Abstract
We use Nielsen's geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and non-equilibrium steady states of the Kitaev model exhibit non-analytical behaviors at the critical points, and thus can be used to detect both {\it equilibrium} and {\it dynamical} topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen's circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and discuss generalizations to higher dimensions. Our result opens up a new avenue for using circuit complexity as a novel tool to understand quantum many-body systems.
Highlights
In computer science, the notion of computational complexity refers to the minimum number of elementary operations for implementing a given task
This concept readily extends to quantum information science, where quantum circuit complexity denotes the minimum number of gates to implement a desired unitary transformation
The quantum state complexity is quantified by imposing a cost functional F [H (t )] on the control Hamiltonian H (t )
Summary
The notion of computational complexity refers to the minimum number of elementary operations for implementing a given task. Nielsen’s approach has been adopted in high-energy physics to quantify the complexity of quantum field theory states [6,7,8,9,10,11,12,13,14,15,16,17,18] The connection between the geometric definition of circuit complexity and quantum phase transitions has so far remained unexplored This connection is important fundamentally, and it is intimately related to the long-standing problem of quantum state preparations across critical points [25,26,27]. (b) phase is characterized by a nonzero winding number and the presence of Majorana edge modes [28,29,30,31,32,33]
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