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

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.

Figures (7)

• Figure 1: Weyl chamber in the positive Cartan coordinate system. The planar regions describing the mirror operation are highlighted.
• Figure 2: Circuit for optimal generation of all two-qubit gates.
• Figure 3: Fractional volume of Weyl chamber covered vs $c_1$ for the families with Cartan coordinates (a) $(c_1, \pi/4, (\pi/2) - c_1)$ for eleven equally spaced values of $c_1$ from $\pi/4$ to $\pi/2$, and (b) $(c_1, c_1/2, 0)$ for eleven equally spaced values of $c_1$ from $0$ to $\pi/2$.
• Figure 4: The region of Weyl chamber covered by two applications of gates from the local equivalence class with Cartan coordinates (a) $(2\pi/7, \pi/4, 3\pi/14)$, (b) $(\pi/3, \pi/4, \pi/6)$, (c) $(2\pi/5, \pi/4, \pi/10)$, (d) $(\pi/8, \pi/16, 0)$, (e) $(\pi/4, \pi/8, 0)$, and (f) $(3\pi/4, 3\pi/8, 0)$.
• Figure 5: (a) $c_1 + c_3 = \pi/2$ planar region of Weyl chamber, and (b) the first half of $c_2 = \pi/4$ planar region of Weyl chamber. In both subfigures, a point and its Cartan coordinates are in the same colour.