Sharp Transitions for Subsystem Complexity

TL;DR

The paper analyzes how quantum circuit complexity for subsystems evolves under time-dependent dynamics, comparing holographic predictions with random quantum circuit models. It reveals a sharp half-system-size transition: subsystems with $p<1/2$ quickly saturate to low complexity, while $p>1/2$ exhibit linear growth over exponentially long times, with a corresponding finite-temperature refinement predicting an additional sub-half transition. The authors combine holographic calculations (complexity via volume) and rigorous random quantum circuit analyses (design-based bounds) to establish the half-size transition and to provide evidence for the time-based transitions, linking geometric entanglement wedge properties to information-theoretic complexity in generic quantum many-body systems. This work offers a unified framework connecting AdS/CFT, entanglement structure, and circuit complexity, yielding concrete predictions for subsystem dynamics and guiding future rigorous proofs in chaotic open quantum systems. The findings have potential implications for understanding bulk locality, black hole interiors, and the complexity growth of mixed states beyond pure-state settings.

Abstract

The circuit complexity of time-evolved pure quantum states grows linearly in time for an exponentially long time. This behavior has been proven in certain models, is conjectured to hold for generic quantum many-body systems, and is believed to be dual to the long-time growth of black hole interiors in AdS/CFT. Achieving a similar understanding for mixed states remains an important problem. In this work, we study the circuit complexity of time-evolved subsystems of pure quantum states. We find that for greater-than-half subsystem sizes, the complexity grows linearly in time for an exponentially long time, similarly to that of the full state. However, for less-than-half subsystem sizes, the complexity rises and then falls, returning to low complexity as the subsystem equilibrates. Notably, the transition between these two regimes occurs sharply at half system size. We use holographic duality to map out this picture of subsystem complexity dynamics and rigorously prove the existence of the sharp transition in random quantum circuits. Furthermore, we use holography to predict features of complexity growth at finite temperature that lie beyond the reach of techniques based on random quantum circuits. In particular, at finite temperature, we argue for an additional sharp transition at a critical less-than-half subsystem size. Below this critical value, the subsystem complexity saturates nearly instantaneously rather than exhibiting a rise and fall. This novel phenomenon, as well as an analogous transition above half system size, provides a target for future studies based on rigorous methods.

Figures (14)

• Figure 1: Hartman-Maldacena transition. Left: early-time (connected) extremal surface crossing the ER bridge. Right: late-time (disconnected) extremal surface terminating outside the horizon. The two surfaces exchange dominance at the thermalization time.
• Figure 2: The one-sided BTZ black hole Banados:1992wn (black circle = boundary, gray circle = horizon). Left: RT surfaces for various subregion sizes. Right: RT surfaces for complementary regions $A$ and $B$. In the two-sided case, only the smaller surface $\gamma_A$ is relevant.
• Figure 3: Schematic dependence of saturation time on subsystem fraction $p$ for the entire range of $p$. The dashed top line should be understood as an approximate expectation, and merely indicates an exponential separation between the saturation times for $p < 1/2$ and $p > 1/2$ (the saturation time should in fact continue to increase with $p$ for $p > 1/2$, which can be understood more precisely on the quantum information side).
• Figure 4: Schematic dependence of saturation time on temperature. The different curves correspond to infinite temperature (blue) and successively lower finite temperatures (orange, red). The blue curve ($\beta = 0$) is well-understood in both random quantum circuits and holography: it describes a $p^{1/(d - 1)}$ power law and starts at $p = 0$ (for instance, in 1D RQCs with $d = 2$, this curve would be linear). The other curves have finite $p_\text{crit}$, and their shape is conjectural. They may reflect the behavior of, e.g., $U(1)$ charge-conserving random circuits.
• Figure 5: $p_\text{crit}$ versus $\beta$ for $d = 2, 3, 4$ with fits (in darker colors) to linear, quadratic, and cubic power laws, respectively. Deviations from power-law behavior are visible at large $\beta$ (small $\mu$): the data points are higher for odd $d$ and lower for even $d$.

Theorems & Definitions (20)

• Conjecture 1
• Definition 1: $\varepsilon$-approximate unitary $k$-designs
• Definition 2: Brickwork random quantum circuits
• Definition 3: Patchwork random quantum circuits
• Remark 1
• Definition 4: Mixed state complexity
• Theorem 1: Complexity growth for $n_A > n_B$
• Theorem 2: Complexity growth for $n_A < n_B$
• proof : Proof of \ref{['thm:complexity_growth_na_gt_nb']}
• Proposition 1