Time Independence Does Not Limit Information Flow. II. The Case with Ancillas

TL;DR

This paper demonstrates that time-independent quantum protocols with local ancillas can achieve the same information transfer capabilities and light cone bounds as time-dependent protocols, using constructions that approximate dynamics efficiently.

Contribution

The authors develop a method to simulate time-dependent Hamiltonian dynamics with time-independent Hamiltonians using local ancillas, matching their run-times and bounds.

Findings

Time-independent protocols with ancillas match the light cone bounds of time-dependent systems.

Constructed Hamiltonians approximate original dynamics with polylogarithmic overhead.

Applications include state transfer in systems with power-law and disordered interactions.

Abstract

While the impact of locality restrictions on quantum dynamics and algorithmic complexity has been well studied in the general case of time-dependent Hamiltonians, the capabilities of time-independent protocols are less well understood. Using clock constructions, we show that the light cone for time-independent Hamiltonians, as captured by Lieb-Robinson bounds, is the same as that for time-dependent systems when local ancillas are allowed. More specifically, we develop time-independent protocols for approximate quantum state transfer with the same run-times as their corresponding time-dependent protocols. Given any piecewise-continuous Hamiltonian, our construction gives a time-independent Hamiltonian that implements its dynamics in the same time, up to error $\varepsilon$, at the cost of introducing a number of local ancilla qubits for each data qubit that is polylogarithmic in the number of qubits, the norm of the Hamiltonian and its derivative (if it exists), the run time, and $1/\varepsilon$. We apply this construction to state transfer for systems with power-law-decaying interactions and one-dimensional nearest-neighbor systems with disordered interaction strengths. In both cases, this gives time-independent protocols with the same optimal light-cone-saturating run-times as their time-dependent counterparts.