Charged complexity and the thermofield double state

Abstract

We establish a systematic framework for studying quantum computational complexity of Gaussian states of charged systems based on Nielsen’s geometric approach. We use this framework to examine the effect of a chemical potential on the dynamics of complexity. As an example, we consider the complexity of a charged thermofield double state constructed from two free massive complex scalar fields in the presence of a chemical potential. We show that this state factorizes between positively and negatively charged modes and demonstrate that this fact can be used to relate it, for each momentum mode separately, to two uncharged thermofield double states with shifted temperatures and times. We evaluate the complexity of formation for the charged thermofield double state, both numerically and in certain analytic expansions. We further present numerical results for the time dependence of complexity. We compare various aspects of these results to those obtained in holography for charged black holes.

Holographic and QFT complexity with angular momentum

“…As we will show explicitly, it is then easy to give an effective description of the rotating TFD state (4.1) in terms of the non-rotating one (4.3), and make use of the available Gaussian state results [85]. This is analogous to what happens for the charged TFD studied in [142], which can also be given an effective description in terms of (4.3).…”

Subsystem complexity after a global quantum quench

“…Besides the reference state, the target state and the set of allowed gates, the circuit complexity can depend also on the tolerance parameter for the target state. Many results have been obtained for the complexity of quantum circuits made by pure states constructed through lattice models [17][18][19][20][21][22][23][24][25] and in the gravitational side of the holographic correspondence. Some proposals have been done also to study the circuit complexity in quantum fields theories [26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Holographic complexity of rotating black holes

“…Compared to the most general Myers-Perry-AdS black holes, these solutions enjoy enhanced symmetry that considerably simplifies the analysis of the causal structure. This particular configuration has some similarities with the charged case [16,50], however, we shall see that there are interesting differences.…”