Complexity in the presence of a boundary

TL;DR

This work examines how a boundary modifies circuit complexity in two-dimensional theories using both holographic and lattice approaches. It finds that the leading UV divergence is unchanged by the boundary for CV, CA, and CV2.0, but CV and CV2.0 acquire boundary-dependent subleading logarithmic terms while CA does not; in addition, holographic subregion complexity can exhibit abrupt jumps as the subregion moves relative to the boundary. On the lattice side, the Dirichlet-boundary harmonic chain reveals a boundary-induced log N term for C1 and a (log N)^2 term for Ckappa=2, illustrating that boundary effects on complexity depend crucially on the chosen framework. Collectively, the results illuminate how boundaries imprint distinct signatures on complexity across holographic prescriptions and discretized field theories, with potential implications for BCFT diagnostics and quantum information in bounded geometries.

Abstract

The effects of a boundary on the circuit complexity are studied in two dimensional theories. The analysis is performed in the holographic realization of a conformal field theory with a boundary by employing different proposals for the dual of the complexity, including the "Complexity = Volume" (CV) and "Complexity = Action" (CA) prescriptions, and in the harmonic chain with Dirichlet boundary conditions. In all the cases considered except for CA, the boundary introduces a subleading logarithmic divergence in the expansion of the complexity as the UV cutoff vanishes. Holographic subregion complexity is also explored in the CV case, finding that it can change discontinuously under continuous variations of the configuration of the subregion.

Figures (6)

• Figure 1: Holographic CV complexity in the AdS$_3$/BCFT$_2$ setup. The red half line corresponds to the constant time slice of the spacetime where the BCFT$_2$ is defined. The holographic CV complexity (see §\ref{['sezCalcoloCV']}) is the volume of the yellow plane, delimited by the UV cutoff $\epsilon$, by the IR cutoff $z_{IR}$ and by the brane $Q$ (solid blue semi-infinite line).
• Figure 2: The future half of the regularized WDW patch for AdS$_3$ with a boundary.
• Figure 3: The complexity $\mathcal{C}_1$ when $\omega \neq 0$ with respect to the massless result (\ref{['C1 massless large N']}). In the left panel we show that for large $N$ this difference is a constant depending on $\omega$ (here $L \omega_R=0.5$). In the right panel the dependence on $\omega$ is shown for $N=10000$.
• Figure 4: The logarithmic term of $\mathcal{C}_2$ when $\omega=0$ for large $N$ (see (\ref{['C2-expansion']})).
• Figure 5: RT surfaces and corresponding spacetime regions relevant for the subregion complexity of an interval of length $\ell$. In figure (a) the interval is attached to the boundary. In cases (b) and (c) it is at finite distance $d$ from the boundary.