Robustness-Runtime Tradeoff for Quantum State Transfer

TL;DR

This work introduces new robust state transfer protocols, charting the landscape between complete state-dependence and state-independence, and gives new minimum runtimes for partially state-dependent protocols which, in certain regimes, are parametrically better than existing bounds.

Abstract

Quantum state transfer is the primitive of transporting an unknown state on one site of a lattice to another. Using power-law interactions, recent state transfer protocols achieve speedup by utilizing the intermediate ancilla sites. However, these protocols require the ancillas to be in a perfectly initialized state, which, due to noise or imperfect control, may not be the case. In this work we introduce the $\textit{robustness}$ of a state transfer protocol, which quantifies the protocol's tolerance to error in the initial ancilla state. In the Heisenberg picture, state transfer grows operators supported on the final site such that they no longer commute with all operators on the starting site. We prove that this robustness tightly bounds the Schatten $p$-norms of these commutators between initial and final-site operators. This generalizes the known cases of $p=\infty$ and $p=2$, which govern completely state-dependent and state-independent state transfer respectively, demonstrating that intermediate values of $p$ govern partially state-dependent state transfer. In conjunction with existing power-law light cones, our result gives new minimum runtimes for partially state-dependent protocols which, in certain regimes, are parametrically better than existing bounds. We introduce new robust state transfer protocols, charting the landscape between complete state-dependence and state-independence.

Figures (5)

• Figure 1: Quantum state transfer is the task of transporting an unknown state from one lattice site to another. The states of intermediate sites may not be fully specified.
• Figure 2: Lower bound from Theorem \ref{['thmmainpnorm']}: minimum commutator $p$-norm of any $\mathcal{S}$-robust state transfer protocol, plotted as a function of $\abs{\mathcal{S}}$ and $p$. Maximal values at state-independence ($\abs{\mathcal{S}}=2^{L-1}$) and operator norm ($p=\infty$) are indicated by the dashed lines.
• Figure 3: The setup for the bridging protocol with qubits on each end of the chain initialized. Blue and red circles respectively indicate ancilla qubits in initialized and uninitialized states at $t=0$.
• Figure 4: Commutator norms for fast GHZ protocol (blue) and its symmetrized version (green). Actual values (solid) compared to lower bounds (dashed).
• Figure 5: The setup with 1 in every $M$ qubits initialized. Blue and red circles respectively indicate ancilla qubits in initialized and uninitialized states at $t=0$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2