Constructing Approximately Diagonal Quantum Gates

TL;DR

This paper introduces a method to generate approximately diagonal quantum gates using iterative sequences, with potential applications in quantum algorithms and topological quantum computing.

Contribution

It proposes a new iterative approach for constructing approximately diagonal 1-qubit gates and provides partial proofs supporting exponential convergence.

Findings

Sequences converge doubly exponentially fast for small integers

Partial proofs support the conjectured convergence

Developed techniques for future resolution of the conjecture

Abstract

We study a method of producing approximately diagonal 1-qubit gates. For each positive integer, the method provides a sequence of gates that are defined iteratively from a fixed diagonal gate and an arbitrary gate. These sequences are conjectured to converge to diagonal gates doubly exponentially fast and are verified for small integers. We systemically study this conjecture and prove several important partial results. Some techniques are developed to pave the way for a final resolution of the conjecture. The sequences provided here have applications in quantum search algorithms, quantum circuit compilation, generation of leakage-free entangled gates in topological quantum computing, etc.