# Circuit Complexity and 2D Bosonisation

## TL;DR

This paper investigates the circuit complexity of free bosons and fermions in 1+1 dimensions, using bosonisation to compare perspectives and analyze how different gate choices affect complexity results.

## Contribution

It demonstrates how the discrepancy in complexity between bosonic and fermionic states can be explained by gate set choices, especially in the context of bosonisation equivalence.

## Key findings

- Different gate sets lead to reconciled complexity results for bosonic and fermionic states.
- Mode-dependent cost functions can unify complexity measures for certain state classes.
- Complexity behavior varies significantly with cutoff and state class, especially for states that are both bosonic- and fermionic-gaussian.

## Abstract

We consider the circuit complexity of free bosons, or equivalently free fermions, in 1+1 dimensions. Motivated by the results of [1] and [2, 3] who found different behavior in the complexity of free bosons and fermions, in any dimension, we consider the 1+1 dimensional case where, thanks to the bosonisation equivalence, we can consider the same state from both the bosonic and the fermionic perspectives. In this way the discrepancy can be attributed to a different choice of the set of gates allowed in the circuit. We study the effect in two classes of states: i) bosonic-coherent / fermionic-gaussian states; ii) states that are both bosonic- and fermionic-gaussian. We consider the complexity relative to the ground state. In the first class, the different results can be reconciled admitting a mode-dependent cost function in one of the descriptions. The differences in the second class are more important, in terms of the cutoff-dependence and the overall behavior of the complexity.