Models of quantum complexity growth

TL;DR

This paper establishes a rigorous link between quantum complexity growth and unitary k-designs, demonstrating that local random circuits exhibit linear complexity growth over time, supporting conjectures about chaotic quantum systems.

Contribution

It provides a formal connection between complexity growth and unitary designs, proving linear complexity growth in local random circuits under a strong complexity definition.

Findings

Local random quantum circuits generate unitary transformations with linearly growing complexity.

The results verify conjectures by Brown and Susskind on complexity growth.

Complexity growth applies under a strong, measurement-based complexity definition.

Abstract

The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a short-cut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary $k$-designs, ensembles which capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind. Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.