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arXiv:2507.18440

Geometric Measures of Complexity for Open and Closed Quantum Systems

Alberto Acevedo, Antonio Falcó

Published Jul 24, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2507.18440

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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.
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 {Λt}t≥0\{Λ_t\}_{t\ge 0}{Λt​}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.
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.
In this paper, we give a complete description for the full isometry groups of a class of right invariant Riemannian metrics on the special unitary group SU(2N)\mathrm{SU}(2^N)SU(2N). These metrics have been used by physicists to study Nielsen's geometric approach for complexities in quantum computations.
Nielsen's geometric approach offers a powerful framework for quantifying the complexity of unitary transformations. In this formulation, complexity is defined as the length of the minimal geodesic in a suitably constructed geometric space associated with the Lie group of relevant operators. Despite its conceptual appeal, determining geodesic distances on Lie group manifolds is generally challenging, and existing treatments often rely on perturbative expansions in the structure constants. In this work, we circumvent these limitations by employing a finite-dimensional matrix representation of the generators, which enables an exact computation of the geodesic distance and hence a precise determination of the complexity. We focus on the su(1,1)\mathfrak{su}(1,1)su(1,1) Lie algebra, relevant for quantum scalar fields evolving on homogeneous and isotropic cosmological backgrounds. The resulting expression for the complexity is applied to de Sitter spacetime as well as to asymptotically static cosmological models undergoing contraction or expansion.
Understanding and mitigating noise in quantum systems is a fundamental challenge in achieving scalable and fault-tolerant quantum computation. Error modeling for quantum systems can be formulated in many ways, some of which are very fundamental, but hard to analyze (evolution by general dynamical map) and others perhaps too simplistic to represent physical reality. In this paper, we present an intermediate approach, introducing the random Schrödinger equation, with a noise term given by a time-varying random Hermitian matrix as a means to model noisy quantum systems. We derive bounds on the error of the synthesized unitary in terms of bounds on the norm of the noise, and show that for certain noise processes these bounds are tight. We then show that in certain situations, minimizing the error is equivalent to finding a geodesic on SU ( n ) with respect to a Riemannian metric encoding the coupling between the control pulse and the noise process, thus connecting our work to the complexity geometry pioneered by Michael Nielsen.