CFT Complexity and Penalty Factors

TL;DR

This paper develops a general framework for quantum circuit complexity in conformal field theories using penalty factors, constructing a metric on state space, and applies it to compute complexities in 1D and 2D CFTs, connecting to holography.

Contribution

It introduces a novel method to incorporate penalty factors into complexity measures for Lie groups, with a geometric approach applicable to CFTs and other quantum systems.

Findings

Derived analytic complexity results when a penalty factor is off

Developed perturbative expansions for small penalty factors

Provided numerical complexity calculations for general penalty choices

Abstract

Quantum complexity of conformal field theory (CFT) states has recently gained significant attention, both as a diagnostic tool in condensed matter systems and in connection with holographic observables probing black hole interiors. Previous studies have primarily focused on cases where all generators of the conformal group contribute equally to the cost of building a circuit. In this work, we present a general framework for studying the complexity of circuits in generic Lie groups, where penalty factors assign relative weights to different generators. Our approach constructs a metric on the coset space of quantum states, induced from a (pseudo-)Riemannian norm on the space of unitary circuits. The geodesics of this metric are interpreted as optimal circuits. The method builds on the formalism of (pseudo-)Riemannian submersions and connects naturally to other prescriptions in the literature, including cost function minimization along stabilizer directions and constructions based on coadjoint orbits. As a concrete application, we compute state complexity for states in one- and two-dimensional CFTs. For specific choices of penalty factors, our prescription yields a positive-definite metric with a viable interpretation as complexity; in other cases, the resulting metric is indefinite. In the viable regime, we derive analytic results when a specific penalty factor is turned off, develop perturbative expansions for small values of the penalty factors, and provide numerical results in the general case. We comment on the relation of our measure of complexity to holography.