Circuit complexity for free fermions

TL;DR

This work develops a Nielsen-geometric framework for circuit complexity in free fermionic quantum field theories, recasting complexity as geodesic length on the orthogonal group $SO(2N)$ and exploiting a $U(N)$ stabilizer to define equivalence classes of circuits. By leveraging Kähler structures and the $ ext{SO}(2N)/ ext{U}(N)$ state manifold, the authors provide a comprehensive method to compute the complexity between arbitrary fermionic Gaussian states, including the ground state and a broad class of excited states of the free Dirac field in four dimensions. They demonstrate that minimal circuits decompose into independent two-mode fermionic squeezings in suitable normal modes, and they quantify the UV divergences and their dependence on reference states and cost functions, drawing parallels and contrasts with bosonic cases and holographic complexity. The results also establish connections to the Fubini-Study metric on the Gaussian-state manifold, and the analysis offers a platform for exploring extensions to other dimensions, chiral fermions, and non-Gaussian states with potential implications for quantum simulations and holography. Overall, the paper provides a rigorous, geometrically grounded framework for fermionic circuit complexity with concrete, per-mode results and clear paths for future generalizations.

Abstract

We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited states. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.

Figures (10)

• Figure 1: This figure illustrates the geometry of the Lie group $\mathcal{G}$ with stabilizer subgroup $\mathrm{Sta}$, whose elements $u$ satisfy $u|\psi_{\mathrm{R}}\rangle=|\psi_{\mathrm{R}}\rangle$. This subgroup induces a fibration of $\mathcal{G}$ into equivalence classes given by displaced stabilizers $[U]=U\,\mathrm{Sta}$. The complexity of a target state $|\psi_\mathrm{T}\rangle=U|\psi_\mathrm{R}\rangle$ is then given by the minimal path $\gamma$ to a point on $[U]$, of which we illustrate two examples. $\gamma_1$ goes from $\mathbb{1}$ to $U$ and $\gamma_2$ from $\mathbb{1}$ to $Uu$ where $u\in\mathrm{Sta}$.
• Figure 2: This plot shows the function $Y(m,\mathbf{p},{\bar{s}})$ in eq. (\ref{['march']}) describing the complexity per mode of a massive Dirac field in its ground state as a function of $|\mathbf{p}|$. Note that there is a single universal curve if we consider this as a function of $|\mathbf{p}|/m$.
• Figure 3: This plot shows the function $\tilde{Y}(m,\mathbf{q},{\bar{r}})$ describing the complexity of the modes excited in the state in eq. (\ref{['excite8']}), i.e., $|\tilde{\psi}\rangle=a^{{\bar{r}}\dagger}_\mathbf{q}\,b^{{\bar{r}}\dagger}_{-\mathbf{q}}\,|0\rangle$, as a function of $|\mathbf{q}|$.
• Figure 4: This plot shows the function $Y(m,\mathbf{p},{\bar{s}})$ for $\mathbf{p}=(p_x,0,p_z)$ and a rotationally reference state $|M,\mathbf{q}=0\rangle$.
• Figure 5: This plot shows the function $Y(m,\mathbf{p},s)$ for $\mathbf{p}=(p_x,0,p_z)$ and a massless reference state $|M=0,\mathbf{q}=(0,0,q)\rangle$.