Direct U(2) approximation via repeat-until-success circuits

TL;DR

This paper introduces a method using repeat-until-success circuits to directly approximate one-qubit unitaries efficiently, avoiding traditional decomposition methods, with potential applications to multi-qubit gate sets.

Contribution

It presents a novel approach leveraging repeat-until-success circuits and lattice algorithms for direct unitary approximation, bypassing Euler decomposition.

Findings

Efficient approximation of arbitrary one-qubit unitaries without Euler decomposition.

Extension of techniques to multi-qubit gate sets like Clifford and CS.

Application of lattice-based algorithms and norm equations in quantum circuit synthesis.

Abstract

We show how to directly and efficiently approximate arbitrary one-qubit unitaries, bypassing the Euler decomposition and the magnitude approximation problem, at the cost of one ancillary qubit. Our technique also applies to approximating unitaries with multi-qubit gate sets such as Clifford and CS, or Clifford and CCZ, as well as to approximating orthogonal matrices using multi-qubit gate sets such as Real Clifford and CCZ. The key tools are repeat-until-success circuits, lattice-based exact synthesis algorithms, integer point enumeration in convex sets, and relative norm equations.