Entangling Power and Quantum Circuit Complexity

Notions of circuit complexity and cost play a key role in quantum computing and simulation where they capture the (weighted) minimal number of gates that is required to implement a unitary. Similar notions also become increasingly prominent in high energy physics in the study of holography. While notions of entanglement have in general little implications for the quantum circuit complexity and the cost of a unitary, in this work, we discuss a simple such relationship when both the entanglement of a state and the cost of a unitary take small values, building on ideas on how values of entangling power of quantum gates add up. This bound implies that if entanglement entropies grow linearly in time, so does the cost. The implications are twofold: It provides insights into complexity growth for short times. In the context of quantum simulation, it allows us to compare digital and analog quantum simulators. The main technical contribution is a continuous-variable small incremental entangling bound.

Figure 1

A schematic picture relating the cost of a circuit with the entanglement over cuts for a system of $n$ constituents. The dark gray triangle represents the Lieb-Robinson cone [32, 33, 36] that depicts at what rate one expects a linear growth of the entanglement entropy over all cuts in nonequilibrium dynamics generated by a local Hamiltonian.