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arXiv:2512.14875

Exact formula for geometric quantum complexity of cosmological perturbations

Satyaki Chowdhury, Jakub Mielczarek

hep-thgr-qcquant-phSubmitted Dec 16, 2025• Announced Dec 18, 2025
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Abstract

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 $\mathfrak{su}(1,1)$ 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.

Comments
45 pages including references, 7 figures

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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.
A bstract According to the pioneering work of Nielsen and collaborators, the length of the minimal geodesic in a geometric realization of a suitable operator space provides a measure of the quantum complexity of an operation. Compared with the original concept of complexity based on the minimal number of gates required to construct the desired operation as a product, this geometrical approach amounts to a more concrete and computable definition, but its evaluation is nontrivial in systems with a high-dimensional Hilbert space. The geometrical formulation can more easily be evaluated by considering the geometry associated with a suitable finite-dimensional group generated by a small number of relevant operators of the system. In this way, the method has been applied in particular to the harmonic oscillator, which is also of interest in the present paper. However, subtle and previously unrecognized issues of group theory can lead to unforeseen complications, motivating a new formulation that remains on the level of the underlying Lie algebras for most of the required steps. Novel insights about complexity can thereby be found in a low-dimensional setting, with the potential of systematic extensions to higher dimensions as well as interactions. Specific examples include the quantum complexity of various target unitary operators associated with a harmonic oscillator, inverted harmonic oscillator, and coupled harmonic oscillators. The generality of this approach is demonstrated by an application to an anharmonic oscillator with a cubic term.
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.
Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our approach is based on the observation that the whole problem refers to the structure of the underlying group. The questions concerning the complexity of particular unitaries are solved by lifting the abstract structure to the operator level by considering the relevant unitary representation. For the class of right-invariant metrics obeying natural invariance condition we solve the geodesic equations on oscillator group. The solution is given explicitly in terms of elementary functions. Imposing the boundary conditions yield a transcendental equation and the length of the geodesic is given in terms of the solutions to the latter. Since the unitary irreducible representations of oscillator group are classified this allows us to compute, at least in principle, the complexity of any unitary operator belonging to the representation.
In this paper, we study circuit complexity for a free vector field of a [Formula: see text] gauge theory in Coulomb gauge, and Gaussian states. We introduce a quantum circuit model with Gaussian states, including reference and target states. Using Nielsen’s geometric approach, the complexity then can be found as the shortest geodesic in the space of states. This geodesic is based on the notion of geodesic distance on the Lie group of Bogoliubov transformations equipped with a right-invariant metric. We use the framework of the covariance matrix to compute circuit complexity between Gaussian states. We apply this framework to the free vector field in general dimensions where we compute the circuit complexity of the ground state of the Hamiltonian.
We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry and selecting a Nielsen complexity metric compatible with the Krylov metric. Up to normalization, the Krylov complexity of a Hermitian operator then equals the length squared of a straight-line trajectory on the manifold of unitaries that connects the identity operator with a precursor operator. The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic. While for general systems we can only establish saturation in the limit of small precursors, we provide evidence that in broad classes of models and for suitable initial operators there is a precise correspondence between Krylov complexity and (the square of) Nielsen complexity for a finite range of precursors.