Circuit complexity in fermionic field theory

We define and calculate versions of complexity for free fermionic quantum field theories in $1+1$ and $3+1$ dimensions, adopting Nielsen’s geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cutoff dependence are discussed.

Figure 1

The $|{\beta }_{+}|$ of first plot is for the geodesic path and the second plot is for the cMERA path. In both cases the target state is the cMERA target state. The above figure shows the plots of $|{\beta }_{+}|$ vs $k$ for $\mathrm{\Lambda }=10$ and $m=1$ for the values of $\sigma =0.2$, 0.4, 0.6, 0.8, 1.0, increasing the peak of the curve signifies increasing $\sigma$. We see that while the cMERA path only acts on momenta $\frac{|k|}{\mathrm{\Lambda }}<\sigma$, the geodesic path alters $|{\beta }_{+}|$ for all the different momenta.