Optimal Construction of Two-Qubit Gates using the Symmetries of B Gate Equivalence Class

TL;DR

This paper explores the symmetries of the B gate equivalence class to optimize the construction of two-qubit gates, proposing parameterized circuits and bounds for efficient quantum gate synthesis.

Contribution

It identifies unique symmetries of the B gate class, develops parameterized universal circuits, and provides bounds on gate counts for arbitrary multi-qubit gates.

Findings

Existence of one-parameter families of local equivalence classes with specific symmetries.

Development of parameterized circuits using these classes for efficient gate construction.

Upper bounds on the number of two-qubit gates needed for arbitrary n-qubit gates.

Abstract

Two applications of gates from the B gate equivalence class can generate all two-qubit gates. This local equivalence class is invariant under the mirror (multiplication with the SWAP gate) operation, inverse (Hermitian conjugate) operation, and the combined inverse and mirror operations. The last two symmetries are associated with the ability of a two-qubit gate to generate the two-qubit local gates and the SWAP gate in two applications. No single local equivalence class of two-qubit gates, except the B gate equivalence class, has these two symmetries. Only the planar regions of the Weyl chamber, describing the mirror operation, contain the local equivalence classes with either one of the two symmetries. We show that there exist one-parameter families of local equivalence classes on these planes, with and without the B gate equivalence class, such that each of them can be used to construct a parameterized universal two-qubit quantum circuit that involves only two nonlocal two-qubit gates. We also discuss the implementation of the gates from a few families of local equivalence classes on superconducting quantum computers for optimal generation of all two-qubit gates. We provide upper bounds on the number of two-qubit gates required to generate an arbitrary $n$-qubit gate for two families, each of which is conjectured to generate all two-qubit gates in two applications. We show that there exists a positive correlation between the area of the convex hull of the squared eigenvalues of the nonlocal part of a parameterized two-qubit gate and the fractional volume of the Weyl chamber covered in two applications of the parameterized two-qubit gate for two families of local equivalence classes.