Abstract
We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen’s approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.
Highlights
CA conjecture identifies the complexity of the boundary state with the gravitational action evaluated at special bulk region called the Wheeler-DeWitt patch, i.e., the causal development of the bulk surface identified in the previous approach
We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen’s geometric approach as in [1]
We find that the complexities, as well as the optimal circuits, derived from these two approaches, are different
Summary
Our goal is to evaluate the complexity of coherent states in a free scalar field theory, applying the techniques of [1]. As a warm-up exercise, we begin here by considering coherent states in the simpler system of two coupled harmonic oscillators. Our focus will be on the F2 cost function (1.7), and on the κ = 2 cost function (1.8) which are extremized by the same trajectories. Our approach here closely follows that in [1] and we refer the reader there for a more detailed discussion
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