Universal Early-Time Growth in Quantum Circuit Complexity

TL;DR

This paper demonstrates that quantum circuit complexity for time-independent Hamiltonians exhibits universal linear growth at early times, with deviations influenced by the algebra of the gates, applicable to various quantum systems including fields.

Contribution

It establishes a universal bound on early-time complexity growth for any Hamiltonian, independent of gate choices, and applies the result to diverse quantum systems and field theories.

Findings

Early-time complexity grows linearly, bounded by the Hamiltonian's algebraic structure.

Deviations from linear growth are negative, reducing the growth rate.

Application to lattice field theories reveals the dependence on lattice spacing.

Abstract

We show that quantum circuit complexity for the unitary time evolution operator of any time-independent Hamiltonian is bounded by linear growth at early times, independent of any choices of the fundamental gates or cost metric. Deviations from linear early-time growth arise from the commutation algebra of the gates and are manifestly negative for any circuit, decreasing the linear growth rate and leading to a bound on the growth rate of complexity of a circuit at early times. We illustrate this general result by applying it to qubit and harmonic oscillator systems, including the coupled and anharmonic oscillator. By discretizing free and interacting scalar field theories on a lattice, we are also able to extract the early-time behavior and dependence on the lattice spacing of complexity of these field theories in the continuum limit, demonstrating how this approach applies to systems that have been previously difficult to study using existing techniques for quantum circuit complexity.