Toward a Definition of Complexity for Quantum Field Theory States

We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form $\mathfrak{s}\mathfrak{u}(1,1)$ algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.

Figure 1

The Poincaré disk, parametrized by real (horizontal) and imaginary (vertical) components of ${\gamma }_{+}$. Examples of geodesics appear as dashed lines. The two dots indicate the reference state (the center) and the target state (on the real axis). The geodesic connecting the two is the straight solid line along the diameter, corresponding to the generator $K(\overrightarrow{k})$. The solid semicircle is the nongeodesic path generated by $B(\overrightarrow{k},M)$ (see Sec. D of Ref. [23]). The $\mathfrak{s}\mathfrak{u}(1,1)$ algebra generates isometries on the hyperbolic plane.