The landscape of complexity measures in 2D gravity

TL;DR

The paper extends holographic complexity beyond the conventional CV and CA proposals by developing a comprehensive complexity=anything framework for 2D dilaton gravity. It builds both a top-down (dimensional reduction) and a bottom-up construction of codimension-1 bulk observables that exhibit hallmark complexity behavior, notably late-time linear growth and the switchback effect, and it discusses the role of multiple extremal surfaces as either distinct locally optimal circuits or additive subsectors. A covariant phase-space perspective via the covariant Peierls bracket is developed to map bulk functionals to boundary complexity, formalizing a bulk–boundary dictionary and clarifying the scheme dependence of holographic complexity. The work provides a flexible, analytically tractable platform to explore quantum complexity in AdS$_2$/CFT$_1$ and related low-dimensional holographies, with potential connections to JT gravity, SYK-like models, and semi-classical quantum corrections. Overall, it lays groundwork for a richer, geometrically grounded understanding of complexity in low-dimensional holographic dualities and outlines concrete directions for refining the bulk–boundary correspondence and extending to more general observables.

Abstract

We investigate the broad landscape of holographic complexity measures for theories dual to two-dimensional (2D) dilaton gravity. Previous studies have largely focused on the complexity=volume and complexity=action proposals for holographic complexity. Here we systematically construct and analyze a wide class of generalized complexity functionals, focusing on codimension-one bulk observables. Two complementary approaches are presented: one inspired by dimensional reduction of codimension-one observables from higher-dimensional gravity, and another that adopts a purely 2D perspective. We verify the resulting observables exhibit hallmark features of complexity, such as linear growth at late times and the switchback effect. We further offer heuristic interpretations of the role of multiple extremal surfaces when they appear. Finally, we comment on the bulk-to-boundary dictionary via the covariant Peierls bracket in 2D gravity. Our work lays the groundwork for a richer understanding of quantum complexity in low-dimensional holographic dualities.

Figures (6)

• Figure 1: Thermofield double state dual to a (neutral) two sided eternal black hole. The diagonal lines (red) denote the black hole event horizon. The extremal codimension-1 hypersurface (in blue) $\Sigma_{\tau}$ time evolves into the extremal surface $\Sigma_{\tau'}$, the evolution of which has a dual description in terms of the TFD evolution.
• Figure 2: Parameter scan to determine the number of maxima in the effective potential for different curvature invariants in $F_{1}=1+\lambda$(scalar). Light blue regions denote when $U$ has a single maxima, dark blue regions represent potentials with two local maxima, purple regions represent potentials with three maxima, and dark purple regions show four maxima. Notice the $R^{2}$ curvature invariant is trivial, having only a single maximum (as is the case for complexity=volume). Meanwhile, $C_{\mu\nu\rho\sigma}^{2}$ produces potentials which can have up to four maxima. The right-most black regions represent disallowed values of charge, when the black hole becomes extremal and $U<0$. (Here we set $L=1$ and $M=1$).
• Figure 3: Effective potential with multiple maxima. The potentials we generate from our calculations have maxima that differ by many orders of magnitude, hence we have used a hand-drawn figure for clarity which still preserves all the features we need. Left: All local maxima are probed by boundary-anchored surfaces. Right: the innermost maximum cannot be probed due to the presence of a larger maximum, which acts as a gravitational barrier.
• Figure 4: Side-by-side comparison of two locally optimal circuits generating the 4-qubit GHZ state $\tfrac{1}{\sqrt{2}}(|0000\rangle + |1111\rangle)$. Left: A CNOT-based approach. Right: A CZ-based approach, common on certain hardware where CZ is the native two-qubit gate. Each uses three two-qubit gates (CNOT or CZ), the minimum needed to entangle four qubits in GHZ form.
• Figure 5: Two independent 3-qubit subcircuits (blue and red). Top: qubits $(q_1,q_2,q_3)$ form a standard GHZ circuit: Hadamard on $q_1$, followed by CNOTs ($q_1 \!\to\! q_2$ and $q_1 \!\to\! q_3$). Bottom: qubits $(q_4,q_5,q_6)$ each receive a Hadamard, then two CZ gates between $(q_4,q_5)$ and $(q_5,q_6)$ introduce phases in the amplitudes where both qubits are 1, yielding a multi-qubit superposition in this sector.