Thoughts on Holographic Complexity and its Basis-dependence

TL;DR

This paper argues that holographic complexity, which relates to the minimal number of quantum gates needed to prepare a state, must be basis-dependent to align with its properties, challenging the basis-invariance assumption.

Contribution

It demonstrates a no-go theorem showing basis-independence cannot reproduce holographic complexity, establishing the necessity of basis dependence in the dual Hilbert space distance.

Findings

Basis-independent distances cannot replicate holographic complexity behavior.

Holographic complexity should be basis-dependent to be consistent with duality.

The minimal gate count defines a discrete Hilbert space distance.

Abstract

In this paper, we argue that holographic complexity should be a basis-dependent quantity. Computational complexity of a state is defined as a minimum number of gates required to obtain that state from the reference state. Due to this minimality, it satisfies the triangle inequality, and can be regarded as a (discrete version of) distance in the Hilbert space. However, we show a no-go theorem that any basis-independent distance cannot reproduce the behavior of the holographic complexity. Therefore, if holographic complexity is dual to a distance in the Hilbert space, it should be basis-dependent, i.e., it is not invariant under a change of the basis of the Hilbert space.