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arXiv:2609.20987

Isometry Groups of Right Invariant Metrics in Geometric Quantum Complexity

Xiaobo Liu, Lei Zheng

math.DGquant-phSubmitted Sep 17, 2026• Announced Sep 21, 2026• 0 citations
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Abstract

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 $\mathrm{SU}(2^N)$. These metrics have been used by physicists to study Nielsen's geometric approach for complexities in quantum computations.

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28 pages

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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.
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.
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.
A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.
This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds. In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.