Circuit Complexity and 2D Bosonisation

TL;DR

The paper investigates circuit complexity for free bosons and fermions in 1+1 dimensions through the lens of 2D bosonisation, demonstrating that discrepancies between bosonic and fermionic descriptions arise from the chosen gate sets and cost function. It analyzes two classes of states: bosonic-coherent/fermionic-Gaussian states and states that are Gaussian in both descriptions, deriving both FS- and Nielsen-based results and highlighting mode-number dependencies and UV-divergence structures. The study shows that bosonic complexity can be cutoff-independent under certain coherent-state constructions, while fermionic complexity often exhibits logarithmic UV growth, and it reveals how a non-linear bosonisation map expands the gate-set, enabling new solvable comparisons. These findings illuminate how gate choices influence complexity in QFT and may offer insights for holographic interpretations and extensions to interacting theories.

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.

Figures (7)

• Figure 1: The base manifold $\mathcal{G}_B$ is made of all the bosonic ground states which are denoted as the red crosses. The vertical blue lines are modules of the fermionic fock space where each represents a bosonic Hilbert space with fixed fermion number. The operators of bosonic type can only move vertically while the fermionic ones span the whole Fock space with a single fermionic operator moving horizontally.
• Figure 2: Complexity $\mathcal{C}_{p=1}$ over ${\alpha\over \sqrt{n} }$ is constant in $|\alpha|$, for $n\in[20,100]$ with an interval of 20, where $n$ labels the bosonic excitation $b_n^\dagger$ in \ref{['eq:onemode']}. The cutoff on the size of the matrices $A(n)$ and $B(n)$ is set equal to $10n$.
• Figure 3: Complexity $\mathcal{C}_{p=2}$ increases when $|\alpha|$ increases for $n\in[2,10]$ with an interval of 2, where $n$ labels the bosonic excitation $b_n^\dagger$ in \ref{['eq:onemode']}. The length of the matrices $A(n)$ and $B(n)$ is cut to be twenty times of the bosonic excitation $n$ for the plot.
• Figure 4: Complexity for two-mode shifts with $n_1=10, n_2=17$. The cutoff is chosen to be $N=200$ for the two-mode shift covariance matrice, which is similar to one of the diagonal blocks in \ref{['eq:delta2']}.
• Figure 5: The complexity for bosonic and fermionic gaussian states with $p=1$ and $p=2$ norms. (a) and (c) represent the bosonic case while (b) and (d) represent the fermionic case. The cutoff is chosen to be $N=40$.