Implementing a Fast Unbounded Quantum Fanout Gate Using Power-Law Interactions

TL;DR

This paper demonstrates how power-law interactions can be used to implement a fast quantum fanout gate, enabling efficient quantum Fourier transforms and Shor's algorithm on lattice systems, and establishes bounds on their classical simulability.

Contribution

It introduces a method to implement a fast quantum fanout gate using power-law interactions and provides new bounds on the time complexity for such operations in constrained systems.

Findings

Quantum fanout gate can be implemented in logarithmic time for certain power-law interactions.

Power-law systems with specific decay rates are hard to simulate classically.

New lower bounds on the time to implement quantum Fourier transform and fanout in constrained systems.

Abstract

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a $D$-dimensional lattice in time logarithmic in the number of qubits for interactions with $\alpha \le D$. As a corollary, we show that power-law systems with $\alpha \le D$ are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques.