Quantum State Complexity in Computationally Tractable Quantum Circuits

TL;DR

This paper investigates the complexity of quantum states generated by a special class of quantum circuits called automaton circuits, showing they can produce highly entangled states with complexity growth similar to Haar random states, using numerical methods.

Contribution

It introduces automaton circuits as a numerically tractable model for studying quantum complexity growth, bridging theoretical insights with numerical evidence.

Findings

Automaton wave functions closely mimic Haar random states.

Complexity, measured via out-of-time ordered correlators, grows linearly over time.

Automaton circuits can be used to study complexity growth beyond scrambling time.

Abstract

Characterizing the quantum complexity of local random quantum circuits is a very deep problem with implications to the seemingly disparate fields of quantum information theory, quantum many-body physics and high energy physics. While our theoretical understanding of these systems has progressed in recent years, numerical approaches for studying these models remains severely limited. In this paper, we discuss a special class of numerically tractable quantum circuits, known as quantum automaton circuits, which may be particularly well suited for this task. These are circuits which preserve the computational basis, yet can produce highly entangled output wave functions. Using ideas from quantum complexity theory, especially those concerning unitary designs, we argue that automaton wave functions have high quantum state complexity. We look at a wide variety of metrics, including measurements of the output bit-string distribution and characterization of the generalized entanglement properties of the quantum state, and find that automaton wave functions closely approximate the behavior of fully Haar random states. In addition to this, we identify the generalized out-of-time ordered 2k-point correlation functions as a particularly useful probe of complexity in automaton circuits. Using these correlators, we are able to numerically study the growth of complexity well beyond the scrambling time for very large systems. As a result, we are able to present evidence of a linear growth of design complexity in local quantum circuits, consistent with conjectures from quantum information theory.