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

TL;DR

This work proves that time-independent Hamiltonians can replicate the light-cone propagation of time-dependent protocols when each data qubit is equipped with local clock ancillas. By extending the clock construction of WWRL to a localized setting, the authors show that saturating Lieb-Robinson bounds remains possible under time-independence, at the cost of a polylogarithmic overhead in the number of local ancillas per site. They provide rigorous error bounds and explicit ancilla-scaling relations, including mollifier-based smoothing and alternative bump-function schemes to handle general piecewise-time-dependent dynamics. The results apply to challenging settings such as long-range power-law interacting systems and disordered 1D chains, yielding time-independent state-transfer protocols with the same optimal run-times as their time-dependent counterparts. Overall, the work demonstrates that locality-based speed limits remain tight under time-independence when suitable local ancilla resources are allowed, with concrete scaling laws for the required clock dimension.

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.

Figures (2)

• Figure 1: A cartoon of the localized WWRL construction acting on 3 qubits. The edges' colors corresponds to the color of their controlling clocks. The clock states are set to be Gaussian wave packets. Note that there is no a priori requirement that edges are controlled by different clocks, but in this cartoon, that happens to be the case.
• Figure S1: Cartoon illustration of the setup of the long-range protocol, where $r_j=2$. The smaller squares in red and magenta, form the collection $\mathcal{C}_1,$ and the blue and cyan squares form the collection $\mathcal{C}_2.$ The cyan and magenta squares are $C_2$ and $C_1$ respectively. The dotted orange regions demarcate the squares in $\Sigma(B)$ for the $B$s in $\mathcal{C}_2,$ and thus the squares not in them correspond to $\chi(B).$ Note that the choice of $\chi(B)$ is arbitrary other than the requirement that $\chi(C_{j+1})=C_j$. For each square $B\in C_2,$ the dot in the square sharing the square's color represents $\varrho(B).$ Note that beyond requiring $\varrho(B)\in B,$ the choice of $\varrho(B)$ is arbitrary as well.

Theorems & Definitions (38)

• Definition S1: Localized WWRL Hamiltonian
• Definition S2: Gaussian States
• Definition S3: Constants
• Definition S4: Partial Difference
• Lemma S1
• proof
• Corollary S1
• Lemma S2
• proof
• Lemma S3