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Geometric Complexity of Quantum Channels via Unitary Dilations

Alberto Acevedo, Antonio Falcó

TL;DR

This work extends Nielsen’s geometric approach to quantum circuit complexity from closed, unitary dynamics to open-system dynamics described by quantum channels. It introduces a dilation-based complexity functional that separates an implementation-dependent cost from an intrinsic channel cost, via a subtractive term that removes purely environmental contributions; a companion noise complexity quantifies the loss of geometric cost relative to an ideal closed evolution. A core result is a coherence-based lower bound on unitary geometric complexity, along with a linear-in-time scaling law for time-homogeneous dilations and GKSL-bound estimates in the Markovian regime. The framework is validated on canonical noise models—pure dephasing, amplitude damping, and depolarizing channels—showing predictable trends that link dissipator strength to the growth of geometric complexity and the associated noise complexity. Together, the results provide a principled way to quantify the cost of noisy implementations and connect GKSL parameters to geometric cost, offering a bridge between open-quantum dynamics and quantum-control complexity theory.

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.

Geometric Complexity of Quantum Channels via Unitary Dilations

TL;DR

This work extends Nielsen’s geometric approach to quantum circuit complexity from closed, unitary dynamics to open-system dynamics described by quantum channels. It introduces a dilation-based complexity functional that separates an implementation-dependent cost from an intrinsic channel cost, via a subtractive term that removes purely environmental contributions; a companion noise complexity quantifies the loss of geometric cost relative to an ideal closed evolution. A core result is a coherence-based lower bound on unitary geometric complexity, along with a linear-in-time scaling law for time-homogeneous dilations and GKSL-bound estimates in the Markovian regime. The framework is validated on canonical noise models—pure dephasing, amplitude damping, and depolarizing channels—showing predictable trends that link dissipator strength to the growth of geometric complexity and the associated noise complexity. Together, the results provide a principled way to quantify the cost of noisy implementations and connect GKSL parameters to geometric cost, offering a bridge between open-quantum dynamics and quantum-control complexity theory.

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.
Paper Structure (52 sections, 13 theorems, 115 equations)

This paper contains 52 sections, 13 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Notations and Conventions
  3. Hilbert spaces, operators, and norms
  4. System--environment split and unitary dynamics
  5. Quantum channels and Kraus/Stinespring representations
  6. Dilation "gauge" freedom and implementation dependence
  7. Reference basis and coherence conventions
  8. Unitary Geometric Complexity
  9. Right--invariant Riemannian metrics on $SU(N)$
  10. Homogeneous versus anisotropic geometries
  11. Example: penalizing nonlocal interactions (multi--qubit setting)
  12. Length, distance, and unitary geometric complexity
  13. Control (Hamiltonian) representation of the length functional
  14. Geodesic equation (Euler--Arnold form)
  15. Complexity Functionals for Channels: Definitions and Invariances
  16. ...and 37 more sections

Key Result

Lemma 1

Let $\mathfrak{D}=(\mathscr{H}_E,\hat{\rho}_E,\hat{H}_{tot})$ be a dilation of $\Lambda_t$ and let $V_E$ be any unitary operator on $\mathscr{H}_E$. Define the transformed dilation data Then $\Lambda_t^{(\mathfrak{D}')}=\Lambda_t^{(\mathfrak{D})}$ for all $t$, and moreover

Theorems & Definitions (37)

  • Definition 1: Geometric complexity of a unitary
  • Lemma 1: Invariance under environment basis changes
  • proof
  • Definition 2: Admissible dilation set
  • Definition 3: Intrinsic channel complexity
  • Remark 1: Why constraints are necessary
  • Remark 2: Attainment of the infimum
  • Definition 4: Postulates for a dilation-based channel complexity
  • Lemma 2: Gauge covariance of the squared discrepancy
  • proof
  • ...and 27 more