Krylov Complexity

TL;DR

Krylov complexity (KC) provides a canonically defined, parameter-free measure of operator and state growth via the Lanczos (Krylov) basis, unifying quantum chaos diagnostics with holographic ideas. The framework links operator spreading, OTOCs, and spectral data to a one-dimensional Krylov chain whose hopping is governed by Lanczos coefficients; it yields optimality theorems that position KC as a robust representative of generalized complexities. The review synthesizes KC's role as a probe of chaos and as a bridge to bulk gravity, notably through DSSYK and JT gravity, with a concrete bulk-boundary dictionary equating KC to bulk length and predicting linear-in-time growth followed by exponential/saturated behavior at Heisenberg times. It also surveys numerical methods, extensions (multiseed, time-dependent, open systems), and nonperturbative bulk aspects, highlighting KC as a promising platform for understanding complexity, chaos, and holography across quantum many-body systems and gravity. Finally, it outlines open problems and directions, including KC in higher-dimensional QFT, refined bulk duals, and experimental realizations.

Abstract

We introduce and review a new complexity measure, called `Krylov complexity', which takes its origins in the field of quantum-chaotic dynamics, serving as a canonical measure of operator growth and spreading. Krylov complexity, underpinned by the Lanczos algorithm, has since evolved into a highly diverse field of its own right, both because of its attractive features as a complexity, whose definition does not depend on arbitrary control parameters, and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times, but also because of its relevance to seemingly far-afield subjects such as holographic dualities and the quantum physics of black holes. In this review we give a unified perspective on these topics, emphasizing the robust and most general features of K-complexity, both in chaotic and integrable systems, state and prove theorems on its generic features and describe how it is geometrised in the context of (dual) gravitational dynamics. We hope that this review will serve both as a source of intuition about K-complexity in and of itself, as well as a resource for researchers trying to gain an overview over what is by now a rather large and multi-faceted literature. We also mention and discuss a number of open problems related to K-complexity, underlining its currently very active status as a field of research.

Figures (21)

• Figure 1: Krylov complexity quantifies operator growth, measured in an optimal basis, here denoted $\{|{\cal O}_n) \}$ and defined in Section \ref{['sec:Lanczos_operator']}. Scrambling dynamics of quantum information proceeds via operator spreading, as shown in the Figure above. This finds its expression in the universal operator growth hypothesis of Parker:2018yvk, relating the scrambling exponent $\lambda_L$ to the slope of the asymptotic growth rate of the Lanczos coefficients, $\alpha$. In situations where the Hilbert space is (effectively) finite, this linear growth cannot last forever, instead the different epochs of behavior that follow, reveal further characteristic properties of the underlying system. Measuring the complexity and quantifying the different stages of this process constitutes the subject of this review. K-complexity shows non-trivial evolution over the full time space covered in the above diagram, while 'naive' size complexity already saturates at its maximal value by the scrambling time, $t_s$.
• Figure 2: Krylov space is a subspace of the full Hilbert space, optimally chosen to accommodate time evolution of states and operators. Without loss of generality we show the operator case. Here the relevant Hilbert space has dimension $D^2$ and one can upper-bound the dimensions of the Krylov subspace by $D^2 - D -1$. Krylov complexity is defined by the spread of an operator or state along the Krylov chain.
• Figure 3: Gate complexity as discrete motion through the space of unitaries (shown in dark blue). At each step we apply a single gate among a small set of universal quantum gates, for example those given in the inset. The total number of gates needed to reach the final state from a given initial state, within a tolerance $\epsilon$ is the sought-after gate complexity. In this case there are six steps, and we assign a complexity of six.
• Figure 4: Nielsen complexity is defined in terms of the optimal path connecting two states in the space of unitaries with respect to some cost function, $F[Y^I]$, depending on a set of parameters $Y^I$. Different cost functions may be appropriate in different contexts (see Section \ref{['subsect:KC_geometry']}).
• Figure 5: $\mathcal{N}=4$ super Yang-Mills in 4d is conjectured to be dual to type IIB string theory on AdS$_5$$\times$$S_5$ containing $N$ units of Ramond-Ramond flux. The salient feature of this holographic duality is that five-dimensional gravity in the "AdS-bulk" is dually described by four-dimensional Yang-Mills theory on the boundary. The compact $S^5$ factor is dually related to the internal symmetries of the SYM theory.