The strength of weak coupling

TL;DR

The paper tackles the challenge of achieving high-fidelity quantum state transfer on graphs when connecting a large base network with weakly coupled pendant edges. It develops an elementary perturbative framework based on the Feshbach-Schur map to prove that HFST is possible and that transfer times can be largely independent of the graph diameter. It further shows robustness against Anderson localization in spin chains and extends the approach to quantum speedups for hitting times and a novel edge-based quantum search. Overall, the T. rex method provides a flexible, perturbation-theory–grounded toolkit for quantum transport and search on complex graphs with broad theoretical and potential practical impact.

Abstract

A paradoxical idea in quantum transport is that attaching weakly-coupled edges to a large base graph creates high-fidelity quantum state transfer. We provide a mathematical treatment that rigorously prove this folklore idea. Our proofs are elementary and build upon the Feshbach-Schur method from perturbation theory. We also show the idea is effective in circumventing Anderson localization in spin chains and finding speedups in hitting times useful for quantum search.

Figures (4)

• Figure 1: The power of weak coupling (T. rex effects) in quantum state transfer on $P_{n}$. Here, $n=55$ as an example and we plot time $t$ (on X axis) versus antipodal fidelity (on Y axis). Left: T. rex arms with strength $\delta = 0.05$. Right: uniform couplings (fidelity did not reach $0.40$; there is no PGST gkss12).
• Figure 2: State transfer protocol robust against Anderson localization: the first and last two vertices (unshaded) are protected from noise while the middle vertices (shaded) are under the influence of Anderson localization. The control parameter $B$ is used to ensure cospectrality.
• Figure 3: The T. rex protocol against Anderson localization on $P_{n}$. Here, $n=55$ where we plot time $t$ (x-axis) versus antipodal fidelity (y-axis). Left: T. rex against Cauchy noise with parameter $0.06$ (as in klmw07) with $\delta=0.002$ and fidelity $0.9997$. Right: T. rex against noise from $U(-2,2)$ with $\delta=0.0067$ and fidelity $0.992$ (localization practically destroys antipodal quantum transport).
• Figure 4: The quotient structure of the barbell graph. The unnormalized barbell has $a=\sqrt{N-2}$, $b=N-3$, and $c=1$.

Theorems & Definitions (21)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• Lemma 3.2
• Claim 3.1
• proof
• Claim 3.2
• proof
• Claim 3.3