoint in the finite-dimensional setting, where a family of reduced dynamics is represented through unitary evolution on a larger system consisting of the original system together with an ancillary environment. After reviewing the reduced-dynamics perspective and the language of quantum channels, we formulate finite-dimensional quantum dynamics as channel-valued dynamical curves and use this framework to discuss Stinespring dilations of such curves. We then present exact dilation results for analytic dynamical curves, explain the singular behavior that can arise at t=0, and describe approximation results showing that Lipschitz-continuous dynamical curves admit approximate finite-dimensional Stinespring dilations. The thesis therefore provides a mathematically focused introduction to dilation-based modeling of quantum dynamics and argues that a change of perspective can lead to new ways of formulating problems in the theory of open quantum systems.
The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps. Although this unattainability principle has been quantified for specific state-to-state transitions, a universal, dynamics-independent bound for implementing a state-agnostic reset map remains elusive. In this work, we unveil the fundamental limits of physical map implementation by deriving a trade-off relation based on geometric complexity. By analyzing continuous paths of maps on a geometric manifold, we prove that the geometric complexity of any classical stochastic map or quantum channel is bounded from below by its execution error. As a consequence, we show that achieving zero error in a state-reset operation requires a divergent geometric complexity -- a unified measure that naturally incorporates disparate physical resources, including infinite time, energetic cost, or control bandwidth. This unattainability principle holds universally across both classical and quantum regimes, establishing a strict geometric limit on the physical realization of reset operations in thermodynamic control and quantum computation.