Complexity measures in QFT and constrained geometric actions

TL;DR

The paper scrutinizes which geometric-cost measures truly bound circuit complexity in quantum field theory, showing that inhomogeneous costs fail as lower bounds and can violate Lloyd’s bound. It argues for a hierarchical choice among homogeneous, basis-independent costs, notably endorsing F_<H^2> (and F_||H||) as the most canonical, since they arise from the standard Hermitian metric and connect to coadjoint-orbit (geometric) actions and the quantum action/Hamilton–Jacobi framework. This canonical perspective yields a natural bridge between complexity, action, and the dynamics of quantum systems, including a unified treatment of chaos via quantum phase-space geometry. The approach also clarifies how penalties outside the gate set can be implemented as constraints (via Lagrange multipliers), ensuring tight lower bounds and a practical route to locality in QFT complexity. Overall, the work consolidates a canonical, action-based view of complexity in QFT and clarifies its interplay with chaos and holography.

Abstract

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.

Figures (1)

• Figure 1: An initial point in the quantum phase space, $\ket{\psi_0}$, and a nearby perturbed version of it, $e^{i\Delta_0} \ket{\psi_0}$, evolve with physical time $t$ to states $\ket{\psi_t}$ and $\ket{\psi_t^{\Delta}}$ respectively. For each $t$, the complexity between both states measured by some continuous metric $F$ is given, for a sufficiently small perturbation, by $F(\Delta(-t))$, where $\Delta(-t)=e^{-iHt}\Delta_0 e^{iHt}$. The dark gray arrow above corresponds to the straight-line geodesic connecting both states. The pale dashed one corresponds to the one directly connecting $\ket{\psi_0}$ with $\ket{\psi_t^{\Delta}}$ ---see discussion around eq. (\ref{['cfs']}).