Berkeley B Gate

A special two-qubit gate that generates any two-qubit unitary with only two applications, the most efficient known primitive.

Definition

The Berkeley B gate is a two-qubit gate with the remarkable property that exactly two B gates, interleaved with single-qubit rotations, can produce any two-qubit unitary. No other single fixed gate does better: a generic gate needs three CNOTs, but only two B gates. It corresponds to the canonical coordinates (π/4, π/8, 0).

Matrix

The B gate is generated by exp(i (π/4)(X⊗X) + i (π/8)(Y⊗Y)) — a specific mixture of two Ising couplings. Its position in the Weyl chamber is chosen precisely so that the set of unitaries reachable by two copies fills the whole chamber.

Why two suffices

The Weyl chamber of two-qubit gates is three-dimensional. Applying one entangler with arbitrary single-qubit gates around it reaches a two-dimensional surface; a second application can then reach any interior point only if the first gate is chosen at the B point. This optimality was proven by Zhang, Vala, Sastry, and Whaley at Berkeley.

import numpy as np
def b_gate():
    a, b = np.pi/8, 3*np.pi/8
    m = np.zeros((4,4), dtype=complex)
    m[0,0]=np.cos(a); m[0,3]=1j*np.sin(a)
    m[3,0]=1j*np.sin(a); m[3,3]=np.cos(a)
    m[1,1]=np.cos(b); m[1,2]=1j*np.sin(b)
    m[2,1]=1j*np.sin(b); m[2,2]=np.cos(b)
    return m

Significance

The B gate is mostly a theoretical benchmark, since it is not native to common hardware. But it sharpens the understanding of two-qubit gate optimality and motivates the Weyl-chamber view of entangling power. See Weyl chamber and canonical gate.